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\begin{document}

\author{Avraham Aizenbud}
%\address{Avraham Aizenbud, Faculty of Mathematics
%and Computer Science, The Weizmann Institute of Science POB 26,
%Rehovot 76100, ISRAEL.}
\address{Avraham Aizenbud,
Massachusetts Institute of Technology
Department of Mathematics
Cambridge, MA 02139 USA.}
\email{aizenr@gmail.com}
\urladdr{\url{http://math.mit.edu/~aizenr/}}

\author{Vladimir Drinfeld}
\address{Vladimir Drinfeld, Department of Mathematics, University of Chicago.}
\email{drinfeld@math.uchicago.edu}
%\RamiQ{Would you like a url to your website to appear below?}}
%\email{\RamiQ{Would you like your e-mail address to appear here?}}

%\urladdr{\url{}}

\title{The wave front set of the Fourier transform of algebraic measures}
\keywords{Wave front set, Fourier transform, Holonomic distributions, $p$-adic, Desingularization \RamiQ {Can you review the keywords?}\\
\indent
 MSC Classes: \RamiCP{46F, 46F10.}\RamiQ{Can you review the MSC Classes?
Can you suggest additional classes? Here is the description of the classes I have put:\\
46-xx  Functional analysis. 46Fxx Distributions, generalized
functions, distribution spaces. 46F10 Operations with distributions.
}
}
%\title[]{}
%\date{\today}
%
%\classification{22E,22E45,20G05,20G25,46F99}
%
%20-xx  Group theory and generalizations 20Cxx Representation
%theory of groups 20C99 None of the above, but in this section
%
%20Gxx Linear algebraic groups (classical groups) 20G05
%Representation theory 20G25 Linear algebraic groups over local
%fields and their integers
%
%
%22-xx  Topological groups, Lie groups 22Exx Lie groups 22E45
%Representations of Lie and linear algebraic groups over real
%fields: analytic methods {For the purely algebraic theory, see
%20G05} 22E50 Representations of Lie and linear algebraic groups over
%local fields
%
%
%46-xx  Functional analysis 46Fxx Distributions, generalized
%functions, distribution spaces 46F10 Operations with distributions
%
%14-xx  Algebraic geometry 14Lxx Algebraic Groups 14L24 Geometric
%invariant theory [See also 13A50] 14L30 Group actions on varieties
%or schemes (quotients) [See also 13A50, 14L24]
%
%\mscclasses{}
%
\maketitle
\begin{abstract}
%\RamiQI{I'll change the abstract at the end.}
We study the Fourier transform of \RamiD{the absolute value of a} polynomial on a finite-dimensional vector
 space over a local field of characteristic 0. We prove that this transform is  smooth on an open dense set.
% \DrinI{Do we really want to formulate the above particular case rather %than the Fourier transform of
% $ \psi (P(x))$, where $P$ is a polynomial?}

%The current work concentrates on the non-Archimedean case. \DrinI{No longer %true.}
\RamiV{We prove this result for the Archimedean and the non-Archimedean case in a uniform way.}
The Archimedean case was proved in \cite{Ber_FT}.
\RamiV{T}he non-Archimedean case  was proved in \cite{HK} \RamiD{and \cite{CL}}.
 Our method is different from those described in \RamiV{\cite{Ber_FT,HK,CL}}.
 It is based on Hironaka's desingularization theorem, unlike  \cite{Ber_FT} \RamiM{which} is based on the theory of D-modules
and \RamiV{\cite{HK,CL}} \RamiM{which} is based on model theory.

% Our method suits, in fact, both the Archimedean and the non-Archimedean %case.
%However,  only the non-Archimedean case is discussed in the current version.
Our method \RamiV{ also gives} bounds on the open dense set where the Fourier transform is  {smooth}.
These bounds are explicit in terms of resolution of singularities.
%,  which, today, has rather explicit descriptions  (see e.g. \cite{Hir_mod}).

We also  prove the same result on the Fourier transform of other \RamiE{measures} of algebraic origins.

\end{abstract}
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\tableofcontents


\section{Introduction}



\subsection{Main results \DrinI{in the non-Archimedean case}}   \label{ss:main}
%$ $
%\DrinD{Here is one of them.}
%The main result of this paper is \RamiE{a generalization of} the following
%theorem:
%$ $\RamiV{For simplicity let us first formulate the main results of the paper in the non-Archimedean case.}
\begin{introtheorem} \label{thm:main_a}
Let $F$ be a non-Archimedean local field of characteristic $0$
 (i.e. a finite extension of the field  of p-adic numbers $\Q_p$).
Let $W$ be a finite-dimensional vector space over $F$. Let \RamiI{$X$ be a smooth algebraic variety over
$F$, let $\phi: X \to W$ be a proper map}
% be a smooth closed algebraic subvariety.
and $\omega$ a \DrinD{regular (algebraic)} top differential form on  {$X$}.
Let $|\omega|$ be the measure on $X$ corresponding to $\omega$ and  {$\phi_*(|\omega|)$ its direct image (which is a measure on $W$).}
%consider it as a distribution on $W.$
%Fix an Haar measure on $W$.
Then there exists  {a dense Zariski open} subset $U\subset W^*$
% \DrinA{stable under homotheties}
 such that the restriction to $U$
of the Fourier transform of $\phi_*(|\omega|)$ \RamiE{is locally constant}.
%$m_{X_{0},\omega_{0}}=p_*(|\omega|)\in \Sc^*(W).$
\end{introtheorem}

\begin{examples*}
$ $
\DrinD{
\begin{itemize}
 \item  Let $X\subset W$ be a smooth closed subvariety and $\omega$ a regular top differential form on
$X$. Consider $|\omega|$ as a measure on $W$. Applying Theorem~\ref{thm:main_a} to the embedding
$\phi :X\hookrightarrow W$\RamiM{,} we see that the Fourier transform of $|\omega|$ is smooth on a dense Zariski open subset.

 \item  More generally, let $X\subset W$ be any closed subvariety and $\omega$ a rational top differential form on $X$. Suppose that for some resolution of singularities $p:\hat X\to X$\RamiM{,} the pullback $p^*(\omega )$ is
regular. Then one can consider $|\omega|$ as a measure on $W$. Its Fourier transform is smooth on a dense Zariski open subset (to see this, apply Theorem~\ref{thm:main_a} to the composition
$\hat X\to X\hookrightarrow W$).
\end{itemize}
}
\end{examples*}

 \DrinD{We deduce Theorem~\ref{thm:main_a} from the following theorem, which says that the singularities of the Fourier transform of  $\phi_*(|\omega|)$ on the whole $W^*$ are ``not too bad".}
%We also \DrinD{show that the behavior of the Fourier transform of  $\phi_*(|\omega|)$ outside
%$U$ is ``not too bad". }
%%%give some bounds on the behavior of the Fourier transform of $|\omega|$ outside $U.$
\DrinD{
\begin{introtheorem} \label{thm:main_b}
In the situation of Theorem~\ref{thm:main_a}\RamiM{,} the wave front set of the Fourier transform of
$\phi_*(|\omega|)$ is contained in
\RamiX{an isotropic}
%a Lagrangian
algebraic subvariety\footnote{An algebraic subvariety of a symplectic algebraic manifold is said to be \RamiX{isotropic}
%Lagrangian
if its nonsingular part is. \RamiX{Note that every algebraic isotropic subvariety of the co-tangent bundle of an algebraic manifold $M$ which is stable with respect to homotheties along the co-tangent space is contained in a union of co-normal bundles of submanifolds of $M$, and in particular in a Lagrangian subvariety. See \S\S\ref{sec:lag} for more details. }} of $T^* (W^*)=W\times W^*$.
\end{introtheorem}
The notion of wave front set is recalled in Appendix~\ref{app:WF} and \S\ref{sssec:wfs}.
}

%\begin{introtheorem} \label{thm:main_b}
%Let $W$ be a finite-dimensional $F$-vector space and $X_{} \subset W$ be a (locally closed) smooth algebraic subvariety
% and let $\omega$ be an algebraic top differential form on it.


%Then the Fourier transform $\cF^*(\mu_{X_{},\omega_{}}) \in \G(W^*)$ is WF-holonomic.

%\end{introtheorem}


% More specifically, we prove that \emph{the wave front set of the Fourier transform of
%$\phi_*(|\omega|)$ is contained in a Lagrangian algebraic subvariety,}
%% {small in an appropriate sense,}
%% included in a Lagrangian variety,
%see Theorem \ref{thm:main_b} below

\begin{rem}
Theorem~\ref{thm:main_b} can be considered as a $p$-adic analog of the following theorem of J.~Bernstein \cite{Ber_FT}:
 if $F$ is Archimedean then the Fourier transform of $\phi_*(|\omega|)$ is a holonomic distribution.
 \end{rem}
% %fact that the Fourier transform of $|\omega|$ is
%% a holonomic distribution, which is proved in \cite{Ber_FT} for the Archimedean case.
%We also prove the following relative version of this statement.

\DrinD{We also prove the following more general, relative version of Theorem~\ref{thm:main_b}.}

%To sum up, we prove the following generalization of Theorem \ref{thm:main_a}:
%\begin{theorem*} [see Theorem \ref{thm:main_bb} below]

\begin{introtheorem} \label{thm:main_bb}
Let $F$ be a non-Archimedean field of characteristic $0$. Let $W$
be a finite-dimensional $F$-vector space and $X,Y$ be smooth algebraic manifolds over $F$. Let
$\phi: X \to Y \times W$ be a proper map
 and let $\omega$ be a \RamiR{regular} top differential form on $X$.
%consider $|\omega|$ as a measure on $X$
Then the wave front set of the partial Fourier transform of
$\phi_*(|\omega|)$ with respect to $W$
is contained in
\RamiX{an isotropic}
%a Lagrangian
 algebraic subvariety of $T^*(Y\times W^*)$.
%\end{theorem*}
\end{introtheorem}


\subsection{\DrinI{The Archimedean analog of Theorem~\ref{thm:main_bb}}}
\DrinI{
%The argument in the Archimedean case is the same as in the non-Archimedean one.
%However, we need to be more careful with the formulations. Namely, in the Archimedean
We have to take in account that in the  Archimedean case the Fourier transform is defined not for general distributions, but only for Schwartz distributions.  Similarly, the partial Fourier transform is defined for distributions that are \emph{partially Schwartz} along the relevant vector space (a precise definition of partially Schwartz distribution can be found in
\S\S\ref{ssec:arc_dist} below).

%Therefore, we need to verify that the distributions that we consider are Schwartz, or in fact partially Schwartz
%along the relevant vector space (see  \S\S\ref{ssec:arc_dist} below for a rigorous definition of the notion of
%partially Schwartz). So we need the following lemma:
%\begin{introlem} \label{lem:Sch}
\begin{introtheorem} \label{thm:main_C_Arch}
Let $F$ be an Archimedean local field (i.e. $\R$ or $\C$). Let everything else be as in Theorem \ref{thm:main_bb}. Then
\begin{enumerate}[(i)]
\item \label{thm:main_C_Arch:1}
the distribution $\phi_*(|\omega|)$ is partially Schwartz along $W$ (so its partial Fourier transform with respect to $W$ is well-defined);
\item  \label{thm:main_C_Arch:2}
the wave front set of the partial Fourier transform of
$\phi_*(|\omega|)$ with respect to $W$
is contained in
\RamiX{an isotropic}
%a Lagrangian
algebraic subvariety of $T^*(Y\times W^*)$.
\end{enumerate}
%\end{introlem}
\end{introtheorem}
}
\RamiW{
\begin{rem}
In fact, the distribution  $\phi_*(|\omega|)$ is Schwartz on the entire space and not only along $W$, but in order to prove it we need to define what it means, and we prefer not to do it in this paper.
\end{rem}
}
%\subsection{Results to be proved in the next version of this article}
\subsection{\RamiV{Stronger versions}}
\RamiV{Our proof of theorem \ref{thm:main_bb} can give an explicit (in terms of resolution of singularities) description of
\RamiX{an isotropic}
%a Lagrangian
 variety that contains the wave front set of the partial Fourier transform of
$\phi_*(|\omega|)$. We provide such description in Theorem \ref{exp} (and an analogous description for Theorem \ref{thm:main_b} in Corollary \ref{cor:exp}). This description implies that this
\RamiX{isotropic}
%Lagrangian
variety ``does not actually depend" on the local field $F$ and is stable under homotheties
%both in $W$ and $
\DrinI{in $W^*$}. Namely we have the following theorem:
\begin{introtheorem} \label{thm:futerO}
Let $K$ be a characteristic $0$ field and $W$ a finite-dimensional $K$-vector space. Let $X,Y$ be smooth algebraic manifolds over $K$. Let $\phi: X \to Y \times W$ be a proper map,
 $\omega$ a \RamiR{regular} top differential form on $X$.%, and $p$ be a regular function on $X$.

Then there exists %a Lagrangian
%\RamiQK{I prefer to say that $L$ is isotropic (here and in the next theorem). %This is equivalent but I do not want to explain why. Is it OK? }
\RamiX{an isotropic}
algebraic subvariety $L \subset T^*(Y\times W^*)$ such that
\begin{enumerate}[(i)]
\item  \label{thm:futer:1}    $L$ is stable with respect to the action of the multiplicative group
\DrinI{on $T^*(Y\times W^*)$ that comes from its action on $W^*$};
\item  \label{thm:futer:2}     \DrinI{f}or any embedding of $K$ into any local field $F$ (Archimedean or not), % and any additive character
% $\theta$ of $F$,
the wave front set of the partial Fourier transform of
 $(\phi_F)_*( |\omega_F|)$ is contained in $L(F)$.
\end{enumerate}

Here $L(F)\subset T^*(Y\times W^*)(F)$ is the set of $F$-points of $L$ and $\omega_F$  is obtained from  $\omega$ by extension of scalars from $K$ to $F$, and $|\omega_F|$ is the corresponding measure on $X(F)$.
\end{introtheorem}
}
\begin{remark*}
The Fourier transform depends on the choice of a nontrivial additive character
 $\psi :F\to\C^{\times}$. But if $L$ satisfies (i) and has property (ii) for some $\psi$, then (ii) holds
 for any $\psi$.
\end{remark*}

%Theorem \ref{thm:futerO} gives us the following corollary:

%\DrinI{It is pretty clear that Theorem~\ref{thm:futerO} is a particular %case of the statement below corresponding to
%$p=0$. It is not so clear why ``Corollary"~\ref{cor:futer} follows from %Theorem~\ref{thm:futerO}. If we call it corollary then we have to explain %\emph{immediately} why it follows from  Theorem~\ref{thm:futerO}. }
%\RamiQK{Can we say immediately after the corollary that we will explain %why it follows from the theorem in \S\ref{sec:ex_boun} (and in \S\ref{sec:arc} %fot the Archemedian case)? If not can we call it a theorem and  add the %something  like following sentence before it? ``We will show that this theorem %implase the following stronger version of itself"?}

\DrinI{We will show that the following variant of Theorem \ref{thm:futerO} easily follows from
Theorem \ref{thm:futerO} itself.}\footnote{\DrinI{More precisely, Theorem~\ref{thm:futer} for
 $\phi :=X \to Y\times W$ and $p:X\to K$ follows from Theorem \ref{thm:futerO} for
  $\phi\times p :X \to Y\times W \times K$. }
}






%\DrinI{Theorem \ref{thm:futerO} is a particular case \DrinJ{NO!} of the next theorem corresponding to $p=0$; %on the other hand, we will show that Theorem~\ref{thm:futer} easily follows from Theorem \ref{thm:futerO}.}

%\RamiW{We will show that this theorem implies the following stronger version of itself:}

\RamiV{
\begin{introthm}  \label{thm:futer}
In the situation of Theorem  \ref{thm:futerO} let $p$ be a regular function on $X$.
Then there exists
\RamiX{an isotropic}
%a Lagrangian
algebraic subvariety $L \subset T^*(Y\times W^*)$ such that
 for any embedding of $K$ into any local field $F$ \DrinI{and any nontrivial additive character
 $\psi :F\to\C^{\times}$},
the wave front set of the partial Fourier transform of
 $(\phi_F)_*(\DrinI{(}\psi \circ p_F\DrinI{)} \cdot |\omega_F|)$ is contained in $L(F)$.
%\begin{enumerate}
%\item [(i)] \label{thm:futer:1}   $L$ is stable with respect to the action of the multiplicative group on $W$;
%\item  [(ii)] \label{thm:futer:2}    \RamiM{F}or any embedding of $K$ into any local field $F$,
%%%%and any additive character $\theta$ of $F$,
%the wave front set of the partial Fourier transform of
% $(\phi_F)_*(\DrinI{(}\psi \circ p_F\DrinI{)} \cdot |\omega_F|)$ is contained in $L(F)$.
%\end{enumerate}
%%%%Here $L(F)\subset T^*(Y\times W^*)(F)$ is the set of $F$-points of $L$,
Here the Fourier transform is performed using the same $\psi$, and $p_F$ is obtained from $p$ by extension of scalars from $K$ to $F$.
%$\psi$ is an additive character,
\end{introthm}
Again, in order for this theorem to make sense for \DrinI{Archimedean} $F$, we will prove the following lemma:
\begin{introlem} \label{lem:Sch2}
In the notations of \RamiW{Theorem} \ref{thm:futer},
let $F$ be an Archimedean local field, with an embedding $K \hookrightarrow F$.  Then the distribution
$(\phi_F)_*(\DrinI{(}\psi \circ p_F\DrinI{)} \cdot |\omega_F|)$ is partially Schwartz along $W$.
\end{introlem}

}

%\RamiQD{This section should be rephrased in a present tense. However, I think that the current formulation of Theorem \ref{thm:futer} is good enough for the introduction. We should just mention that we will give a more exact formulation that includes the specific bound in section \ref{sec:ex_boun}.}

%   \RamiI{
%   Our method can give stronger results, which we plan to add to this paper in the near future.
%   In particular, we will prove Theorem \ref{thm:main_bb} for Archimedean and non-Archimedean fields $F$ simultaneously.
%   We will also give some bounds on the Lagrangian variety which contains the wave front set (and, correspondingly, on
%   the set $U$ from Theorem \ref{thm:main_a}). These bounds will not depend on the field $F$, but only on the field of definition
%   of the initial data. We will also show that one can replace the absolute value with an arbitrary multiplicative or
%   additive character.
%
%   To sum up, we will prove the following generalization of Theorem \ref{thm:main_bb}:
%   \begin{theorem} \label{thm:futer}
%   Let $K$ be a characteristic $0$ field and $W$ a finite-dimensional $K$-vector space. Let $X,Y$ be smooth algebraic manifolds over $K$. Let $\phi: X \to Y \times W$ be a proper map,
%    $\omega$ a \RamiR{regular} top differential form on $X$, and $p$ be a regular function on $X$.
%
%   Then there exists a Lagrangian algebraic subvariety $L \subset T^*(Y\times W^*)$ such that
%   \begin{enumerate}
%   \item\label{thm:futer:1} [(i)]   $L$ is stable with respect to the action of the multiplicative group on $W$;
%   \item \label{thm:futer:2} [(ii)]    \RamiM{F}or any embedding of $K$ into any local field $F$ (Archimedean or not), and any additive character
%    $\theta$ of $F$, the wave front set of the partial Fourier transform of
%    $(\phi_F)_*(\theta \circ p_F \cdot |\omega_F|)$ is contained in $L(F)$.
%   \end{enumerate}
%
%   Here $L(F)\subset T^*(Y\times W^*)(F)$ is the set of $F$-points of $L$, $\omega_F$ and $p_F$ are obtained from  $\omega$ and $p$ by extension of scalars from $K$ to $F$, and $|\omega_F|$ is the corresponding measure on $X(F)$.
%   \end{theorem}
%
%   \begin{remark*}
%   The Fourier transform depends on the choice of a nontrivial additive character
%   $\psi :F\to\C^{\times}$. But if $L$ satisfies (i) and has property (ii) %   for some $\psi$, then (ii) holds
%   for any $\psi$.
%   \end{remark*}
\begin{example*}
Let $p$  be a polynomial on an $F$-vector space $W$.
%Let $\theta$ be an additive character of $F$.
\DrinI{Theorem \ref{thm:futer} implies that the Fourier transform of the function $x\mapsto\psi (p(x))$ is smooth on a dense Zariski open  subset.}
\end{example*}



%\subsection{Related results}
\subsection{Method of the proof and comparison with related results}
$ $

%\RamiI{The main tool used in our proof is Hironaka's desingularization theorem.}
The Archimedean counterpart of Theorem \ref{thm:main_a} above was proved by J.~Bernstein \cite{Ber_FT} using
D-module theory.
%It is, in  fact, a direct application of the theory of holonomic  D-modules that was developed there and in
%related works.  It also proves that the Fourier transform of $\phi_*(|\omega|)$   is  a holonomic distribution.
%%%However, the proof there does not give any \Ramada{effective} bounds on the set $U$.
In the non-Archimedean case % {a version of}
%a special case of this
Theorem \ref{thm:main_a} was proved by Hrushovski - Kazhdan \cite{HK} and Cluckers - Loeser \cite{CL}
using model theory.


\DrinI{
In this work we give a proof of Theorems \ref{thm:main_a}-\ref{thm:main_C_Arch}
based on Hironaka's desingularization theorem. The proof  is simple and effective modulo desingularization
and treats Archimedean and non-Archimedean local fields in a uniform way. Since the proof is effective it also
yields Theorems \ref{thm:futerO}-\ref{thm:futer}, which seem to be new.}
%The \RamiX{proof works both for the Archimedean and non-Archimedean case.
%Also since it is effective it gives us Theorem \ref{thm:futerO},}
%%same method allows to prove the Archimedean counterpart of Theorem \ref{thm:main_a}
%% and to prove
%%Theorem \ref{thm:futer},
%which seems to be a new result.
Note that although Hironaka's desingularization theorem is far from being elementary,
it now has understandable proofs (e.g., see ~\cite{Hir_mod}).

%Our method has the following advantages on the methods of \cite{Ber_FT,HK,CL}:
%\begin{itemize}
%\item Our method suits both the Archimedean and the non-Archimedean case,
% see theorem \ref{thm:futer}.
%\item Our method can give bounds on the open dense set $U$.
% These bounds are explicit in terms of
% resolution of singularities,
% which, today, has rather explicit descriptions (see e.g. \cite{Hir_mod}).

%In particular, one can show that $U$ can be chosen to
% depend only on the field of definition of the initial data
% and not on the local field itself, see theorem \ref{thm:futer}.
%\item Similarly to the results in \cite{Ber_FT}, but unlike the results
% in \cite{HK,CL}, our method gives bounds %on the wave front set.
%\end{itemize}



% There result has two additional assumptions:
%\begin{itemize}
%\item The subvariety $X$ have to coincide with $W$.
%\item The residue characteristic of $F$ has to be large enough.
%\end{itemize}
%The first assumption is  not crucial for their argument. On the other hand, the second one %seems to be essential and causes difficulties in the applications.

%The proof in \cite{HK} \Ramada{and \cite{CL} } is based on model theory
% and
%\Ramada{the bounds on $U$ it provides are less effective. In particular,
% the bounds depend, \RamiE{in some cases,} on
%the local field.}
%they do not show that that this }
% also does not give any \Ramada{effective} bound on $U$.

 The present paper is not the first time when Hironaka's theorem is used to replace D-module theory  in the non-Archimedean case. A well-known example is one of the earliest applications of the theory of D-modules --  the regularization and analytic continuation of the distribution $p^\lambda$ where $p$ is a polynomial and
 $\lambda$ is a complex number (see \cite{ber_pl}).
 This result has an
 alternative proof based on Hironaka's theorem, which is valid both in the Archimedean and the
non-Archimedean
cases, see \cite{BG} and \cite{Ati}.
\subsection{Idea of the proof}
$ $

%As was mentioned earlier, we prove a stronger result (see Theorem \ref{thm:main_bb}).
%In particular we show that
%if $Y$ is a smooth algebraic variety and $X$  {is a smooth} closed submanifold
% of $Y \times W$ then
%the wave front set of the partial Fourier transform of $|\omega|$ with respect
% to $W$ is small in an appropriate sense.
\DrinD{Theorem~\ref{thm:main_a} is deduced from Theorem \ref{thm:main_bb}. The latter
has}
%\RamiI{We prove directly Theorem \ref{thm:main_bb}, and then deduce Theorem~\ref%{thm:main_a}.
% Theorem \ref{thm:main_bb} has}
two advantages:
\begin{enumerate}
  \item Since we are discussing the wave front set, Theorem \ref{thm:main_bb} is more flexible with respect to changes of $X$ and $Y$.
 \item Since we are discussing a relative version, Theorem \ref{thm:main_bb} can be  %studied
\DrinD{approached} locally \DrinD{with respect to \RamiI{$Y$}}.
\end{enumerate}
Using those facts\RamiM{,} we can reduce Theorem \ref{thm:main_bb}  to the special case (see Proposition \ref{thm:main_c})
when the map $X \to Y$ is an \DrinD{open} embedding.
 Furthermore, using Hironaka's theorem we can assume \DrinD{that $\omega$ and $\phi$ behave
 ``nicely" in the neighborhood of $Y-X$.}

 %a good asymptotic behavior on the boundary of the image of $X$ in $Y$.

Using (1) and (2) again\RamiM{,} we can reduce further (see Lemma \ref{lem:red_to_dim_1}) to the case when $W$ is
1-dimensional.
By localizing the problem on $Y$\RamiM{,} we reduce Proposition \ref{thm:main_c} to
%its local model, which is rather simple in view of the assumptions
%provided by Hironaka's theorem.
\DrinD{a simple local model, which has  a symmetry with respect to an action of a large torus.
This symmetry allows to prove  Proposition \ref{thm:main_c} for the local model.}
%We prove  Proposition \ref{thm:main_c} for this local model in Lemma \ref{lem:dim_1} using
%the fact that the local model possesses a symmetry with respect to the standard action of a torus
%on an affine space.


%Let us consider the relative counterpart of Theorem \ref{thm:main_a}.
%Namely, let $Y$ be a (smooth) algebraic variety and let $X$ be a (locally closed) submanifold of $Y \times W$.
% We are interested in the regularity of the partial Fourier transform of $|\omega|$ with respect to $W$.
%
%Now, consider the special case of this problem when the projection $X \to Y$ is an embedding.
% Note that in the absolute situation (when $Y$ is a point) this special case  becomes trivial,  since it means that $X$ is one point. However, in the relative situation  it turns out that this special case implies Theorem \ref{thm:main_a}. Furthermore, using Hironaka's theorem we can assume a good asymptotic behavior on the boundary of the image of $X$ in $Y$.

%It turns out that it is more convenient to prove this theorem in the analytic case under these
%asymptotic assumptions (see Theorem \ref{thm:main_c} below for the precise formulation of this special case).

%In order to prove Theorem \ref{thm:main_c},
%we first note that its statement is local on $Y$. Using this, we can reduce the situation to the case when $W$
%is one-dimensional (see Lemma \ref{lem:red_to_dim_1}).
%Moreover, using the assumptions on the  asymptotic behavior given by Hironaka's theorem,
% the locality allows us to pass to an explicit local model.
% This passage is more convenient in the analytic category.
%The explicit local model is solved using the fact that it possesses a symmetry with respect to  the standard action of a torus on an affine space and that this action has a finite number of orbits (see Lemma \ref{lem:dim_1}).



\subsection{Structure of the paper}
$ $
%\RamiQD{This subsection should be edited in the end}
In \S\ref{sec:prel} we will fix notations and give the necessary preliminaries for the paper.
In  {\S\RamiAM{\S}\ref{ssec:ag} we recall two algebro-geometrical tools used in this paper.}
Namely, in \S\RamiAM{\S}\RamiAM{\S}\ref{sssec:rs} we  review Hironaka's theory of resolution of singularities
(see  \cite{Hir}, or \cite{Hir_mod} for a more recent overview), and
 in \S\RamiAM{\S}\RamiAM{\S}\ref{sssec:nag}  we recall Nagata's compactification theorem.
In \S\RamiAM{\S}\ref{ssec:dist} we review the theory of distributions and in particular, the notion of the wave front
 set. Most of the results there are from \cite{Hef} and  \cite{Aiz}. {The rest we provide in Appendix \ref{app:WF}.}

In \S\ref{sec:WF-Holonomic} we introduce the notion of WF-holonomic distributions and state some of its basic
properties. This notion can be viewed as a partial analytic counterpart of the algebraic notion of  holonomic
 distributions, which is defined via the theory of D-modules. We use this notion in order to formulate our main result. \RamiAM{In \S\S\ref{sec:lag}, we recall the basic facts from symplectic geometry that we use in order to work with WF-holonomic distributions. We provide proofs and references for those results in Appendix \ref{app:symp_geom}. }
% We prove the basic properties of WF-holonomic distributions  in Appendix %\ref{subapp:WFHol}.

{In  \S\ref{sec:pf_main_bb}-\ref{sec:pf_futer} we prove the main results of the paper in the non-Archimedean case.

In  \S\ref{sec:pf_main_bb} we prove  Theorem \ref{thm:main_bb} (which implies
Theorem \ref{thm:main_b} and Theorem \ref{thm:main_a}).
}

%\RamiJ{In  \S\ref{sec:pf_main_bb} we prove the main result of the paper --- Theorem \ref{thm:main_bb},
%which implies
%Theorem \ref{thm:main_b} and Theorem \ref{thm:main_a} \RamiAM{--- for the non-Archimedean case}.}

\RamiAM{In \S\ref{sec:ex_boun} we {prove} Theorem \ref{exp}, Corollary \ref{cor:exp} and  Corollary  \ref{c:explcit_open},  which are ``explicit" versions of Theorems \ref{thm:main_bb}, \ref{thm:main_b} and \ref{thm:main_a} respectively {(e.g., Theorem~\ref{thm:main_a} claims the existence of a dense open $U$ on which a certain distribution is smooth, while  Corollary  \ref{c:explcit_open} provides a concrete $U$ with this property).} We also explain how Theorem \ref{exp} implies Theorem \ref{thm:futerO}.
% (in the non-Archimedean case).

In \S\ref{sec:pf_futer} we deduce Theorem \ref{thm:futer} from Theorem \ref{thm:futerO}.
% (in the non-Archimedean case).


In \S\ref{sec:arc} we explain how to adapt the proofs {from \S\ref{sec:pf_main_bb}-\ref{sec:pf_futer}} %of the results above
for the Archimedean case.}

% and prove some implications between them.
%Theorem \ref{thm:main_b} (a stronger version of Theorem \ref{thm:main_a})
% {and Theorem \ref{thm:main_bb} (a relative version of Theorem \ref{thm:main_b}).}
%, and formulate Theorem \ref{thm:main_c} which is a special case of a relative analytic version of \ref{thm:main_b}. We also show that Theorem \ref{thm:main_c} implies Theorem \ref{thm:main_b}.

%In section  {\S\ref{sec:pf_main_bb}}  we prove Theorem  {\ref{thm:main_bb}},
% which implies the main results of the paper.

% \DrinK{
% Rami, please mention Corollary \ref{c:explcit_open} and give a precise reference to it (the formulation of Corollary \ref{c:explcit_open}  does not involve wave fronts and is more understandable than that of other statements from \S\ref{sec:ex_boun}).
% }


{In \RamiAM{Appendix} \ref{app:WF} we
% prove
\RamiAM{elaborate on}
 the results stated in  \S\S\S \ref{sssec:wfs}.}

\RamiAM{In Appendix  \ref{app:symp_geom} we elaborate on the results stated in  \S\S \ref{sec:lag}}
\RamiD{
\subsection{Acknowledgments}
We thank Patrick Gerard for a useful reference and Vladimir Hinich, for fruitful discussions.
We thank Joseph Bernstein, Ehud Hrushovski, David Kazhdan and Michael Temkin for their useful remarks.
\RamiAM{We would also like to thank Inna Entova-Aizenbud for proofreading the drafts for this work.}

Part of the work on this paper was done while Avraham Aizenbud participated in the program
 ``Analysis on Lie Groups" at the Max Planck Institute for Mathematics (MPIM) in Bonn.
Avraham Aizenbud is also partly supported by NSF grant DMS-1100943.
Vladimir Drinfeld is partly supported by NSF grant DMS-1001660.
}

\section{Preliminaries} \label{sec:prel}
\subsection{\RamiM{Notation} and conventions}\label{ssec:not}
$ $
\RamiAM{Below is a list of notations {and conventions} throughout the paper. The somewhat nonstandard among them are (\ref{it:notstd1}), (\ref{it:notstd2}), (\ref{it:notstd3}).
%Most of them are standard, though (\ref{it:notstd1}), (\ref{it:notstd2}), (\ref{it:notstd3}) are specific to our %problem or not commonly used notations.
}
% %\RamiQH{ }
% %We will use the following
% %\Rami{standard}
% %notation \Rami{and conventions} throughout the paper:
% %
% %\RamiQD{I think that we should try to write the paper uniformly for the %Archimedean and non-Archimedean case. This will cause some changes in this %subsection,
% %\S\S\ref{ssec:dist},\S\S\ref{ssec:dist} and Appendix \ref{app:WF}, but I %hope that nowhere else.}
% %
% \RamiQF{I marked the non-standard notations below. %so when we will finish the more essential issues we can think
% %how to divide it.
% The unmarked notations are not necessarily absolutely standard, but they are one of the possible
%  standard notations and they can be understood from the context rather easily.}
% \RamiQW{We should discuss how to divide it. We can put them here in 2 groups. We can
% send one of the groups to the appendix or somewhere else. The disadvantage of those options that its brakes the
% natural order of the items. We can use different fonts for the non-standard ones.
% We can also enumerate the items and say in advance what are the non-standard ones.
%  What do you think is the best? Do you have other ideas? The same refers to Notation 2.3.1.}
%
%  \DrinK{I suggest to divide \S\ref{ssec:not} into subsubsections. Each sub subsection should have a suggestive title (e.g., ``The local field $F$", ``Varieties and manifolds", ``(Co)tangent and (co)normal bundle", ``Differential forms and measures"). In each group please try to emphasize the ``nonstandard" notations. If some notations are used only in specific sections it may be better to introduce them there.}
\subsubsection{\RamiAM{The local field $F$}}\label{sssec:not_f}
\begin{enumerate}%[(\ref{sssec:not_f}.1)]
 \setcounter{enumi}{0}
\item\label{it:notstd1}  We fix a local  field $F$ of characteristic $0$. \RamiT{It will be non-Archimedean in the entire paper except \S\ref{sec:arc} and the Appendices.}
\RamiU{\item We always {equip $F$ with the \emph{normalized}} absolute value %$|\cdot |$ on $F$
(this is the multiplicative {quasi-}character $x\mapsto |x|$ %$|\cdot |$
given by the action on Haar measures). }
\item \label{it:notstd2} We fix \RamiAG{a non-trivial} additive character \RamiE{$\psi:F \to \C^{\times}$}.
\end{enumerate}
\subsubsection{\RamiAM{Varieties and manifolds}}%\label{}
$ $
\RamiQAB{the numbering shift is manual. Should be checked in the end}
\begin{enumerate}%[(\ref{sssec:not_f}.1)]
 \setcounter{enumi}{3}

\item All the algebraic varieties and analytic varieties which we consider are \RamiE{reduced, separated and} defined over $F$.
\item {We will treat $F$-algebraic varieties as $\bar{F}$-algebraic ones equipped with an $F$-structure.}
\item  {We will treat $F$-vector spaces both as algebraic varieties and analytic varieties.} %\DrinK{I think it is not necessary to say this.} \RamiQAB{let's discuss it}
\item When we say ``an analytic variety", we mean an $F$-analytic variety in the classical sense of \cite{ser} and not in the sense of rigid geometry or Berkovich geometry.
\item For an algebraic variety $X$,  we will denote by $\fpt{X}$ the set of $F$ points of $X$
 considered as an analytic variety (and, in particular, as a topological space).
 \DrinD{By abuse of notation, the map $X(F) \to Y(F)$ corresponding to a morphism of algebraic varieties
 $\phi:X \to Y$ will also be denoted by $\phi$.}
%  {For a morphism $\phi:X \to Y$ between two algebraic varieties we will  denote, by abuse of
%notations, the corresponding map $X(F) \to Y(F)$  also by $\phi$ }
%\item We will consider $F$-vector spaces both as algebraic varieties and %analytic varieties.
\item We will use the word ``manifold" to indicate smoothness, e.g. ``algebraic manifold" will mean smooth algebraic variety.
\item When we want to speak in general about algebraic and analytic varieties or manifolds, we will just say variety or manifold.
\RamiR{\item We will use the word ``regular" only in the sense of algebraic geometry and not in the sense of analytic geometry.}
\item We will usually use the same notation for a vector bundle and its total space.
%\DrinK{I think it is not necessary to say this.} \RamiQAB{let's discuss it}
\item \RamiAN{For a vector bundle $E$ over a manifold $X$, we will identify $X$ with the zero section inside $E$.}
\end{enumerate}
\subsubsection{\RamiAM{The \RamiAN{(co)}tangent \RamiAN{and the (co)normal} bundle} %\RamiQAB{or may be better ``vector bundles"?}
}%\label{}
$ $
\RamiQAB{the numbering shift is manual. Should be checked in the end}

\begin{enumerate}%[(\ref{sssec:not_f}.1)]
 \setcounter{enumi}{13}
% \item
%  \DrinK{Yes, please remove this notation  from here and
% introduce it when we use it.} For a vector bundle $E$ over a variety $X$, we denote by $P_E:E \to X$ the projection.
% \RamiE{We will also identify $X$ with the zero section inside $E$.}
  \item For a manifold $X$\RamiM{,} we denote by  $T X=T (X)$ and  $T^* {X}=T^* {(X)}$ the tangent and co-tangent bundles, respectively. For a point $x \in X$, we denote by $T_x X=T_x(X)$ and  $T^*_x X=T^*_x(X)$ the tangent and co-tangent spaces, respectively.
\item \label{it:notstd3} For a (locally closed) submanifold $Y \subset X$, we denote by
$N_Y^X:=(T_X|_Y)/T_Y$ and $CN_Y^X:=(N_Y^X)^*$ the
normal and co-normal bundle to $Y$ in $X$, respectively.
\end{enumerate}
\subsubsection{\RamiAM{Group {and Lie algebra} action{s}}}%\label{}

$ $
\RamiQAB{the numbering shift is manual. Should be checked in the end}

\begin{enumerate}%[(\ref{sssec:not_f}.1)]
 \setcounter{enumi}{15}
\item For a group $G$ acting on a set $X,$ and a point $x \in X,$ we denote by $Gx$ or by $G(x)$ the orbit of $x$ and by $G_x$ the stabilizer of $x$.
%
% we also denote by $X^G$ the set of $G$ invariant elements and for an element %$g \in G$ denote by $X^g$ the set of $g$ invariant elements
\item An action of a Lie algebra $\g$ on a
%({analytic or} algebraic etc)
manifold $M$ is a Lie algebra homomorphism from $\g$ to the Lie algebra of vector fields on $M$.
Note that an action of an ({analytic or} algebraic) group on $M$ defines an action of its Lie algebra on $M$.
\item For a Lie algebra $\g$ acting on $M$, an element $\alpha \in \g$ and a point $x \in M$, we denote by $\alpha(x) \in T_xM$ the value at $x$ of the vector field corresponding to
$\alpha$. We denote by $\g x \subset T_xM$ or by $\g (x) \subset
T_xM$ the image of the map $\alpha \mapsto \alpha(x)$ and by $\g_x
\subset \g$ its kernel.
% We denote $M^{\g}:=\{x \in M|\g x=0 \}$
%and $M^{\alpha}:=\{x \in M|\alpha(x)=0 \}$, analogously to the
%group case

\end{enumerate}
\subsubsection{\RamiAM{Differential forms}}%\label{}

$ $
\RamiQAB{the numbering shift is manual. Should be checked in the end}

\begin{enumerate}%[(\ref{sssec:not_f}.1)]
 \setcounter{enumi}{18}
\item For a top differential form \Rami{$\omega$ on \RamiAM{a} %an analytic
  manifold $M$},
 we define \Rami{its} absolute value \Rami{$|\omega|$ to be the corresponding measure on $M$} \RamiAM{(or on $M(F)$ in the algebraic case).}

% \item For a top differential  form $\omega$ on an algebraic manifold $X$,
%  we denote $|\omega|:=|\omega(F)|$ to be the corresponding measure on $X(F).$
%   \DrinK{Do we ever use the notation $|\omega(F)|$?}
\end{enumerate}
\subsection{ {Algebraic geometry}} \label{ssec:ag}
\subsubsection{Resolution of singularities} \label{sssec:rs}
$ $
In this paper we will need Hironaka's theory of \RamiM{resolution} of singularities. This theory was established in \cite{Hir}. A more recent overview can be found in \cite{Hir_mod}. \\ Let us summarize here the results we need.
\begin{definition}
Let $X$ be an algebraic variety.
\begin{itemize}
\item A \DrinD{\emph{resolution of singularities}} of $X$ is a proper \RamiD{map $p:Y \to X$ such that $Y$ is smooth and $p$ \RamiE{is a} birational equivalence.}
%\end{definition}
%\begin{definition}
%Let $X$ be an algebraic variety
%\item A divisor in $X$ is a subvariety  $D \subset X$ such that for any  irreducible component $X_0  \subset X$, we have $\dim (D \cap X_0)=\dim (X_0)-1$.
%\end{definition}
%\begin{definition}
%Let $X$ be an algebraic variety
\item \DrinD{A subvariety $D \subset X$ is said to be a \emph{normal crossings divisor}
(or \emph{NC divisor}) if }
%one of the following equivalent conditions satisfied:
%\begin{itemize}
%\item
for any $x\in D$ there exists an \et neighborhood $\phi: U \to X$ of $x$
\RamiD{and an \et map $ {\alpha}:U \to \A^n$ such that $\phi^{-1}(D)= {\alpha}^{-1}(D')$, where $D' \subset \A^n$}
is a union of coordinate hyperplanes.% in $U$.


 {
\item %A divisor with strict normal crossings is a subvariety $D \subset X$ such that
\DrinD{A subvariety $D \subset X$ is said to be a \emph{strict normal crossings divisor}
(or \emph{SNC divisor}) if }
for any $x\in D$ there exists \RamiM{a} Zariski neighborhood $U \subset X$ of $x$
and an \et map $ {\alpha}:U \to \A^n$ such that $D \cap U= {\alpha}^{-1}(D')$, where $D' \subset \A^n$
 is a union of coordinate hyperplanes.% in $U$.
}
%\item for any $x\in D$ there exist an Etale neighborhood $\phi: U \to X$ of $x$ %such that $U$  is a Zariski open subset of an affine space and $phi^{-1}(D)$ %is a union of coordinate hyperspaces in $U$.
%\end{itemize}
\item \DrinD{We say that a resolution of singularities $p:Y \to X$ \emph{resolves} (resp. \emph{strictly resolves}) a closed subvariety $D \subset X$ if $p^{-1}(D)$ is an NC divisor (resp. an SNC divisor).}
%A resolution of singularities of $X$ that resolves a \Ramada{closed subvariety} $D \subset X$ is a resolution of
%singularities  $p:Y \to X$ such that $p^{-1}(D)$ is a divisor with normal crossings.
% {
%\item A resolution of singularities of $X$ that strictly resolves a closed subvariety $D \subset X$ is a resolution
%of singularities  $p:Y \to X$ such that $p^{-1}(D)$ is a divisor with strict normal crossings.}
\end{itemize}
\end{definition}

%The following lemma is standard (see  e.g. \cite{AG_HC} for a similar argument)


%%%%%%%% I do not think that we need it any more

%We will need the following analytic counterpart of the notion of divisor with normal crossings:

%\begin{definition}
%Let $X$ be an analytic  variety.
% A subvariety $D \subset X$ is called a divisor with normal crossings if
%for any $x\in D$, there exists an open analytic embedding $\phi: U \to X$ of $x$ such that $U$  is an
%open analytic  subset of an affine space and $\phi^{-1}(D)$ is a union of coordinate hyperplanes in $U$.
%\end{definition}


%This notion is related to the algebraic notion by the following theorem:\\ %In order to apply the algebraic notion of divisor with normal crosings to %the analitic world we will need
%\begin{theorem}[cf. \cite{ser}, Theorem 2 in \S 9 of Chapter III in part
%II] \label{InvFunct} Let $\phi: M \to N$ be an \et map of analytic
%manifolds. Then locally  {(on $M$) it is} an isomorphism.
%\end{theorem}
%\begin{cor}
%If $D \subset X$ is a divisor with normal crossings, then
%\RamiD{$D(F) \subset X(F)$ is a divisor with normal crossings.}

%%for any $x\in D(F)$ there exists an analytic neighborhood $x \in  U \subset X(F)$ of $x$ such that $U$  is isomorphic to an open subset of an analytic affine space and $(D \cup U)$ corresponds to a union of coordinate hyperspaces in $U$.
%\end{cor}
%%%%%%%% I do not think that we need it any more (end)

\begin{theorem}[Hironaka] \label{thm:Hir}
Let $X$ be an algebraic variety and
\DrinD{$U\subset X$ a dense nonsingular open subset. Then there exists a resolution of singularities
$p:\tilde X \to X$ that resolves $X-U$ such that the map $p^{-1}(U) \to \RamiM{U}$ is an isomorphism.}
% {??to ask Temkin??}
%$D \subset X$ be a \Ramada{closed subvariety}.
%Then there exists a resolution of singularities of $X$ that resolves $D$.
% {Moreover, if all the singularities of $X$ are contained in $D$ then we can choose a resolution
%$p:\tilde X \to X$ such that $p|_{p^{-1}(X-D)}:p^{-1}(X-D) \to (X-D)$ is an isomorphism.}
\end{theorem}
 {
There is a standard procedure to resolve a normal crossings divisor further to a strict normal crossings divisor,
 see e.g. \cite{Jon}.
This gives the following corollary.

\begin{corollary}
%\DrinD{Theorem~\ref{thm:Hir} remains valid if one replaces ``resolves" by ``strictly resolves".}
\DrinD{In Theorem~\ref{thm:Hir} one can replace ``resolves" by ``strictly resolves".}
%Let $X$ be an algebraic variety and $D \subset X$ be a closed subvariety. Then there exists a resolution of
%singularities of $X$ that strictly resolves $D$.??to ask Temkin??
%
% {again, if all the singularities of $X$ are contained in $D$ then we can choose a resolution
%$p:\tilde X \to X$ such that $p|_{p^{-1}(X-D)}:p^{-1}(X-D) \to (X-D)$ is an isomorphism
\end{corollary}
}
\subsubsection{ {Nagata's  compactification theorem}} \label{sssec:nag}
 {
We will need the following theorem:

\begin{theorem}[Nagata (see e.g. \cite{Con_Nag})] \label{thm:Nag}
Let $\phi: X \to Y$ be a morphism of algebraic \RamiM{varieties}.
Then there exists a factorization $\phi=\phi' \circ i: X \to X' \to  Y$ such that  $i: X \to X'$ \RamiM{is} an  open
 embedding and $\phi':X' \to Y$ is proper.
\end{theorem}
% (Nagatas corollary)\begin{corollary} \label{cor:Nag}
% (Nagatas corollary)Let  $X,Y\subset \overline{Y}$ \RamiM{be}  smooth algebraic varieties such that $Y$ is open dense in $ \overline{Y}$.
% (Nagatas corollary) Let $\phi:X \to Y$ be a proper map.  Then there exist an algebraic variety $\overline X$, a map
% (Nagatas corollary)$\overline \phi: \overline X \to   \overline Y $  and an open \RamiM{embedding} $i:X \hookrightarrow\overline X$
% (Nagatas corollary) such that the following diagram commutes:
% (Nagatas corollary)%$$
% (Nagatas corollary)%\xymatrix
% (Nagatas corollary)%{        X  \ar@{->}^{\phi}[r]   \ar@{->}^{i}[d]        & Y  \ar@{->}^{}[d] \\
% (Nagatas corollary)%         \overline X    \ar@{->}^{\overline \phi}[r]    &  \overline Y  \\
% (Nagatas corollary)%}
% (Nagatas corollary)%$$
% (Nagatas corollary)
% (Nagatas corollary)$$
% (Nagatas corollary)\xymatrix
% (Nagatas corollary){        X  \ar@{->}^{i}[r]   \ar@{->}_{\phi}[d]        & \overline X  \ar@{->}^{\overline \phi}[d] \\
% (Nagatas corollary)         Y \    \ar@{->}^{}[r]    &  \overline Y  \\
% (Nagatas corollary)}
% (Nagatas corollary)$$
% (Nagatas corollary)
% (Nagatas corollary)\end{corollary}


%\begin{corollary} \label{cor:Nag}
%Let $W$ be a finite-dimensional $F$ vector space and $X,Y$ be a smooth algebraic varieties.  Let $\phi:X \to Y \times W$
%be a proper map. Then there exist an algebraic variety $\overline X$, a map
%$\overline \phi: \overline X \to  Y \times \overline W $  and an open imbedding $i:X \to \overline X$
% such that the following diagram is commutative:
%$$
%\xymatrix
%{        X  \ar@{->}^{\phi}[r]   \ar@{->}^{i}[d]        & Y \times W  \ar@{->}^{}[d] \\
%         \overline X    \ar@{->}^{\overline \phi}[r]    & Y \times \overline W \\
%}
%$$
%\end{corollary}


%This theorem together with Hironaka's theorem gives the following corollary
%\begin{corollary}
%Let $W$ be a finite dimensional $F$ vector space and $X$ be a smooth algebraic variety.  Let $\phi:X \to W$
%be a proper map. Then there exist a smooth algebraic variety $\tilde X$ and a map
%$\tilde \phi: \tilde X \to \overline{W}$ such that the map
% $\tilde \phi|_{\tilde \phi^{-1}(W)}: {\tilde \phi^{-1}(W)} \to W$ factors through a birational ....??
%\end{corollary}
}
\subsection{Distributions \RamiAE{in the non-Archimedean case}}\label{ssec:dist}
$ $\\
\RamiAE{We recall here the facts that we need about distributions in the non-Archimedean case. The Archimedean case will be discussed in \S\S\ref{ssec:arc_dist}. }

We will use the language of $l$-spaces and distributions on them. For an overview of this theory we refer
 the reader to \cite{BZ}.
%\RamiQD{In the Archimedean case we should choose a framework. Although many
% of the statements here are valid for general distributions, we have to
%  be able to discuss Schwartz distributions. So I suggest to restrict our
%   discussion to Schwartz distributions. This will slightly restrict our
%    generality, but only in a semantic way. Since all the statements here
%     which are valid for general distributions can be easily deduced from
%      the Schwartz case.
%
%In any case, we should choose in what generality we discus Schwartz distributions.
% I believe that the natural generality is the generality of Nash manifolds.
%  It have one disadvantage: some people are not familiar with it, therefor
%   I suggest to put a remark that except in the appendix, one can replace
%    the word ``Nash", by ``smooth real algebraic".

%If we write everything uniformly, then I think that we need some name which
% will mean ``analytic manifold''   in the
%in the non-Archimedean case and ``Nash manifold'' in the Archimedean case. %What do you suggest? I once used the name ``B-analytic space'' in such a %situation.
%}

% \DrinK{
% I suggest to organize the beginning of \S\ref{ssec:dist} (before the subsubsection on Fourier transform) into two subsubsections: ``Functional spaces" and ``Pullbacks and pushforwards". The subsubsection on functional spaces will mostly contain notation, and I don't see here anything that could be skipped.
% \RamiQAB{Done}
%
% On the other hand, I think that in the subsubsection on pullbacks and pushforwards the actual definitions of the operators can be skipped. I think the main goal here is to recall \emph{which} operators are defined rather than \emph{how} to define them. Another important goal here is to introduce the \emph{nonstandard} notations.
% \RamiQAB{let's discuss it.}
%}

\RamiAM{Let us briefly recall the basic notations and constructions of this theory;  all the notations except numbers (\ref{it:dist_notstd1}), (\ref{it:dist_notstd2}), (\ref{it:dist_notstd3}) are standard.}

%We will use the following notations.

%\begin{notation}
\subsubsection{\RamiAM{Functional spaces}}\label{sssec:fun_sp}
Let $X$ be an $l$-space,{ i.e. a locally compact totally disconnected topological space. }
\begin{enumerate}
\item Denote by $C^\infty(X)$ the space
of smooth functions on $X$ (i.e. locally constant \RamiE{complex valued} functions).

\item Denote by $\Sc(X)$ the space
of \RamiK{\emph{Schwartz functions}} on $X$,
\RamiE{i.e. smooth, compactly
supported functions.} {Define the space of distributions} $\Sc^* (X):= \Sc(X)^*$ to be the dual
space to $\Sc(X)$\RamiT{, endowed with the weak dual space topology}. \RamiE{We will also denote by $C(X)$ the space of complex valued, continuous functions on $X$.}

\RamiE{\item By ``locally constant sheaf'' over an $l$-space we mean a locally constant sheaf of
\DrinA{finite-dimensional vector spaces over $\C$. }
%finite rank.
In fact, we will
need only locally constant sheaves of rank $1$.}
%\RamiQD{If we write everything uniformly, then here we need some name which %will mean ``locally constant sheaf''
%in the non-Archimedean case and ``a (Nash) vector bundle'' in the Archimedean %case. What do you suggest? I once used the name ``vector system'' in a similar %(but more general) situation.}

\item For any locally constant sheaf $E$ over $X$, we denote by
$\Sc(X,E)$ the space of compactly supported sections of $E$, by $\Sc^*(X,E)$ its dual space,
 and by $C^\infty(X,E)$ the space of sections of $E$.
\RamiE{
We will also denote by %$C(X,E):= C^\infty(X,E) \otimes_C^\infty(X) C(X,E)$
\DrinA{
%maybe
$C(X,E):= C^\infty(X,E) \otimes_{C^\infty(X)} C(X)$
%?
}the space of
 continuous sections of $E$.}


%\item For any finite dimensional complex vector space $V$ we denote
%$\Sc(X,V):=\Sc(X,X \times V)$ and $\Sc^*(X,V):=\Sc^*(X,X\times
%V)$, where $X \times V$ is a constant sheaf.

%\item Let $Z \subset X$ be a closed
%subset. We denote
%$$\Sc^*_X(Z):= \{\xi \in \Sc^*(X)|\Supp(\xi)\subset Z\}.$$
%For a locally closed subset $Y \subset X$, we denote
%$\Sc^*_X(Y):=\Sc^*_{X\setminus (\overline{Y} \setminus Y)}(Y)$. In
%the same way, for any locally constant sheaf $E$ on $X$ we define
%$\Sc^*_X(Y,E)$.

% {\item
%For a distribution $\xi \in \Sc_c^*(X,E)$ define its support $\supp(\xi)$ to be the union of all open subsets
% $U \subset X $ such that $\xi|_U=0$}

\item
 {Let $\Sc_c^*(X,E)$  be the space of compactly supported distributions.
%, i.e. distributions whose restriction to an open set with compact compliment vanishes.
}
Note that we have a canonical embedding
 {$\Sc_c^*(X,E) \hookrightarrow (C^\infty(X,E))^*$.}
\item\label{it:dist_notstd1}
%\RamiQW{non-standard}
Suppose $X$ is an analytic variety. Then we define $D_X$ to be the sheaf of locally constant measures on $X$
 (i.e. measures that are locally isomorphic to the Haar measure on $F^n$).
 We set $\G(X):=\Sc^*(X,D_X)$ \RamiE{to be the space of generalized functions} and $\G(X,E):=\Sc^*(X,D_X \otimes
E^*)$ \RamiE{to be the space of generalized sections of $E$.}
\RamiE{Similarly, we define  $\G_c(X)$ and $\G_c(X,E)$. Note that we have natural embeddings $C^\infty(X,E) \subset C(X,E) \subset \G(X,E)$ and $\Sc(X,E) \subset \G_c(X,E)$.
We will identify these
spaces with their images and we will refer to the generalized  \RamiM{sections which} lie in $C^\infty(X,E)$  as ``smooth'' and those
which lie in $C(X,E)$ as ``continuous''}.
\end{enumerate}
\subsubsection{\RamiAM{Pullback and pushforward}}\label{sssec:op_dist}
\RamiAM{Let $\phi:X \to Y$ a \RamiE{continuous} map of $l$-spaces.}
\RamiQAB{the numbering shift is manual. Should be checked in the end}
\begin{enumerate}%[(\ref{sssec:not_f}.1)]
 \setcounter{enumi}{6}
\item %For a \RamiE{continuous} map of $l$-spaces $\phi:X \to Y$,
\RamiAM{W}e define $\phi^*:\RamiE{C^\infty(Y) \to C^\infty(X)}$
 to be the pullback
and $\phi_*:=(\phi^*)^* {|_{\Sc_c^*(X,E)}}:\RamiE{\Sc_c^*}(X) \to \RamiE{\Sc_c^*}(Y)$ to be the pushforward.
Similarly, we define \RamiE{$\phi^*:C^\infty(Y,E) \to C^\infty(X,\RamiM{\phi^*(E)})$ and
$\phi_*:\Sc_c^*(X,\DrinD{\phi}^*(E)) \to \Sc_c^*(Y,E)$}
for any locally constant sheaf
$E$.

\item \RamiAN{Assume that $\phi$ is proper. This allows us to extend the    pushforward
 to a map $\phi_*:\Sc^*(X) \to \Sc^*(Y)$ in the following way.} %For a proper map of $l$-spaces $\phi:X \to Y$, {we
%\RamiAM{If $\phi$ is proper,}
{\RamiAN{N}ote that $\phi^*(\Sc(Y)) \subset \Sc(X)$
and consider $\phi^*|_{\Sc(Y)}$
 as a map from $\Sc(Y)$ to $\Sc(X)$. So we can} \DrinA{define the pushforward
 $\phi_*:=(\phi^*|_{\Sc(Y)})^*:\Sc^*(X) \to \Sc^*(Y)$ extending the above map
  $\phi_*:\Sc_c^*(X) \to \Sc_c^*(Y)$.}
 %extend the pushforward $\phi_*$ by $\phi_*:=(\phi^*|_{\Sc(Y)})^*:\Sc^*(X) \to \Sc^*(Y)$}.
 Similarly, we define
%$\phi^*:\Sc(Y\RamiE{,E}) \to \Sc(X\RamiE{,E})$ and
$\phi_*:\Sc^*(X,\DrinD{\phi}^*(E)) \to \Sc^*(Y,E)$
for any locally constant sheaf
$E$.

 {\item
We can generalize the above two definitions in the following way.
Let% $\phi:X \to Y$ be a continuous map of $l$-spaces and
$\xi \in \Sc^*(X)$. \RamiM{Assume  $\phi|_{\supp(\xi)}$ is proper}.
\DrinD{Then $\phi^*(f)\cdot\xi$ has compact support for any $f\in\Sc (Y)$, so we can define
$\phi_*(\xi)\in\Sc^*(Y)$ by
$$\langle\phi_*(\xi) , f\rangle=\int\limits_X\phi^*(f)\cdot\xi:=\langle\phi^*(f)\cdot\xi,1\rangle,
\quad f\in\Sc (Y).$$
Similarly, for any locally constant sheaf $E$ on $Y$
one defines $\phi_*(\xi)$ if $\xi \in \Sc^*(X,\phi^*(E))$ is such that
$\phi|_{\supp(\xi)}$ is proper.}
%Then we can define $\phi_*(\xi)$ by
%$\langle\phi_*(\xi) , f\rangle=\langle(\phi|_{\phi^{-1}(\supp(f))})_*(\xi|_{\phi^{-1}(\supp(f))}) , f|_%{\supp(f)}\rangle$.
}
\item \RamiE{Let $\phi:X \to Y$ be an analytic submersion of analytic manifolds. \RamiAN{Let us extend the pullback $\phi^*:C(Y)\to C(X)$ to a map $\phi^*:\G(Y) \to \G(X)$ in the following way. N}ote that
 $\phi_*(\Sc(X,D_X)) \subset \Sc(Y,D_Y)$, and consider $\phi_*|_{\Sc(X,D_X)}$
 as a map from $\Sc(X,D_X)$ to $\Sc(Y,D_Y)$.
 \DrinA{The pullback $\phi^*:=(\phi_*|_{\Sc(X,D_X)})^*:\G(Y) \to \G(X)$ extends the map $\phi^*:C(Y)\to C(X)$.}
%We extend the pullback $\phi^*:=(\phi_*|_{\Sc(X,D_X)})^*:\G(Y) \to \G(X)$.
}
%For an analytic {submersion} $\phi:X \to Y$ of analytic manifolds we denote by
 %$\phi_*:\Sc(X,D_{X}) \to \Sc(Y,D_{Y})$ the {pushforward.
%One can define it easily using the local model for a submersion of analytic manifolds.} \RamiQ{(Should I add more details here?)}
\item \label{it:dist_notstd2}%\RamiQW{non-standard}
For an analytic {submersion} $\phi:X \to Y$ of analytic manifolds\RamiM{,}
we define $D_Y^X:=\phi^*(D_Y^*)\otimes D_X$.
For a locally constant sheaf $E$ over $Y$\RamiM{,} we denote  $\phi^{\RamiAN{!}}(E):=\phi^*(E)\otimes D_Y^X$.
\RamiQ{(Do you know any standard notation for this notion?)}
%\item For an analytic {submerion} $\phi:X \to Y$ of analytic manifolds % and a locally constant sheaf $E$ over $Y$.
\RamiE{As before, we have the pushforward $\phi_*|_{\Sc(X,\phi^{\RamiAN{!}}(E))}:\Sc(X,\phi^{\RamiAN{!}}(E)) \to \Sc(Y,E)$
and the pullback $\phi^*:=(\phi_*|_{\Sc(X,\phi^{\RamiAN{!}}(E))})^*:\G(Y,E) \to \G(X,\phi^*(E))$.
}


\item \label{it:dist_notstd3} Let $T:X \to Y$ be an isomorphism of analytic manifolds. Note that $T_*=(T^{-1})^*$ both for
functions and for distributions. In this case, we will use the notation $T$ for both of these maps.
% \DrinK{This notation is also non-standard.}


%\item For an analytic {submersion} $\phi:X \to Y$ of analytic manifolds %we denote by $\phi^*:=(\phi_*)^*:\G(Y) \to \G(X)$ the pullback, similarly %we denote $\phi^*:\G(Y,E) \to \G(X,\phi^*(E))$ for any locally constant %sheaf
% $E$.
\end{enumerate}
%\end{notation}


% Recall that an $l$-spaces is locally compact totally disconnected topological space $X$, a Schwartz function on it is a locally constant compactly supported function on $X$, the space of Schwartz function on $X$ is denoted  by $\Sc(X).$  and a distribution on $X$ is a linear functional on $\Sc(X)$ and the space of  distribution on $X$ is denoted  by $\Sc^*(X)$.
%
% \RamiQ{I've changed my mind: I think it is better to talk about generalized sections of
%sheaves and not only about distributions. It makes life significantly easier in lemma  \ref{lem:red_to_dim_1}.}
%
%
% More generally for a ??(may be we should say locally constant)?? sheaf  $E$ %on $X$  we denote by $\Sc(X)$


%??The next 2 sections seem to be too long for the preliminaries, so maybe %it worth to move the proofs (and even some technical definitions and formulations) %to an appendix??

%\DrinA{It seems that the notation $\Sc^*_X(Z):= \{\xi \in \Sc^*(X)|\Supp(\xi)\subset Z\}$ has disappeared.
%I hope you have checked that it is not used in the article.}

\subsubsection{Fourier transform} \label{sssec:ft}

\begin{definition}\label{def:FT}
$ $
\begin{itemize}
\item Let $W$ be an $F$-vector space. We define the %\RamiK{\emph{Fourier transform}}
\term{Fourier transform}{FT}
$$\cF^{}:\Sc(W,D_W) \to \Sc(W^*)$$ by $$\cF(f)(\phi)=\int f \cdot(\psi \circ \phi), \quad \phi\in W^*.$$
%(here $\phi$ is a linear map $W\to F$ and $\psi \circ \phi$ is a map $W\to ??? $).
We also define $$\cF^{*}:\Sc^*(W) \to \G(W^*),$$
 to be the dual map (when $W$ is replaced with $W^*$)
%If there is no possible ambiguity we %will denote both of them by $\cF$
\item Let $X$ be an \RamiF{analytic manifold}. Similarly, we define  the partial Fourier transforms
 $$\cF_{W}:\Sc(X \times W,D_X^{X \times W}) \to \Sc(X \times W^*)$$
 \RamiQ{(Should I add more details here?)} and
$$\cF_{W}^{{*}}:\Sc^*(X \times W) \to \RamiF{\Sc^*(X \times W^*,D_X^{X \times W^*})}
  \DrinD{=\G(X\times W^*,D^{X\times W^*}_{W^*})}.$$
\end{itemize}

\end{definition}

We formulate here some standard properties of the Fourier transform \RamiM{which} we will use in the paper. %The following proposition is standard
\begin{proposition} \label{prop:F_prop}
Let $W$ \RamiAA{and $W'$} be  $F$-vector space\RamiAA{s} and $X$ be an \RamiF{analytic manifold}.
Let $\xi \in \Sc^*(X \times W)$.
\begin{enumerate}
\item Let $U \subset X$ \RamiT{be an open set.} Then $\cF_{W}^{{*}}(\xi)|_{\RamiT{U\times W}}=\cF_{W}^{{*}}(\xi|_{\RamiT{U\times W}})$.
%\RamiQH{We also use a similar statement for \et maps, but it follows formally %from this one so I do not think we should mention it.}
\item Let $f\in C^\infty(X)$ be a locally constant function. Then $\cF_{W}^{{*}}(f\xi)=f\cF_{W}^{{*}}(\xi).$
\item Let $p:X \to Y$ be a proper map of $l$-spaces. Then $\cF_{W}^{{*}}(p_*\xi)=p_*\cF_{W}^{{*}}(\xi).$
\RamiAA{\item \label{prop:F_prop:4}
Let $\eta \in \Sc^*(X \times W\times W')$. Then $\cF_{ W\times W'}(\eta)=\cF_{W}(\cF_{W'}(\eta)).$}
\end{enumerate}
\end{proposition}

In order to formulate the last \RamiF{properties}, we will need the following \RamiF{notation}.

\begin{notation}\label{not:F_prop}
\RamiF{
Let $W,L$ be $F$-vector spaces and $X$ be an analytic manifold. Let $\nu:X \to Hom(L,W)$ be a continuous map. Then
%Let $\xi \in \Sc^*(X \times W)$.
\begin{enumerate}
\item  $\nu^t:X \to Hom(W^*,L^*)$ \DrinH{denotes} the map given by
$\nu^t(x)=\nu(x)^t$;
% composition of $\phi$ with the
%transposition.

\item   $\rho_\nu:X \times L \to X \times W$  \DrinH{denotes} the map given by $\rho_\nu(x,y)=(x,\nu(x)(y))$;
\DrinH{in particular, we use this notation when $L=W$ and $\nu:X \to F\subset \End(W)$ is a scalar function;}
%\RamiR{In case we have a fixed space $W$, we will use the same notation when $\nu:X \to F$ is a scalar
%function when we consider $F$ embedded in $\End(W)$  in the obvious way};

\item  $Mon(L,W) \subset Hom(L,W)$  \DrinH{denotes} the space of linear embeddings from $L$ to $W$.
%Let $L$ be another $F$-vector space and Let $Mon(L,W)$ denote the space of linear embeddings from $L$ to $W$.
%Let $\nu:X \to Mon(L,W)$ be a continuous map. This gives an embedding $\phi_\nu:X \times L \to X \times W$. cons
\end{enumerate}
}

%Let $W$ be an $F$-vector space and $X$ be an $l$-space.
% For a continuous  function
 %$f:X \to GL(W)$.
% $f:X \to F^\times$, we define a transformation $\rho_f:X \times W \to X \times W$ by $\rho_f(x,w):=(x,f(x)w)$.
\end{notation}

\begin{proposition} \label{prop:F_prop2}

Let $W,L$ be $F$-vector spaces and $X$ be an analytic manifold. Let $\nu:X \to Mon(L,W)$ be a continuous map.
Then
\RamiG{
the following diagrams are commutative:

\begin{equation}
\xymatrix
{        \Sc(X \times W^*,D_X^{X \times W^*})  \ar@{->}^{\quad \quad \cF_{W^*}}[r] \ar@{->}^{(\rho_{{\nu^t}})_*}[d]& \Sc(X \times W) \ar@{->}^{(\rho_{\nu})^* }[d]  \\
         \Sc(X \times L^*,D_X^{X \times L^*})  \ar@{->}^{\quad \quad \cF_{L^*}}[r]                                 & \Sc(X \times L)   \\
}
\end{equation}

\begin{equation}
\xymatrix
{        \Sc^*(X \times W^*,D_X^{X \times W^*})  \ar@{<-}^{\quad \quad \cF^*_{W}}[r] \ar@{<-}^{(\rho_{{\nu^t}})^*}[d]& \Sc^*(X \times W) \ar@{<-}^{(\rho_{\nu})_* }[d]  \\
         \Sc^*(X \times L^*,D_X^{X \times L^*})  \ar@{<-}^{\quad \quad \cF^*_L}[r]                                 & \Sc^*(X \times L)   \\
}
\end{equation}
%\item
%\xymatrix{ & \parbox{60pt}{\center{(1)}} & \parbox{100pt}{\center{(2)}} & \parbox{100pt}{\center{(3)}} & \\
%(1)&   & \parbox{60pt}{$\fv\otimes \Sc(X' \times V)$}\ar@{->}[r]^{Id_{\fv}\otimes(Id_{X'} \times p)_*}\ar@{->}^{d \tilde{\pi}_\nu}[d] & \parbox{60pt}{$\fv \otimes \Sc(W \times X')$\ar@{->}^{d\pi_\nu}[d]} & \\
%(2)& \parbox{60pt}{$\fl \otimes \Sc(X' \times V)$}\ar@{->}^{d\sigma}[r]\ar@{->}^{Id_\fl\otimes\bar{I}}[d]&\parbox{45pt}{$\Sc(X' \times V)$}\ar@{->}^{(Id_{X'} \times p)_*}[r]\ar@{->}^{\bar{I}}[d]
%& \parbox{50pt}{$\Sc(W \times X')$} \ar@{->}[r]\ar@{->}^{I}[d]
%& 0 \\
%(3)& \parbox{40pt}{$\fl \otimes \Sc(X')$}\ar@{->}^{\tau_\mu}[r]\ar@{->}[d]& \parbox{25pt}{$\Sc(X')$}\ar@{->}^{r}[r]\ar@{->}[d] & \parbox{30pt}{$\Sc(X'_0)$} \ar@{->}[r]\ar@{->}[d] & 0 \\
% &                   0                                       &           0                                   &                   0                          &
%}
%\item $ (\rho_{\nu})^* \circ \cF_{W^*}  =   \cF_{L^*} \circ (\rho_{{\nu^t}})_* :\Sc(X \times W^*,D_X^{X \times W^*}) \to \Sc(X \times L)$
%\item $\cF_W^* \circ (\rho_{\nu})_*= (\rho_{{\nu^t}})^* \circ \cF^*_L :\Sc^*(X \times L) \to \RamiF{\Sc^*}(X \times W^*,D_X^{X \times W^*})$
}
\end{proposition}
Note that since $\rho_\nu$ is an embedding, and $\rho_{\nu^t}$ is a submersion,
the inverse and the direct images in the diagrams
are defined.



%Let $W$ be an $F$-vector space and $X$ be an $l$-space. Let $f:X \to F$ be an invertible continuous function, and
%let $\xi \in \Sc^*(X \times W)$.
% Then
 %  $\cF_{W}^{{*}}(\rho_f(\xi))=
 %  %\frac{1}
 %  {|\det \circ f|^{2}}\rho_{f^{-1}}(\cF_{W}^{{*}}(\xi)).$
%\RamiE{$\cF_{W}^{{*}}(\rho_f(\xi))=
 %\frac{1}
% \rho_{f^{-1}}(\cF_{W}^{{*}}(\xi)).$
%}

%$\cF_{W}^{{*}}(\rho_f(\xi))=
% %\frac{1}
% {|f|^{2\dim W}}\rho_{f^{-1}}(\cF_{W}^{{*}}(\xi)).$



% }
\subsubsection{The wave front set} \label{sssec:wfs}
{As it was mentioned earlier, we will prove a stronger version of Theorem \ref{thm:main_a}, which has to
do with the wave front set. The  wave front set is an important invariant of a distribution $\xi$ on an analytic manifold $X$, which was introduced  in  \cite{Hor} in the Archimedean case and then adapted in \cite{Hef} to the non-Archimedean case.

The wave front  set is a closed subset of $T^*X$. We will denote it by $\WF(\xi)$.
%\RamiAE{The wave front set was originally introduced in the Archimedean case (see \cite{Hor}). It was
%adapted in \cite{Hef} to the non-Archimedean case.}
% In the non-Archimedean case\RamiM{,} the  wave front set was defined in %\cite{Hef}.
The definition of $\WF(\xi)$ will be recalled in Appendix \ref{app:WF}.
Here we list the properties of the wave front set that will be used in this paper.
 Most of them are\RamiAE{ adaptations of results from \cite{Hor}. Some are} proved in \cite{Hef} and \cite{Aiz}, the rest will be proved in Appendix \ref{app:WF}.

%\RamiQP{I need to change the introductory paragraph above.}

\begin{proposition} \label{prop:WF_prop}
$ $
Let $X$ be an analytic variety and $E$ a \RamiF{locally
constant sheaf} over it. Let $\xi \in \G(X,E)$. Then we have:
\begin{enumerate}
\item \label{prop:WF_prop:1} $P_{T^*(X)}(WF(\xi))=WF(\xi) \cap X =\Supp(\xi).$ \RamiE{Here we identify $X$ with the zero section inside $T^*{X}$ \RamiAM{and $P_{T^*(X)}:T^*{X}\to X$ is the projection}.}
\item \label{prop:WF_prop:2} $WF(\xi) \subset X$ if and only if $\xi$ \RamiE{is smooth}.
%is given by a locally constant section.
\item \label{prop:WF_prop:3} Let $U \subset X$ be an open set. Then $WF(\xi|_U) = WF(\xi) \cap T^*(U).$
\item  \label{prop:WF_prop:4} Let $\xi' \in \G(X,E)$ and $f,f' \subset C^\infty(X).$  Then
$$WF(f \xi+f' \xi') \subset WF(\xi) \cup
WF(\xi').$$
\item \label{it:G_act} Let $G$ be an analytic group acting on $X$ and $E$. Suppose $\xi$ is
$G$-{invariant.}
Then $$ WF(\xi) \subset \{(x,v) \in T^*X(F)|v(\g x)=0\}=\bigcup_{x \in X} CN_{Gx}^X,$$
where $\g$ is the Lie algebra of $G$.
% and $\g x$ is the tangent space to the $G$-orbit of $x$
\end{enumerate}

\end{proposition}
In order to formulate the rest of the properties we will need the following notions:
\begin{definition}\label{def:dir_and_inv_image}
Let $X$ be an analytic variety. Let $A \subset T^*(X).$
\begin{enumerate}
\item \label{def:dir_and_inv_image:1}
  \RamiCP{We say that $A$ is \RamiK{\emph{conic}} if it is stable with respect to the homothety action
of $F^\times$ on $T^* X$, given by $\rho_\lambda(x,v)=(x,\lambda v)$.}

\item \label{def:dir_and_inv_image:2}
If $p: Y \to X$ is an analytic map we define $p^*(A)\subset T^*(Y)$ by
$$p^*(A):=\{(y,v) \in T^*(Y)|\exists w \in (dp^*)^{-1}(v)\subset T^{*}_{p(y)}Y \text{ with } (p(y),w) \in A \}.$$
\item \label{def:dir_and_inv_image:3}
 If  $p:  X \to Y $ is an analytic map we define $p_*(A)\subset T^*(Y)$ by
 $$p_*(A):= \{(y,v) \in T^*(Y)|\exists x \in p^{-1}(y) \text{ with } (x,(d_xp)^*(v)) \in A \}.$$
%\item \label{def:dir_and_inv_image:4}
% \DrinK{I have moved the following notation to \S\ref{ss:notation_explicit}.}
%\RamiCPD{We define $S_\phi:=\phi_*(X)\subset T^*Y$ where $X\subset T^*X$ %is considered as the zero section.}

\end{enumerate}
\end{definition}
\RamiAE{
\begin{rem}\label{rem:simp_gem}
We can describe the procedures of direct and inverse images in terms of symplectic geometry.

Namely, let $\pi: M \to N$ be a map of manifolds.
It gives rise to a correspondence $\Lambda_\pi \subset T^*(M) \times T^*(N)$ by $\Lambda_\pi=\{((x,v),(y,w))|y=\pi(x), v=d\pi^*(w) \}$.

Now let $S$ and $T$ be
 a symplectic manifold and $\Lambda \subset S \times T$ be a  correspondence.
 For a subset $Z \subset S$, we set $\Lambda(Z)=\{y\in T|\exists x\in Z \text{ such that }(x,y)\in \Lambda\}$.

This gives the following alternative definition for direct and inverse images: \begin{itemize}
\item for a subset $Z \subset T^*(M)$, we have $\pi_*(Z)=\Lambda_\pi(Z)$.
\item for a subset $Z \subset T^*(N)$, we have $\pi^*(Z)=\Lambda^{-1}_\pi(Z)$. Here $\Lambda^{-1}_\pi$ is $\Lambda_\pi$ considered as a subset  of $T^*(N) \times T^*(M).$
\end{itemize}

\end{rem}
}
%\RamiQD{Do you think that we should put here the remark that says that $p_*$ %can be defined in the language of symplectic geometry, or  to leave it in %\S\ref{sec:ex_boun}?}
%
\begin{proposition} \label{prop:WF_prop2}
$ $
Let $X$ be an analytic variety. Then we have:
\begin{enumerate}
\item $WF(\xi)$ is conic.
\item \label{prop:WF_prop2:2} Let \RamiL{$E$ be a locally
constant sheaf over $X$,  let $\xi \in \G(X,E)$ and let}
$p: Y \to X$ be a analytic submersion. Then $WF(p^*(\xi)) \subset p^*(WF(\xi))$.
\item \label{prop:WF_prop2:3} Let $ {q}: X \to Y$ be an analytic map\RamiL{, let $E$ be a locally
constant sheaf over $Y$  and let $\xi \in \Sc(X,q^*\RamiM{(E)})$.}
\RamiM{Assume $q|_{\supp (\xi)}$ is proper.}
%submersion.
Then $WF( {q}_*(\xi)) \subset  {q}_*(WF(\xi))$.
\end{enumerate}
\end{proposition}
}
\RamiT{In \S\ref{sec:pf_futer} we will need the following more complicated properties of the wave front set:
\RamiAB{
\begin{notation}
Let $X$ be an analytic manifold.
For a closed conic set $\Gamma \subset T^*X$ we denote by $\G_\Gamma(X)$ the space of generalized functions whose wave front set is in $\Gamma$. We will consider this space equipped with its natural topology which we describe in Appendix \ref{app:WF}.  We will use similar notations for other types of generalized sections.
\end{notation}
}
%\RamiQP{
%The topology is defined in \cite[Definition 8.2.2.]{Hor} in the Archimedean case. I'll give a uniform definition in the Appendix.
%}


\begin{proposition} \label{prop:WF_prop3}
We have the following generalization of Proposition \ref{prop:WF_prop2} \eqref{prop:WF_prop2:2}.
Let $p: Y \to X$ be an analytic map of analytic manifolds, let
$$N_p= \{(x,v)\in T^{*}X| x = p(y)  \text{ and } d_{\RamiAB{y}}^* p(v)=0 \text{ for some } y\in Y \}.$$
Let $E$ be a locally
constant sheaf over $X$.
\RamiAB{
Let $\Gamma \subset T^{*}X$ be a conic closed subset such that $\Gamma \cap N_p \subset X$.
}

%Let  $$\G_p(X,E)=\{\xi \in \G_p(X,E)|WF(\xi) \cap N_p \subset X \}.$$
Then the  map $p^{*}:C^\infty(X,E)\to C^\infty(Y,p^*(E))$
 has a unique continuous extension
%\RamiQO{Here I need to clarify about the topology}
to a
map
$p^{*}:\G_\RamiAB{\Gamma}(X,E)\to \G(Y,p^*(E))$.
 Moreover
for any $\xi \in \G_\RamiAB{\Gamma}(X,E)$ we have
$WF(p^*(\xi)) \subset p^*(WF(\xi))$.
\end{proposition}
%\RamiQH{This is Theorem 2.8 of [Hef] and Theorem 8.2.4 in \cite{Hor}; I'll add it to the appendix.}
}
% \RamiAE{
% \begin{rem}\label{rem:iver_im}
% Here is an explicit procedure to compute $p^*(\xi)$: We may assume that $X$ is a vector space and $E$ is trivial. Let $f_1\in\Sc(X)$ and $f_2\in\Sc(X,D_X)$ \RamiQT{(in the Archimedean case we will take compactly supported functions)}  such that $f_1(0)=1$ and $\int f_2=1$. Let $\lambda_{i} \in F$ be a sequence that converges to infinity. Let $\xi_i:= \rho_{\lambda_{i}}(f_{1}) \cdot (\rho_{\lambda_{i}^{-1}}(f_{2}) * \xi)$, where $\rho_{\lambda_{i}}$ is the homothety  as defined in Notation \ref{not:F_prop}. Note that $\xi_i$ are Schwartz \RamiQT{(in the Archimedean case -- smooth and compactly supported)} functions. Now, $p^*(\xi)$ is the weak limit of $p^*(\xi_i)$.
% \end{rem}
% }
% -
\RamiAD{
\begin{definition}\label{def:cont_dep}
Let $\xi\in \G(X \times Y)$ be a generalized function on a product of analytic manifolds. We will say that $\xi$ depends continuously on $Y$ if
for any $f \in \Sc(X,D_X)$ the generalized function $\xi_f \in \G(Y) $ given by $\xi_f(g)=\xi(f \boxtimes g)$ is continuous.
%\RamiQS{In the Archimedean case we need additional conditions.}
%   the following are satisfid:
%   \begin{enumerate}
%   \item
%   \item
%   \end{enumerate}
In this case we \DrinI{define} $\xi|_{X \times \{y\}} \in  \G(X \times \{y\}) $   by  $\xi|_{X \times \{y\}}(f):=\xi_f(y)$.
\end{definition}

%\DrinK
{
\begin{rem}
In the situation of Definition~\ref{def:cont_dep} the generalized functions $\xi_y:=\xi|_{X \times \{y\}}$, $y\in Y$, form a \emph{continuous family} (i.e., the map $Y\to\G(X)$ defined by $y\mapsto\xi_y$ is continuous). Thus one  gets a bijection between generalized functions on $X \times Y$ depending continuously on $Y$ and continuous families of generalized functions on $X$ parametrized by~$Y$.
\end{rem}
}

\begin{proposition}   \label{prop:Treves}
Let $\xi\in \G(X \times Y)$ be a generalized function on a product of analytic manifolds. Assume that
{
\begin{equation} \label{e:wavefront_condit}
WF(\xi) \cap CN_{X \times \{y\}}^{X \times Y}\subset X \times Y
\end{equation}
}
and that $\xi$ depends continuously\footnote{In fact, \eqref{e:wavefront_condit} implies that $\xi$ depends continuously on $Y$ (cf. the discussion after Proposition 6.11 in \cite{tre}).
We will not need this implication.} on $Y$.
%$\xi$ depends continuously on $Y$ and \RamiAM{moreover \footnote{\RamiAM{The fact that $\xi$ depends %continuously on $Y$ is implied by the fact that
%$WF(\xi) \cap CN_{X \times \{y\}}^{X \times Y}\subset X \times Y$ (cf. the discussion after Proposition 6.11
%in \cite{tre}). We will not need this implication.}  }}
%$WF(\xi) \cap CN_{X \times \{y\}}^{X \times Y}\subset X \times Y$.
Then
$\xi|_{X \times \{y\}}=j^*(\xi)$ where $j:X \times \{y\}\hookrightarrow X \times Y$ is the embedding.
\end{proposition}
\RamiAM{
}
% \begin{proof}
% \RamiAE{This follows from Remark \ref{rem:iver_im}, when we  chose $f_1$ and $f_2$ to be compatible with the product $X \times Y$.
% }
% % \RamiQS{This is statement 3 of your question. It seems that we only need this statement. I have not found such a statement in Treves. I also do not understand how Melrose wants to deduce it from 1,2. However it follows from the construction in Treves. So we can just say this. Do you think that it is enough? If not, I can write  some proof, e.g. repeat the construction from Treves and then deduce the statement, or alternatively to give a proof similar to the current proof of Proposition \ref{prop:WF_prop5}. What do you prefer?     } \DrinJ{I answered this question in a separate e-mail.}
% \end{proof}
}
%\RamiQT{Do you think that \RamiQW{the proposition and/or the proof}  belongs %here or in the appendix?}

Combining Proposition~\ref{prop:WF_prop3}  and Proposition~\ref{prop:Treves} we get the following
\begin{cor}   \label{c:Treves}
In the situation of Proposition~\ref{prop:Treves} one has
$$WF(\xi|_{X \times \{y\}})\subset j^*(WF(\xi)).$$
\end{cor}

%\RamiQS{The rest of the subsection is probably obsolete}
\RamiAbs{below are leftovers which we will probably not need.
\RamiAA{
\begin{proposition} \label{prop:WF_prop4}
Let $\phi:X \to Y$ be a proper map of  analytic manifolds.
Let $Z \subset Y$ be a closed submanifold. Assume that $\phi$ is transversal to $Z$, i.e. for any $x\in X$ such that $\phi(x)\in Z$, we have $d_x(T_x(X))+T_{\phi(x)}Z=T_{\phi(x)}Y$. Let $W=\phi^{-1}(Z)$. Note that it is a submanifold.
Let $\xi\in \G(X)$ be such that $WF(\xi) \cap CN_{W}^X \subset X.$ Then
\begin{enumerate}
\item $WF(\phi_*(\xi))\cap CN_{W}^Y \subset Y.$
\item $\phi_*(\xi)|_Z=(\phi|_W) _*(\xi|_{W})$. Here the restriction means the inverse image with respect to the corresponding embedding
\end{enumerate}

The analogous result holds for generalized sections of locally
constant sheaves.
\end{proposition}
}

\RamiQO{I'll try to find a reference. If I don't find,  I'll write a proof.}
\RamiAB{
\begin{proposition} \label{prop:WF_prop5}
Let $p:Y\to X$ be a proper map of  analytic manifolds. Let $W$ be a finite dimensional  $F$-vector space. Let

$$p':=p\times Id_W:Y \times W\to X \times W \text{ and } p'':=p\times Id_{W^*}:Y \times W^*\to X \times W^*.$$
Let $\Gamma_1 \subset T^*(X \times W),\Gamma_2\subset T^*(X \times W^*)$. Assume that
$$\Gamma_1 \cap N_{p'} \subset X \times W \text{ and } \Gamma_2 \cap N_{p''} \subset X \times W^*.$$
Let $\xi \in \Sc_{\Gamma_1}^*(X\times W)$ be such that $WF(\cF_W(\xi)) \subset \Gamma_2$.
%Let $\xi \in \Sc_{p}^*(X\times W):=\G_{p}(X\times W,D_{X\times W})$.
Then $(p')^*(\cF^{*}_W(\xi))=\cF^{*}_W((p')^*(\xi))$.
\end{proposition}
}

\RamiAB{
\begin{proof}
Without loss of generality we may assume that $X$ is a vector space and the projection of the support of $\xi$ to $X$ is compact. Fix Haar measures on $X$ and $W$.

Let $\lambda \in  F$ such that $|\lambda| >1$. Let $\alpha\in C^\infty_c(X)$ and $\beta\in \Sc(W)$ be such that
$$\alpha(1)=\beta(1)=1,\quad \int \alpha=\int \beta=1$$
and $\cF(\beta)\in C^\infty_c(W^{*}).$
Let $$\alpha_{i}=\rho_{\lambda^{i}} \alpha, \quad\beta_{i}=\rho_{\lambda^{i}} \beta.$$ Let $\xi_i=  \beta_{i}\cdot (\beta_{-i}*\alpha_{-i}*\xi) \cdot (|\lambda|^{i\cdot \dim(X \times W)})$.
\RamiQP{I hope that the normalization $ (|\lambda|^{i\cdot \dim(X \times W)})$ is correct. I'll check it again.}
It is easy to see that $\xi_i$
 and $\cF_W(\xi_i)$ are smooth compactly supported. \RamiQP{In the Archimedean case it is Schwartz and not compactly supported.}

 Similarly to \cite[Theorem 8.2.3]{Hor}  it is also easy to see that $\xi_i \to \xi$ and   $\cF_W(\xi_i) \to \cF_W(\xi)$ in the topologies
on $C^{-\infty}_{\Gamma_1}(X\times W, D_{X\times W})$ and
$C^{-\infty}_{\Gamma_2}(X\times W^*,D_{X}\boxtimes \C_{W^*})$. This implies the assertion.
\end{proof}
}



\RamiQP{This proof belongs to the appendix, I put it here for the time being
}
\RamiQP{Currently the proof (and the formulation) is only valid for the non-Archimedean case, but it can be easily adapted to the Archimedean case
}
}

\section{WF-holonomic distributions}\label{sec:WF-Holonomic}
%\setcounter{lemma}{0}
\RamiQ{
$ $
It seems that we  need to weaken the notion of WF-holonomic to make it more appropriate to the analytic world. Although
 it should not matter for Theorem \ref{thm:main_b}, it is relevant for Theorem \ref{thm:main_c}.
Since we are proving Theorem \ref{thm:main_b} via Theorem \ref{thm:main_c}, it is  more convenient to use only one
 notion of WF-holonomic. The only disadvantage is  that we will prove regularity only  in an analytically open dense
 set and not a Zariski open  dense set. It should be fixed in the next version, when we give an explicit bound.}

  \DrinD{
 \subsection{Recollections on isotropic and Lagrangian conic subvarieties of $T^*(X)$}\label{sec:lag}
 Let $M$ be a symplectic algebraic manifold and $V\subset M$ a {constructible subset\footnote{{A  subset of an algebraic variety is said to be constructible if it is a finite union of locally closed subsets. A theorem of Chevalley says that the image of a constructible subset under a regular map is constructible. (A similar statement for preimages is obvious.)}}.}
 %locally closed algebraic subvariety.
 We say that $V$ is \emph{isotropic} (resp. \emph{Lagrangian}) if there is an open dense {subset  $V'\subset V$ which is a smooth isotropic (resp. Lagrangian) locally closed subvariety  in $M$}.
 %(In this case one can take $V'$ to be the set of all nonsingular points of $V$.)

%I could say that $V$ is isotropic (or Lagrangian) if and only if
%each of its irreducible component is?

 \begin{rem}    \label{r:easy_facts}
 The closure of an  isotropic (resp. Lagrangian) subset is isotropic (resp. Lagrangian).
 The union of two  isotropic (resp. Lagrangian) subset is isotropic (resp. Lagrangian).
 \end{rem}


\begin{proposition} \label{prop:sub_iso}
If $V\subset M$ is isotropic then so is any {constructible subset} $Z\subset V$.
\end{proposition}

The statement is nontrivial because $Z$ may be contained in the set of singular points of $V$.
For a proof, see, e.g., \cite[Proposition 1.3.30]{CG} and  \cite[\S 1.5.16]{CG}.

Now let $M=T^*(X)$, where $X$ is a smooth algebraic manifold. The multiplicative group acts
on $M$ by homotheties. A subvariety of $M$ is said to be \emph{conic} if it is stable with respect to this action. If $A\subset X$ is a smooth algebraic subvariety then the conormal bundle
$CN_A^X$ and its closure $\overline{CN_{A}^X}$ are conic Lagrangian subvarieties of $T^*(X)$.
It is well known that any closed conic Lagrangian subvariety of $T^*(X)$ is a finite union of
varieties of the form $\overline{CN_{A}^X}$. Here is a slightly more general statement.



\begin{lemma}    \label{lem:isotr_discrip}
Let $X$ be an algebraic manifold and $C \subset T^*(X)$ a closed conic algebraic subvariety.
Then the following properties of $C$ are equivalent:

\begin{enumerate}
\item $C$ is isotropic; \label{1}
\item $C$ is contained in a Lagrangian subvariety
of $T^*(X)$; \label{2}
\item  There is a finite collection of smooth locally closed subvarieties $A_i \subset X$ such that
$$C \subset \bigcup_i \overline{CN_{A_i}^X}\,;$$ \label{3}
\item There is a finite collection of smooth locally closed subvarieties $A_i \subset X$ such that
$$C \subset \bigcup_i {CN_{A_i}^X}\,.$$ \label{4}
\end{enumerate}
\end{lemma}
This lemma is standard. For completeness, we include its proof in
\RamiAF{Appendix \ref{app:symp_geom}.} %\RamiQU{It will be Appendix B.}
%Appendix \ref{subapp:WFHol}.

%\begin{rem}
%Property (4) from Lemma~\ref{lem:isotr_discrip} implies the existence of a Zariski open dense
%$U\subset X$
%such that $C\cap T^*(U)$ is contained in the zero section of $T^*(U)$.
%\end{rem}

\medskip

Now let $S\subset T^*X(F)$ be \emph{any} conic subset (not necessarily an algebraic subvariety). Its Zariski closure $\bar S\subset T^*X$ is also conic.

\begin{lemma}   \label{l:closure}
$\bar S$ has the equivalent properties from Lemma~\ref{lem:isotr_discrip} if and only if
there is a finite collection of smooth locally closed subvarieties $A_i \subset X$ such that
$S \subset \bigcup\limits_i {CN_{A_i}^X (F)}\,$.
\end{lemma}

\begin{proof}
If $\bar S \subset \bigcup\limits_i  CN_{A_i}^X$\RamiM{,} then
$S \subset \bigcup\limits_i {CN_{A_i}^X (F)}\,$. If
$S \subset \bigcup\limits_i {CN_{A_i}^X (F)}\,$\RamiM{,} then
$\bar S \subset \bigcup\limits_i  \overline{CN_{A_i}^X}$.
\end{proof}
}


{The following lemma is well known (see Appendix \ref{app:symp_geom} for a proof).}
%\RamiQAB{I recall that you saed that it is in \cite{CG}. S}
\RamiAF{
\begin{lemma}\label{lem:iso_dir_invs_im}
Let $p:X\to Y$ be a morphism of algebraic manifolds. {Let $T\subset T^*X$ and $S\subset T^*Y$ be constructible subsets.}
\begin{enumerate}
\item If $T$ is isotropic then $p_*(T)$ is.
\item If $S$ is isotropic then $p^*(S)$ is.
\end{enumerate}
\end{lemma}
}
{For the definition of $p_*$ and $p^*$, see Definition~\ref{def:dir_and_inv_image} and Remark~\ref{rem:simp_gem}. Note that since $T$ and $S$ are constructible so are  $p_*(T)$ and $p^*(S)$.}
%For a proof of the lemma, see Appendix \ref{app:symp_geom}.}


% \RamiAE{\begin{rem} \label{rem:symplectic_rem}
% Note that we also can chos the identification $(v,w)\mapsto (-w,v)$.
% \end{rem}
% }

%\RamiT{
% \begin{rem} \label{rem:symplectic_rem}
% Since both $CN_{E^\bot}^{X \times V^*}$ and $CN_E^{X \times V}$ are stable with respect to homotheties, both in $V$ and $V^*$, we can replace the symplectic identification $V \times V^*{\buildrel{\sim}\over{\longrightarrow}} V^* \times V$ with the usual flip. \DrinJ{I think we came to the conclusion that this is false, didn't we?}
% \end{rem}
% }
% \RamiQH{We can erase the minus sign in the formulation of the Lemma and incorporate the remark into the proof. Do you think it will be better?
%  The reason I want this remark (in that way or the other) is that otherwise the formulations in \S\ref{sec:ex_boun} become even more messy than they are now.
% }
 %\RamiQD{I suggest putting the proof in the appendix. Maybe we should make ``appendix B -- symplectic geometry of the co-tangent bundle" and put
 %there the content of appendix \ref{subapp:WFHol} and this proof.}
 \subsection{WF-holonomic distributions}


\newWFHol{
\begin{definition}
Let $X$ be an \RamiJ{algebraic manifold over $F$} \DrinD{and let $E$ be a locally constant sheaf on $X(F)$}.
 A distribution $\xi \in \Sc^*(X\RamiJ{(F)},E)$ is said to be \emph{\RamiJ{algebraically} WF-holonomic} if \RamiAF{the \RamiAH{Zariski} closure of $WF(\xi)$ is isotropic.} %{there exists a finite collection of locally closed smooth algebraic subvarieties %$A_i \subset X$ such
% that $WF(\xi)\subset \bigcup\limits_i {CN_{A_i}^X\RamiJ{(F)}}.$
%We generalize this notion to generalized sections of locally constant
%\RamiE{sheaves} \RamiJ{over $X(F)$} in the natural way.}
\end{definition}
\begin{remark}
%\DrinK
{
By Lemmas~\ref{lem:isotr_discrip} and \ref{l:closure}, $\xi$ is algebraically WF-holonomic if and only if
$WF(\xi) \subset \bigcup\limits_i {CN_{A_i}^X (F)}$ for some smooth locally closed subvarieties
$A_1,\ldots, A_n\subset X$.
}
%\RamiAM{Lemma~\ref{lem:isotr_discrip} gives several equivalent definitions for this notion.}
%By Lemma~\ref{l:closure},
%%%%\ this means that the Zariski closure of $WF(\xi)$
%\RamiAM{it is enough to require that $WF(\xi) \subset \bigcup\limits_i {CN_{A_i}^X (F)}$ for some smooth
%locally closed subvarieties $A_1,\ldots, A_n\subset X$.}
\end{remark}
%has the
%equivalent properties from Lemma~\ref{lem:isotr_discrip}.
%   \DrinK{To say this, we don't need Lemma~\ref{l:closure}!  Probably we wanted to say the following: By Lemma~\ref{l:closure}, this means that
% $WF(\xi) \subset \bigcup\limits_i {CN_{A_i}^X (F)}$ for some smooth locally closed subvarieties
% $A_1,\ldots,A_n\subset X$.}

% $\bar S$ has the equivalent properties from Lemma~\ref{lem:isotr_discrip} if and only if
% there is a finite collection of smooth locally closed subvarieties $A_i \subset X$ such that
% $S \subset \bigcup\limits_i {CN_{A_i}^X (F)}\,$.


\RamiJ{
\begin{remark}
One can also define a more general notion of ``analytically WF-holonomic distribution'' for analytic manifolds. However, we
will not discuss it in this paper. So we will use the expression ``WF-holonomic'' as a shorthand for
 ``algebraically WF-holonomic''.
\end{remark}
}


%\RamiQ{This property is one of the equivalent properties that was introduced in the previous draft. It seems that this is the only one we use. It might still be interesting to have the discussion of the equivalent properties. Its translation to the analytic case seems to be easy but not automatic, since the theorem in Chris and Ginzburg is not formulated (there) for the analytic case. Therefore I propose to postpone it until the next version. Is it OK?}
}
\begin{remark}
{In general, the notion of WF-holonomicity is not as powerful as the notion  of holonomicity given by the theory
 of D-modules.
 For example, it is not true that the Fourier transform of a  WF-holonomic distribution on an affine space is WF-holonomic. Yet if the variety $X$ is compact\RamiM{,} then the notion of WF-holonomicity
 seems to be a good candidate for  replacing the notion of holonomicity in the non-Archimedean
case.
}
\RamiQ{Is this remark appropriate? }
\end{remark}

The next lemma {follows immediately from \DrinD{statements (2) and (3) of } Proposition~\ref{prop:WF_prop}}.
\begin{lemma}\label{lem:hol_reg}
Let $X$
be an \RamiJ{algebraic $F$-manifold} and $E$ a \RamiF{locally
constant sheaf} over \RamiJ{$X(F)$}.
%A WF-holonomic distribution is  {smooth} on an open dense set. More precisely, if
If $\xi \in \G(X (F),E)$ is WF-holonomic
%generalized section of a \RamiE{sheaf} $E$ on \RamiJ{$X(F)$, where $X$ is algebraic manifold.}
then there exists \RamiJ{a Zariski} open dense subset $U \subset X$
such that $\xi|_{\RamiJ{U(F)}}$
\RamiE{is smooth}.
%is given by a locally constant section.
\end{lemma}

\RamiAF{The fact that inverse and direct images preserve \RamiAH{isotropicity} (Lemma \ref{lem:iso_dir_invs_im}) and the properties of the wave front set (Propositions \ref{prop:WF_prop2} and \ref{prop:WF_prop3}) imply the following proposition:}

\RamiL{
\begin{proposition} \label{prop:WF_Hol}
$ $
Let $X$
be an \RamiJ{algebraic $F$-manifold}.
\begin{enumerate}
\item Let $E$ be a locally
constant sheaf over $X(F)$,  let $\xi \in \G(X,E)$ be a WF-holonomic generalized section and let
$p: Y \to X$ be \RamiAF{a morphism.
Assume that $\WF(\xi) \cap N_p \subset X$. }
%an algebraic submersion.
Then $p^*(\xi)$ is WF-holonomic.
\item Let $ {q}: X \to Y$ be a \RamiR{regular} map, let $E$ be a locally
constant sheaf over $Y(F)$  and let $\xi \in \Sc(X,q^*\RamiM{(E)})$
 be a WF-holonomic distribution.
Assume that the map $\supp(\xi)\to Y(F)$ induced by $q$ %$q|_{\supp(\xi)}$
is proper (as a continuous map).
%submersion.
Then $ {q}_*(\xi)$
is WF-holonomic.
\end{enumerate}
\end{proposition}
}



%In Appendix \ref{subapp:WFHol} we will deduce this proposition from the %properties of the wave front set
% (see Proposition \ref{prop:WF_prop2}).

\RamiG{
We will also use the following corollary of Proposition \ref{prop:WF_prop} \RamiAH{\eqref{it:G_act}}%\DrinD{(5)}.
\begin{corollary} \label{cor:WF_Hol}
$ $
Let $X$ be an \RamiJ{algebraic manifold} and $E$ a \RamiF{locally
constant sheaf} over \RamiJ{$X(F)$}. Let an \RamiJ{algebraic} group $G$ act on $X$ and \RamiJ{let $G(F)$ act on}
 $E$. Let $U \subset X$ be a
$G$-stable
open set and $Z=X-U$. Let $\xi \in \G(X\RamiJ{(F)},E)^{\RamiJ{G(F)}}$. Suppose $Z$ has a finite number of
$G$-orbits and $\xi|_{\RamiJ{U(F)}}$ is smooth. Then $\xi$ is WF-holonomic.
\end{corollary}
By twisting the action of $G$ on $E$ by a quasi-character, we obtain the following \RamiM{version} of
Corollary~\ref{cor:WF_Hol}.
\begin{corollary} \label{cor:WF_Hol2}
Let $X,G,E,Z,U$ be as in Corollary \ref{cor:WF_Hol}.
% Let $\chi$ be a character of $G\RamiJ{(F)}$ and
\RamiJ{Let
$\xi \in \G(X\RamiJ{(F)},E)^{G\RamiJ{(F)}}$}. \RamiM{Suppose $\xi|_{\RamiJ{U(F)}}$} is smooth and the line $\C\xi\subset\G(X\RamiJ{(F)},E)$
is $G\RamiJ{(F)}$-stable (i.e., $\xi \in \G(X\RamiJ{(F)},E)^{G\RamiJ{(F)},\chi}$ for some quasi-character
$\chi :G\RamiJ{(F)}\to\C^{\times}$\RamiJ{)}. Then
$\xi$ is WF-holonomic.
\end{corollary}
}
%\subsection{ {Algebraically WF-holonomic distributions}}
% {
%For algebraic manifolds we have a slightly stronger notion of WF-holonomicity. Namely we have the following definition

%\begin{definition}
%Let $X$ be an algebraic variety. Let $E$ be a locally constant sheaf over $X(F)$.
%We call a distribution $\xi \in \Sc^*(X(F),E)$ algebraically WF-holonomic if
%there exists a finite collection of (locally closed) smooth algebraic subvarieties $A_i \subset X$ such that
% $WF(\xi)\subset \bigcup_i {CN_{A_i}^X}(F).$
%\end{definition}

%\begin{remark}
% The natural analogs of the results of this section hold for  algebraically WF-holonomic with analogous proofs.
%\end{remark}
%}


%\section{Main results}\label{sec:main}
%
%\setcounter{lemma}{0}
%
%%In this paper we prove a stronger version of Theorem \ref{thm:main_a}.
%%In order to formulate it, we will need some preparation.
%
%%\begin{notation}\label{not:m}
%%Let $W$ be a finite dimensional $F$-vector space and $X \subset W$ be a closed algebraic
%subvariety. Let $p:\tilde{X} \to X$ be a resolution of singularities of $X$. Let $X_0 \subset X$ be %an open dense regular subvariety and let $\omega_0$ be a \RamiQF{(erazed algebraic)} \RamiE{(regular)} top
%differential form on it. Let $\tilde{\omega}_0:=p^{*}(\omega_{0})$ be its pullback to
%$p^{-1}(X_0)$, and suppose there exists a regular top differential form $\omega$ on $\tilde{X}$ %such that $\omega|_{p^{-1}(X_0)}=\tilde{\omega}_0$. Consider $|\omega|\in \Sc^*(\tilde{X}(F)).$

%%We denote $m_{p,\tilde{X},X,\omega}=p_*(|\omega|)\in \Sc^*(W)$.
%%%$m_{X_{0},\omega_{0}}=p_*(|\omega|)\in \Sc^*(W).$
%%\end{notation}
%%\begin{lemma}\label{lem:indep}
%%Let $W,X ,p, \tilde{X}, X_0, \omega_0,\omega$ be as in %the above notation (
%%Notation \ref{not:m}. Let $p':\tilde{X}' \to X$ be another resolution of singularities of $X$. Let
%%$\tilde{\omega}'_0:=p'^{*}(\omega_{0}).$

%%Then
%%\begin{enumerate}
%%\item There exists a regular top differential form $\omega'$ on $\tilde{X'}$ such that $\omega'|_{p'^{-1}(X_0)}=\tilde{\omega}'_0$.
%%\item $m_{p,\tilde{X},X,\omega}=m_{p',\tilde{X}',X',\omega}$.
%%\end{enumerate}
%%\end{lemma}
%%\RamiE{
%%\begin{proof}
%%(1) is a standard fact from algebraic geometry. One can easily deduce it from \RamiE{Proposition} 1 on page 210 of
%%\cite{Mum}, see also Theorem 8.19 of \cite{Har} for a similar argument.
%%\RamiQB{
%%I've looked in \cite{Mum}, \cite{Har} and Shafarevich -- ``basic algebraic geometry''. I did not find the
%% exact statement we need. In \cite{Har} and Shafarevich (page 207), they discuss this statement, but only for
%%projective varieties. The proof seems to be the same but the statement is different. In \cite{Mum}, he proves a statement that immediately implies that a proper birational equivalence is an isomorphism outside co-dim 2, but it seems that
%%he does not
%%deduce the statement that we need.
%%
%%Can I leave it the way it is now? Or do you prefer I deduce the statement that we need from \cite{Mum}?}
%%
%%(2) follows immediately from (1).
%%%rad book mamford
%%\end{proof}
%%}

%%%For the proof we will need the following standard lemma.
%%%\begin{lemma}\label{lem:pul_back_reg}
%%%Let $\pi:X \to Y$ be a proper birational equivalence of algebraic manifolds.
%%%Let $f$ be a rational function on $Y$ such that $\pi^{*}(f)=f \circ \pi$ is regular. Then $f$ is regular.
%%%\end{lemma}
%%%\begin{proof}
%%%Since $\pi$ is proper, the ring of regular functions $\bar F[X]$ is finite over $\bar F[Y]$. Also,   $\pi$ is  a birational equivalence, so the field of rational functions $\bar F(X)$ coincides with $\bar F(Y)$. Thus $\bar F[Y] \subset \bar F[X] \subset \bar F(Y)$. Since $X$ is smooth, it is also normal. Thus $\bar F[Y] = \bar F[X] $.
%%%\end{proof}
%%%\begin{proof}[Proof of Lemma \ref{lem:indep} ]
%%%Let $Y=\tilde{X} \times_X \tilde{X}'$ and let $\tilde{Y}$ be a resolution of singularities of $Y.$ Let $q:\tilde{Y} \to \tilde{X}'$ be the neutral map. Consider $\tilde{\omega}'_0$ as a rational top differential form on $X'$. Clearly, $q^*(\tilde{\omega}'_0)$ is regular, hence so is $\tilde{\omega}'_0$. This means that  there exists a regular top differential form $\omega'$ on $\tilde{X'}$ such that $\omega'|_{p'^{-1}(X_0)}=\tilde{\omega}'_0$. The fact that  $m_{p,\tilde{X},X,\omega}=m_{p \circ q,\tilde{Y},X,q^*(\omega')}=m_{p',\tilde{X}',X',\omega}$ is obvious.
%%%\end{proof}

%%\begin{notation}\label{not:mu}
%%In view of  Lemma \ref{lem:indep}, we denote
%%$\mu_{X_{0},\omega_{0}}:= m_{p,\tilde{X},X,\omega}.$
%%\end{notation}



%The following theorem implies Theorem \ref{thm:main_a}
%\DrinD{by virtue of} Lemma \ref{lem:hol_reg}.

% {
%\begin{introtheorem} \label{thm:main_b}
%Let $W$ be a finite-dimensional $F$-vector space and $X$ an algebraic manifold. Let $\phi: X \to W$ be a
%proper map and let $\omega$ be an algebraic top differential form on $X$.
%%consider $|\omega|$ as a masure on $X$
%Then the Fourier transform $\cF^*(\phi_*(|\omega|) \in \G(W^*)$ is
%% {algebraicly}
% {algebraically WF-holonomic}.
%\end{introtheorem}
%}

%%\begin{introtheorem} \label{thm:main_b}
%%Let $W$ be a finite-dimensional $F$-vector space and $X_{} \subset W$ be a (locally closed) smooth algebraic subvariety
%% and let $\omega$ be an algebraic top differential form on it.


%%Then the Fourier transform $\cF^*(\mu_{X_{},\omega_{}}) \in \G(W^*)$ is WF-holonomic.

%%\end{introtheorem}

% {
%We will prove the following more general, relative version of this theorem:
%\begin{introtheorem} \label{thm:main_bb}
%Let $W$ be a finite-dimensional $F$-vector space and $X,Y$ be algebraic manifolds. Let
%$\phi: X \to Y \times W$ be a proper map
% and let $\omega$ be an algebraic top differential form on $X$.
%%consider $|\omega|$ as a masure on $X$
%Then the partial Fourier transform
%$$\cF_W^*(\phi_*(|\omega|) \in \DrinD{\Sc^*( {Y(F)} \times W^*,D_{Y(F)}^{Y(F)\times W^*})=\G( {Y(F)}\times W^*,D^{Y(F)\times W^*}_{W^*})}$$  is
%% {algebraicly}
% {algebraically WF-holonomic}.
%\end{introtheorem}
%}
\section{ Proof of \DrinI{Theorems \ref{thm:main_a}-\ref{thm:main_bb}}}\label{sec:pf_main_bb}
%Theorem \ref{thm:main_bb}}}

%\RamiQD{I suggest to leave this section as it is (after changing the name and the first sentence). Although it
%follows from the next one, I think that it is much easier to read, most of the readers will not be interested in
%reading the next one, and even for those who are interested in the next section, this section will make the
%understanding of the next one easier. }
\DrinD{In this section we prove the theorems formulated in \S\ref{ss:main}.

%The following theorem implies Theorem \ref{thm:main_a}.

Using the notion of \RamiJ{WF-holonomic} distribution from \S\ref{sec:WF-Holonomic}, one can reformulate Theorem~\ref{thm:main_bb} as follows.

\begin{theorem}   \label{t:C}
Let $W$ be a finite-dimensional $F$-vector space and $X,Y$ be algebraic manifolds. Let
$\phi: X \to Y \times W$ be a proper map
 and let $\omega$ be a \RamiR{regular} top differential form on $X$.
Then the partial  Fourier transform%\FT
\footnote{{Partial Fourier transform was introduced in
Definition \ref{def:FT}. The symbols $\Sc^*$ and $\G$ were introduced in \S\S\S\ref{sssec:fun_sp}. For the symbols
$D_{Y(F)}^{Y(F)\times W^*}$ and $D^{Y(F)\times W^*}_{W^*}$, see  \S\S\S\ref{sssec:op_dist}.}}
%Fourier transform
$$\cF_W^*(\phi_*(|\omega|)) \in \Sc^*( {Y(F)} \times W^*,D_{Y(F)}^{Y(F)\times W^*})=\G( {Y(F)}\times
 W^*,D^{Y(F)\times W^*}_{W^*})$$  is
WF-holonomic.

%\DrinK{In the footnote please replace question marks by appropriate references.}

%\RamiQV{
%I put reference only to the definition of the Fourier transform. The other notations  (like  $D_{Y(F)}^{Y(F)\times
%W^*}$) are defined slightly before it and are used already there. Do you want me to put references to those
%notations in Definition \ref{def:FT}?
%}

% \RamiQV{
% Do you want the link to be hidden (black) or visible (blue)?
% } \RamiQW{Do you also want that the symbol ``$\cF$" be a link? Hidden  or visible?}
% \DrinK{I am skeptical about using such high-tech devices. E.g., many readers prefer to print out the files and to read the print-outs.}
%
% \RamiQV{
% The expresson $\in \Sc^*( {Y(F)} \times W^*,D_{Y(F)}^{Y(F)\times W^*})=\G( {Y(F)}\times
%  W^*,D^{Y(F)\times W^*}_{W^*})$ is not necessary. If you think it does not contribute to clarity we can erase it.
%  \DrinK{Keep it.}
%  }
\end{theorem}

Theorem~\ref{thm:main_b} is a particular case of Theorem~\ref{t:C} when
$Y$ is a point. Theorem \ref{thm:main_a} follows from Theorem~\ref{thm:main_b}
by virtue of Lemma \ref{lem:hol_reg}. Thus it remains to prove Theorem~\ref{t:C}.
}



\subsection{Reduction to the key special case}  \label{ss:Reduction to key}
$ $
\RamiAM{
\begin{notation}
For a vector space $W,$ we denote by $\overline W$ the projective space of one dimensional subspaces of $W \oplus F$. We consider $W$ as an open subset of $\overline W$.
\end{notation}}
\RamiR{Using Hironaka's \DrinG{theorem and Nagata's theorem}, we}
 will deduce Theorem \ref{t:C} from \DrinG{Proposition~\ref{thm:main_c}, which is, in fact, a special case of  Theorem \ref{t:C}. To formulate Proposition~\ref{thm:main_c}, we need some notation.}

% (old vertion of prop 4.1.1)\begin{proposition} \label{thm:main_c}
% (old vertion of prop 4.1.1)Theorem \ref{t:C} holds under the following additional assumptions:
% (old vertion of prop 4.1.1)\begin{itemize}
% (old vertion of prop 4.1.1) \item The composition \DrinG{$X\overset{\phi}\longrightarrow Y\times W\overset{}\longrightarrow Y$
% (old vertion of prop 4.1.1) is an open embedding $i:X\hookrightarrow  Y$.}
% (old vertion of prop 4.1.1)% $i:=pr_Y \circ \phi$ is an open embedding, where $pr_Y:Y \times W \to Y$ is the projection.
% (old vertion of prop 4.1.1) \item There exists a map $\tilde \phi:Y \to \overline W$ such that \DrinG{the diagram
% (old vertion of prop 4.1.1) $$
% (old vertion of prop 4.1.1)\xymatrix
% (old vertion of prop 4.1.1){        X  \ar@{->}^{i}[r]   \ar@{->}_{\phi_W}[d]        & Y \ar@{->}^{\tilde \phi}[d] \\
% (old vertion of prop 4.1.1)         W \    \ar@{->}^{}[r]    &  \overline W  \\
% (old vertion of prop 4.1.1)}
% (old vertion of prop 4.1.1)$$
% (old vertion of prop 4.1.1)commutes (in this diagram  $\phi_W$ is the composition $X\overset{\phi}\longrightarrow Y\times W\to W$).}
% (old vertion of prop 4.1.1) % $pr_W \circ \phi= \tilde \phi \circ i$,where $pr_W:Y \times W \to W$ is the projection.
% (old vertion of prop 4.1.1)\item $ \overline{i(Z)} \cup Y_\infty$ is an SNC divisor, where
% (old vertion of prop 4.1.1)$Z \subset X$ is the zero locus of $\omega$ and $Y_\infty:=Y-\DrinG{i(X)}=\tilde \phi^{-1}(W_\infty)$.
% (old vertion of prop 4.1.1)\end{itemize}
% (old vertion of prop 4.1.1)\end{proposition}
% (old vertion of prop 4.1.1)\DrinG{Are we saying here that the equality $Y-\DrinG{i(X)}=\tilde \phi^{-1}(W_\infty)$ \emph{obviously} holds or do
% (old vertion of prop 4.1.1)we \emph{require} it to hold? Probably it holds automatically for the following reason: the proper map
% (old vertion of prop 4.1.1)$\phi :X\to Y\times W$ factors as $X\hookrightarrow \tilde \phi^{-1}(W)\to Y\times W$, so $X$ is closed in
% (old vertion of prop 4.1.1)$ \phi^{-1}(W)$, so $ \phi^{-1}(W)=X$ and therefore $\phi^{-1}(W_\infty)=Y-\DrinG{i(X)}$.
% (old vertion of prop 4.1.1)But this argument cannot be considered as quite obvious.
% (old vertion of prop 4.1.1)\RamiQE{I prefer to require it. As you say it is almost automatic (it is if $i(X)$ dense in $Y$ which we can always assume), but
% (old vertion of prop 4.1.1)I do not think that we want to go into those details. If we decide to keep this proposition I'll rewrite it in a
% (old vertion of prop 4.1.1) way that it is clear that we require it.}
% (old vertion of prop 4.1.1)Do we really need the equality
% (old vertion of prop 4.1.1)$Y-\DrinG{i(X)}=\tilde \phi^{-1}(W_\infty)$?}
% (old vertion of prop 4.1.1)\RamiQE{It depends what do you mean by ``really''. The proposition holds without this assumption since it is a special case of the
% (old vertion of prop 4.1.1)main theorem (and since it is almost automatic). However I think this makes the proposition more convenient to prove and to use}
% (old vertion of prop 4.1.1)\DrinG{Is $\tilde \phi$ a good notation for the map $Y\to \overline W$? After all, $ \tilde \phi$ is not quite a
% (old vertion of prop 4.1.1)``variant" of $\phi$.}
% (old vertion of prop 4.1.1)\RamiQE{$\tilde \phi$ encodes all the information about $\phi$. Would you prefer the notation $\Phi$?}
% (old vertion of prop 4.1.1)
% (old vertion of prop 4.1.1)\DrinG{Is it possible to replace the horizontal arrows of the diagram by the symbol ``$\hookrightarrow$" (to make it clear that they are embeddings)?} \RamiQE{I'll take care of this}


%its special case. For its formulation we will need the following notation:


\DrinB{
\begin{notation}\label{not:eta}
Let $Y$ be an  algebraic
%analytic
manifold and $W$ a vector space. Let $\phi:Y\to \overline{W}$ be an
 {algebraic}
%analytic
map.
 We set $Y_0 := \phi^{-1}(W)$  and $Y_{\infty}:=\phi^{-1}(W_\infty)$. Assume $Y_0$  is
dense {in $Y$}. Let $\omega$ be a  {rational}
%meromorphic
top differential form on $Y$ which is
  {regular}
%analytic
 on $Y_0$ and let
 $\omega_0:=\omega|_{Y_0}$.  Define $i: Y_0 \hookrightarrow Y\times W$ by $i(y):=(y,\phi (y))$ and
set
 {$$\eta_{\phi,\omega}:=\DrinD{i}_*(|\omega_0|)\in\Sc^*(Y(F)\times W)\, .$$
%$$\eta_{\phi,\omega}:=i_*(|\omega_0|)\in\Sc^*(Y\times W)\, .$$
We also set $$\hat{\eta}_{\phi,\omega}:= \cF^{*}_{W}(\eta_{\phi,\omega})\in
\DrinD{\Sc^*(Y(F) \times W^*,D_{Y(F)}^{Y(F)\times W^*})=
\G(Y\RamiM{(F)} \times W^*,D^{Y(F)\times W^*}_{W^*})}.$$}
\end{notation}
%In the current subsection we deduce Theorem \ref{thm:main_bb} from prp
%In order to study this proposition, let us introduce the following notation.

\begin{rem}\label{rem:eta}
$i_*(|\omega_0|)$ is a well-defined measure because the embedding $i$ is closed; to see this,
represent $i: Y_0 \hookrightarrow Y\times W$ as the composition
\DrinC{$$Y_0 {\buildrel{\sim}\over{\longleftarrow}}\Gamma_{\phi}\cap (Y_0\times W)=
\Gamma_{\phi}\cap (Y\times W)\hookrightarrow Y\times W,$$}
%$Y_0 {\buildrel{\sim}\over{\longleftarrow}}\Gamma_{\phi}\cap (Y_0\times W)\hookrightarrow Y\times W$,
where $\Gamma_{\phi}\subset Y\times\overline{W}$ is the graph of $\phi$.
\end{rem}
}


%\begin{notation}\label{not:eta}
%Let $Y$ be an analytic manifold and $W$ be a vector space. Let $\phi:Y\to \overline{W}$ be an analytic map.
% We set $Y_0 := \phi^{-1}(W)$  and $Y_{\infty}:=\phi^{-1}(W_\infty)$. Assume $Y_0$  is
%dense {in Y}. Let $\omega$ be a meromorphic top differential form on $Y$ which is analytic on $Y_0$ and let
%$\omega_0:=\omega|_{Y_0}$. Consider the closed imbedding $i: Y_0 \to Y\times W$. We set $$\eta_{\phi,\omega}:=i_*(|\omega_0|)\in\Sc^*(Y\times W)$$
%\end{notation}
% Proposition \ref{thm:main_c} can be reformulated in the following way:
\RamiR{
Now we can formulate the key special case of Theorem \ref{t:C}:}
\begin{proposition} \label{thm:main_c}
Let $Y, W, \phi,\omega, Y_\infty,Y_0$ be as in Notation \ref{not:eta}. Let $Z \subset Y$ be the zero locus of
$\omega$. \RamiM{Assume  $Z \cup  Y_\infty$} is an SNC \RamiP{divisor.}
Then the partial Fourier transform
$\RamiM{\hat \eta_{\phi,\omega}}$
%\in \RamiF{\Sc^*}(Y\times W^*,D^{Y\times W^*}_Y )\DrinD{=\G(Y\times W^*,D^{Y\times W^*}_{W^*})}$
is \RamiJ{WF-holonomic}.
\end{proposition}

We will prove this proposition in section \ref{sec:pf_C}.
\RamiR{
\begin{remark}
Note that Proposition \ref{thm:main_c} is indeed a special case of Theorem \ref{t:C}. Namely, if we substitute $X,Y,\phi:X \to Y \times W$ and $\omega$
 from Theorem \ref{t:C} by $Y_0,Y,i:Y_0 \to Y \times W$ and $\omega_0$ from Notation \ref{not:eta},
 then we obtain the assertion of Proposition \ref{thm:main_c}.
\DrinG{Substituting $\phi$ for $i$} is possible because, as mentioned in Remark \ref{rem:eta}, the map $i$ is a closed embedding and hence proper.
\end{remark}
}

%\begin{proposition} \label{prop:C_imp_B}
%Theorem C implies theorem B.
%\end{proposition}

\RamiF{In some cases, one can describe
$ {\hat \eta_{\phi,\omega}}$
%$\cF^{*}_{W}(\eta_{\phi,\omega})$
explicitly. Namely, we have the following
straightforward calculation:}
\begin{lemma} \label{lem:exp_comp}
%\RamiQH{(I need to check in the Archimedean case)}
Let $(Y,W,\phi,\omega)$ be as above. Suppose $\Im \phi \subset W$.
Define  $f_\phi \in C^{\infty}(Y\RamiM{(F)}\times W^*)$ by $f_{\phi}(y,\xi):=\psi(\langle \xi,\phi(y) \rangle).$

Then the generalized section $ {\hat \eta_{\phi,\omega}}$ \DrinD{is equal}
 {to} the continuous section
%is given by the smooth section
$f_{\phi} \cdot  {pr_{\RamiS{W}}}^{*}(|\omega|)$, where $pr_{\RamiS{W}}:Y\times W \to Y$ is the projection.
\end{lemma}
\DrinC{
\begin{rem} \label{rem:exp_comp}
In fact, the formula
\begin{equation}   \label{e:always_true'}
 {\hat \eta_{\phi,\omega}}=f_{\phi} \cdot  {pr}^{*}(|\omega|)
\end{equation}
holds without assuming that $\Im \phi \subset W$ if the r.h.s. of \eqref{e:always_true'} is understood appropriately.
 More precisely, Definition~\ref{def:FT} and the definition of $\eta_{\phi,\omega}$
(see Notation~\ref{not:eta})
immediately imply that the scalar product of $ {\hat \eta_{\phi,\omega}}$ with any
$h\in  \Sc (Y\DrinD{(F)}\times W^*,D^{Y\RamiM{(F)}\times W^*}_{Y\RamiM{(F)}} )$ equals
the
\RamiR{iterated}
%double
integral $$\int\limits_{Y_0}\int\limits_\RamiW{W^*} hf_{\phi} \cdot  {pr}^{*}(|\omega|)$$
(the latter makes sense because after integrating along $\RamiW{W^*}$\RamiM{,} one gets a measure on $Y_0\RamiM{(F)}$
\emph{with compact support}).
\end{rem}
}


Now let us prove Theorem  {\ref{t:C}} using   {Proposition} \ref{thm:main_c}.

%\subsubsection{Proof of Theorem  {\ref{t:C}}}
 {
\begin{proof}[Proof of Theorem  {\ref{t:C}}]
% Embed $W$ in its projectivization $\overline{W}.$
%Let $\overline {X}$ and  $\overline {\phi}:X \to Y \times \overline W$  %be as in
% Corollary \ref{cor:Nag}
%(here we use Corollary \ref{cor:Nag} with $Y \times  W$ instead of $Y$ and %$Y \times \overline W$ instead of $\overline Y$)
% and identify  ${X}$ with its image in $\overline {X}$.

% \DrinG{
% The first sentence is confusing. What if we remove Corollary \ref{cor:Nag} %altogether and just say: }

  \DrinG{Applying Nagata's Theorem~\ref{thm:Nag} to the composition
$X\overset{\phi}\longrightarrow Y\times W\hookrightarrow  Y\times \overline W$ we get a commutative diagram
%\RamiR{
\[
\xymatrix
{        X   \ar@{}[r]|-*[@]{\hookrightarrow}   \ar@{->}_{\phi}[d]        & \overline X \ar@{->}_{\overline \phi}[d] \\
         Y\times W \    \ar@{}[r]|-*[@]{\hookrightarrow}   &  Y\times\overline W  \\ }
\]
in which the map $\overline X\to Y\times \overline W$ is proper and the map $X\RamiR{\hookrightarrow} \overline X$ is an open embedding.
Identify  $X$ with its image in $\overline X$.}

%\RamiQE{OK.}
%\DrinG{Is it possible to replace the horizontal arrows of the diagram by %the symbol ``$\hookrightarrow$"
%(to make it clear that they are embeddings)?}
%
%\RamiQE{I'll take care about it}


Let $Z \subset X$ be the zero locus of $\omega$, $\overline Z$ be its closure in $\overline X$ and
$\overline {X}_\infty:=\overline {X}-X$.
Let $\Xi :=\overline Z \cup \overline {X}_\infty$ and $U:=\overline X-\Xi\DrinH{\subset X}$. %\RamiM{.}
Let $\RamiS{\rho}:\tilde X \to \overline{X}$  be a resolution of singularities of $\overline{X}$ that strictly resolves $\Xi$\RamiM{,}
such that
$\RamiS{\rho}_{\RamiS{\rho}^{-1}(U\RamiI{)}}:\RamiS{\rho}^{-1}(U) \to U$
is an isomorphism.
Identify $U$ with $\RamiS{\rho}^{-1}({U})$.

Let $\pi_Y:Y \times \overline W \to Y$ and $\pi_{\overline W}:Y \times \overline W \to \overline W$
% and $\pi_{\tilde X}:\tilde X \times  W \to W$
 be the projection\DrinH{s}.
\RamiR{Let $\alpha_0:\tilde X \to Y$ be the composition
$$\tilde X \overset{\RamiS{\rho}}{\to}  \DrinH{\overline X \overset{\overline\phi}{\to} Y} \times
 \overline W \overset{\pi_{ Y}}{\to} Y$$
and $\alpha:\tilde X \times W \to Y \times W$ be $\alpha_0 \times Id_{\DrinH{W}}$.
 }
%Let $\alpha:\tilde X \times W \to Y \times W$ be given by $\alpha(x,v)=
%(\pi_Y(\overline{\phi} (p(x))),v)$.
Clearly $\alpha$ is proper. Let $\beta: \tilde X \to \overline W$ be
% given by
%$\beta(x)=\pi_W(\overline{\phi} (p(x)))$.
\RamiR{the composition
$$\tilde X \overset{\RamiS{\rho}}{\to}  \DrinH{\overline X \overset{\overline\phi}{\to} Y} \times  \overline W \overset{\pi_{\overline W}}{\to} \overline W$$}
Let $\omega'=\omega|_U$\RamiM{,} and consider $\omega'$ as a rational form on $\tilde X$. Let $Z' \subset \tilde X$
 be its zero locus.
Note that
$Z' \cup \beta^{-1}(W_\infty)=\RamiS{\rho}^{-1}(\Xi)$ is an SNC divisor.
% Since $\overline {X}$ is complete, $Y$ is complete
% and thus $\pi$ is proper.
We have:
$$\phi_*(|\omega|)=\alpha_*(\eta_{\beta,\omega'})$$
and hence
$$\cF^*_{W}(\phi_*(|\omega|))=\pi_*(\hat \eta_{\beta,\omega}).$$
By  {Proposition} \ref{thm:main_c}, the distribution
$\hat \eta_{\beta,\omega}$
 is  WF-holonomic. Thus by Proposition  \ref{prop:WF_Hol}
% {(and Remark \ref{??})} , ??I should change the proposition??
$\cF^*_{W}(\phi_*(|\omega|))$
is WF-holonomic\RamiM{.}
\end{proof}
}

\subsection{Proof of \DrinH{ Proposition \ref{thm:main_c}} } \label{sec:pf_C}
\setcounter{lemma}{0}
%\begin{theorem}
%\end{theorem}
The proof of  {Proposition}   \ref{thm:main_c} is based on
% {Hironaka's theorem and} %\RamiF{two key}
the key Lemmas~\ref{lem:red_to_dim_1} and \ref{lem:dim_1} below.
\subsubsection{\DrinB{\RamiR{T}he key lemmas}}
\DrinB{
Recall that if $W$ is a finite-dimensional vector space over $F$\RamiM{,} then  $\overline{W}$ stands for the space of lines in $\RamiG{ W \oplus F}$. The image in $\overline{W}$ of a nonzero vector
$(w,a)\in W\oplus F$ will be denoted by $\RamiG{(w:a)}$.

\begin{lemma} \label{lem:red_to_dim_1}
%\RamiQH{(I need to check in the Archimedean case)}
Let $Y$ be an  {algebraic} manifold and $W$  a vector space over $F$,  {with}  $\dim W<\infty$.
Let $\phi :Y\to \overline{W}$ be a map defined by $\phi(y)=\RamiG{( {\alpha}(y):p(y))}$,
 where $ {\alpha}:Y\to W$ and
$p:Y\to F$ are \RamiJ{regular}\RamiF{, $p \neq 0$ on a dense subset} and $ {\alpha}$ has no zeros. Let $\omega$ and
$\eta_{\phi,\omega}\in\Sc^*(Y {(F)}\times W)$ be as in Notation \ref{not:eta},
so $  {\hat \eta}_{\phi,\omega}\in
 \RamiF{\Sc^*}(Y {(F)}\times W^*,D^{Y {(F)}\times W^*}_{Y {(F)}})$.
Considering $\frac{1}{p}$ as a map $Y \to \mathbb{P}^1=\bar{\A}^1$, we also get
$\eta_{\frac{1}{p},\omega}\in\Sc^*(Y {(F)}\times F)$ and
$ {\hat \eta}_{\frac{1}{p},\omega}\in \RamiF{\Sc^*}(Y {(F)}\times F,
\DrinD{D^{Y(F)\times F}_{Y (F)}})$.
Then
\begin{equation}   \label{e:equality_in_question}
 {\hat \eta}_{\phi,\omega}  =g^*( {\hat \eta}_{\frac{1}{p},\omega}),
\end{equation}
where $g:Y \times W^{*} \to Y \times F$ is defined by
\DrinD{
\begin{equation}   \label{e:def_of_g}
g(y,\xi ):=(y,\langle \xi , {\alpha}(y)\rangle \,), \quad\quad y\in Y,\;\xi\in W^*.
\end{equation}
}


%Then
%$$\cF_{W}(\eta_{\phi,\omega})  =g^*(\cF_{F}(\eta_{\frac{1}{p},\omega})).$$

\end{lemma}
}
Note that since $ {\alpha}$ has no zeros $g$ is a submersion, so we have a well-defined map
$g^*:  \RamiF{\Sc^*}(Y\RamiM{(F)}\times F,\DrinD{D^{Y(F)\times F}_{Y (F)}} )\to \RamiF{\Sc^*}(Y\RamiM{(F)}\times W^*,
\DrinD{D^{Y(F)\times W^*}_{Y (F)}})$ \DrinH{and the r.h.s. of \eqref{e:equality_in_question} makes sense.}


%\begin{lemma} \label{lem:red_to_dim_1}
%Let $Y$ be an analytic manifold and $W=F^n$ and let $\tilde{\phi}:Y \to F \oplus W$ given by $$\tilde{\phi}(y) = (p(y), 1 ,f_2(y),...,f_n(y)).$$ let $pr:F \oplus W \to \overline{W}$ be the \RamiB{quotient map} and let ${\phi}=pr \circ \tilde{\phi}$. Let $g:Y \times W^{*} \to Y \times F$ defined by
%$$g(y,x_1,...,x_n)=(y, x_1+\sum_{i=2}^n x_i f_i(y)).$$
%Then
%\RamiB{
%$$\cF^{\RamiF{*}}_{W}(\eta_{\phi,\omega})  =g^*(\cF^{\RamiF{*}}_{F}(\eta_{\frac{1}{p},\omega}))$$
%(here we consider $\frac{1}{p}$ as a map $Y \to \mathbb{P}^1=\bar{\A}^1$)}.
%\end{lemma}

%\DrinD{A proof will be given in \S \ref{ssec:pf_red_to_dim_1}.

\begin{rem}
If $p$ has no zeros then Lemma~\ref{lem:red_to_dim_1} is obvious. To see this, note that by
Lemma~\ref{lem:exp_comp}, in this case
$$ {\hat \eta_{\phi,\omega}}=f_{\phi} \cdot \RamiS{pr_{W}}^{*}(|\omega|), \quad
g^*( {\hat \eta_{\frac{1}{p},\omega}})=\RamiS{g^*(f_{\frac{1}{p}} \cdot pr_{F}^{*}(|\omega|))=g^*(f_{\frac{1}{p}}) \cdot pr_{W}^{*}(|\omega|)},$$
where $f_{\phi}:Y\RamiM{(F)}\times W^*\to\C$ and $f_{\frac{1}{p}}:Y\RamiM{(F)}\times F\to\C$ are defined by
\begin{equation}   \label{e:fphi}
f_{\phi}(y,\xi )=\psi  {\left(  \frac{\langle\xi,  {\alpha}(y)\rangle}{p(y)}  \right)} ,\quad\quad y\in Y,\;\xi\in  W^*,
\end{equation}
\begin{equation}   \label{e:f1/p}
f_{\frac{1}{p}}(y,\nu )=\psi  {\left(  \frac{\nu}{p(y)}  \right)} ,\quad\quad y\in Y,\;\nu\in  F,
\end{equation}
so \eqref{e:equality_in_question} follows from the equality $f_{\phi}=g^*(f_{\frac{1}{p}})$,
which is obvious by \eqref{e:def_of_g}, \eqref{e:fphi}, and \eqref{e:f1/p}.
(The case where $p$ has zeros is not much harder in view of Remark \ref{rem:exp_comp}.)
\end{rem}

\RamiR{Let us give a complete proof now.}
%\DrinH{Do we need this sentence? (The previous arguments were also rigorous.)}
%\RamiQG{Can I replace the word ``rigorous" with ``complete"? The sentence in the parentheses just means that the proof below is not really needed if the reader does not require complete formal proof (the same as in Lemma \ref{lem:equv})}
%{\bf  ?? Rami, please improve graphics in the r.h.s. of \eqref{e:fphi} and
% \eqref{e:f1/p} ??\linebreak}.

\begin{proof}[\RamiR{Proof of Lemma \ref{lem:red_to_dim_1}}]
\RamiF{
Let $Y_0=p^{-1}(F-\DrinD{ \{ 0\} } ),\omega_0=\omega|_{Y_0}$,  $i_{\frac{1}{p}}:Y_0 \to Y \times F$ be the graph of $\frac{1}{p}$ and
$i_{\phi}:Y_0 \to Y \times W$ be the graph of $\DrinD{\phi }$.

\RamiR{Recall that $Mon(F,W)$ \DrinH{stands for} the space of monomorphisms from $F$ to $W$.} Let $\nu:Y \to Mon(F,W)$ be given by $\nu(y)(\lambda)=\lambda \cdot {\alpha}(\DrinD{y} )$.
Let $\rho_\nu:Y \times  F \to Y \times  W$ be the corresponding map (as in \DrinD{N}otation \ref{not:F_prop}).
\RamiR{The map $i_{\phi}$ \RamiS{is equal to} the composition
%\DrinH{Does ``is" mean here ``equals" or ``denotes"?}
$$Y_{\RamiT{0}} \overset{ i_{\frac{1}{p}}}{\to} Y \times  F \overset{\rho_\nu}{\to} Y \times  W.$$
Thus,}
%We have
$$(\rho_\nu)_* \eta_{\frac{1}{p},\omega}= (\rho_\nu)_* ( (i_{\frac{1}{p}})_*(|\omega_0|))=
%=(\rho_\nu \circ i_{\frac{1}{p}})_*(|\omega_0|))=
(i_{\phi})_*(|\omega_0|)= \eta_{\phi,\omega}\, .$$
%
\RamiR{Note that $\rho_{\nu^t}=g$.} Thus by Proposition \ref{prop:F_prop2},
$${\hat \eta_{\phi,\omega}} = (\rho_{\nu^t})^*  (\cF^{*}_{F}(\eta_{\frac{1}{p},\omega}))
 =g^*(\cF^{*}_{F}(\eta_{\frac{1}{p},\omega}))\DrinD{\, .}$$
 }
\end{proof}

\begin{lemma}  \label{lem:dim_1}
Let $Y$ be the affine space with coordinates $y_1, \dots, y_n.$ Let $p:Y \to F$ defined by
%$p(y_1, \dots, y_n)=a \prod y_i^{l_i}$
$p= \prod_{i=1}^n y_i^{l_i}$,
where
%$a \in F^\times$ and
$l_i \in \Z_{\geq 0}$. Let $\omega$ be the top differential form on $Y$
given by $\omega= (\prod_{i=1}^n y_i^{r_i}) d{y_1} \wedge \dots \wedge d{y_n}$, where $r_i \in \Z$.
 Suppose $r_i \geq 0$ whenever $l_i=0$,
\RamiG{
so $\omega$ is regular on the set $Y_0:=\{ y\in Y | p(y)\neq 0\}$ and therefore
$\eta_{\frac{1}{p},\omega}$ is well-defined.
}

Then $\RamiT{\hat \eta}_{\frac{1}{p},\omega}$ is  {\RamiJ{WF-holonomic}}.
\end{lemma}
\RamiG{
%Lemma \ref{lem:dim_1}
\RamiR{This lemma} follows from the next \DrinH{one} combined with Corollary \ref{cor:WF_Hol2}.
\begin{lemma}\label{lem:equv}
%%% Let $Y$ be the affine space with coordinates $y_1, \dots, y_n.$ Let $p:Y \to F$ be defined by
%%%%%$p(y_1, \dots, y_n)=a \prod y_i^{l_i}$
%%%$p= \prod_{i=1}^n y_i^{l_i}$ where  $l_i \in \Z_{\geq 0}$. Let $\omega$ be a top differential form on $Y$
%%%given by  $\omega= (\prod_{i=1}^n y_i^{r_i}) d{y_1} \wedge \dots \wedge d{y_n}$ where $r_i \in \Z$.
%%% Suppose $r_i \geq 0$ whenever $l_i=0$.
%%%Consider the action $\pi(t)$ of $T$ on $Y \times  F$ given by
%%% $$\pi(\alpha_1, \cdots,\alpha_n)\cdot (y_1, \cdots,y_n,\xi)=
%%%(\alpha_1 y_1, \cdots,\alpha_n y_n, \xi \prod_{i=1}^n \alpha_i^{l_i}).$$
%%%Then
\DrinH{In the situation of Lemma~\ref{lem:dim_1} one has}
$$\pi(\DrinH{\alpha_1, \cdots,\alpha_n})(\RamiT{\hat \eta}_{\frac{1}{p},\omega})=|\prod_{i=1}^n \alpha_i^{-1-r_i}|\RamiT{\hat \eta}_{\frac{ 1}{p},\omega}, \quad \DrinH{(\alpha_1, \cdots,\alpha_n)\in (F^{\times})^n ,}$$
\DrinH{where $\pi$ denotes the following action of $(F^{\times})^n$ on $Y \times  F$:}
$$\pi(\alpha_1, \cdots,\alpha_n)\cdot (y_1, \cdots,y_n,\xi)\DrinH{:= }
(\alpha_1 y_1, \cdots,\alpha_n y_n, \xi \prod_{i=1}^n \alpha_i^{l_i}).$$
\end{lemma}

\DrinD{
\begin{remark}
 By Lemma \ref{lem:exp_comp} and Remark \ref{rem:exp_comp},
$$\RamiT{\hat \eta}_{\frac{1}{p},\omega}=g\cdot |d{y_1} \wedge \dots \wedge d{y_n}|,$$
where $g$ is the function
$$g(y,\xi )=\psi(\xi \cdot \prod_{i=1}^n y_i^{-l_i} ) \cdot
\prod_{i=1}^n |y_i|^{r_i} $$
considered %understood
 as a \emph{generalized} function on the \emph{whole} $Y\RamiM{(F)} \times F$ (namely, to compute its scalar product with
 any test function, one integrates first with respect to $\xi$ and then with respect to $y$). So
 Lemma~\ref{lem:equv} just says that the equality
$$g(\alpha_1^{-1}y_1,\ldots ,\alpha_n^{-1}y_n,\xi\prod_{i=1}^n \alpha_i^{-l_i})
=\prod_{i=1}^n |\alpha_i|^{-r_i}\cdot g(y_1,\ldots y_n,\xi )$$
holds in $\G (Y\RamiM{(F)}\times F)$ (not merely on the locus $y_i\ne 0$).
This is clear. On the other hand, a formal proof of the lemma is given below.
\end{remark}

}
}

%\begin{lemma*}
%{Let $Y$ be the affine space with coordinates $y_1, \dots, y_n.$ Let $p:Y \to F$ defined by
%%$p(y_1, \dots, y_n)=a \prod y_i^{l_i}$
%$p= \prod_{i=1}^n y_i^{l_i}$
%where  $l_i \in \Z_{\geq 0}$. Let $\omega$ be a top differential form on $Y$ given by $\omega= (\prod_{i=1}^n y_i^{r_i}) d{y_1} \wedge \dots \wedge d{y_n}$ where $r_i \in \Z.$ Suppose  $r_i \geq 0$ whenever $l_i=0.$
%
%Then $\cF_W^{*}(\eta_{\frac{1}{p},\omega})$ is WF-holonomic.}
%\end{lemma*}
\begin{proof}[\RamiG{Proof of Lemma \ref{lem:equv}}]
Consider the action $\pi_1$ of $T:=(F^\times)^n$ on $Y$ given by
$\pi_1(\alpha_1, \cdots,\alpha_n)\cdot (y_1, \cdots,y_n)=  (\alpha_1 y_1, \cdots,\alpha_n y_n).$
 Let $t=(\alpha_1, \cdots,\alpha_n)$.
Clearly,
$$\pi_1(t)(p)=( \prod_{i=1}^n \alpha_i^{-l_i})\cdot p\quad  \quad \text{ and } \quad \quad \pi_1(t)(\omega)=(\prod_{i=1}^n \alpha_i^{{-1-r_i}})  \cdot \omega.$$
Thus by
%Lemma \ref{lem:loc_hom} and
%\RamiQG{
%(Lemmas \ref{lem:loc_hom}, (1),(2) are ``used" in the middle equality of the second line. However the lemma is trivial, and since it now appears below, I think it is OK just not
% to mention it)
%}
Proposition \ref{prop:F_prop2},  we have
\begin{multline*}
\pi_1(t)\cF^{*}_F(\eta_{\frac{1}{p},\omega}) =\cF^{*}_F(\pi_1(t)^{}\eta_{\frac{1}{p},\omega})=\cF^{*}_F(\eta_{\frac{1}{\pi_1(t)^{}(p)},\pi_1(t)^{}(\omega)})=
\\ =\cF^{*}_F\left(\eta_{\frac{ \prod_{i=1}^n \alpha_i^{l_i}}{p},(\prod_{i=1}^n \alpha_i^{-1-r_i})\omega}\right)=
   \RamiE{|\prod_{i=1}^n \alpha_i^{-1-r_i}|\cdot \cF^{*}_F(\rho_{\prod_{i=1}^n \alpha_i^{l_i}}(\eta_{\frac{1}{p},\omega}))=}
\\
\RamiE{= |\prod_{i=1}^n \alpha_i^{-1-r_i}|\cdot \rho_{\prod_{i=1}^n \alpha_i^{-l_i}}\left(\cF^{*}_F(\eta_{\frac{1}{p},\omega})\right).}
%|\prod_{i=1}^n \alpha_i^{r_i+1}|\rho_{\prod_{i=1}^n \alpha_i^{-l_i}}\left(\cF^{*}_F(\eta_{\frac{1}{p},\omega})\right).
\end{multline*}
%    \begin{multline*}
%    t\cF^{*}_W(\eta_{\frac{1}{p},\omega}) =|\alpha_1 \cdots\alpha_n|^{-2}\cF^{*}_W(t^{-1}\eta_{\frac{1}{p},\omega})=|\alpha_1 \cdots\alpha_n|^{-2}\cF^{*}_W(\eta_{\frac{1}{t^{-1}(p)},t^{-1}(\omega)})=\\ =|\alpha_1 \cdots\alpha_n|^{-2}\cF^{*}_W\left(\eta_{\frac{ \prod_{i=1}^n \alpha_i^{-l_i}}{p},(\prod_{i=1}^n \alpha_i^{r_i-1})\omega}\right)=|\alpha_1 \cdots\alpha_n|^{-2}(\prod_{i=1}^n \alpha_i^{r_i-1})\cF^{*}_W(\eta_{\frac{ \prod_{i=1}^n \alpha_i^{-l_i}}{p},\omega})=??\cF^{*}_W(\eta_{\frac{1}{p},\omega}).
%    \end{multline*}
%Now consider the action $\pi_2(t)$ of $T$ on $Y \times  F$ given by
% $\pi_2(\alpha_1, \cdots,\alpha_n)\cdot (y_1, \cdots,y_n,\xi)=  (\alpha_1 y_1, \cdots,\alpha_n y_n, \xi \prod_{i=1}^n \alpha_i^{l_i}).$
%We get
\RamiG{This implies
$$\pi(t)^{}\cF^{*}_W(\eta_{\frac{1}{p},\omega})=|\prod_{i=1}^n \alpha_i^{-1-r_i}|\cdot \cF^{*}_W(\eta_{\frac{ 1}{p},\omega}).$$
}
%$$\pi_2(t)^{}\cF^{*}_W(\eta_{\frac{1}{p},\omega})=|\prod_{i=1}^n \alpha_i^{r_i+1}|\cF^{*}_W(\eta_{\frac{ 1}{p},\omega}).$$
%Thus, Proposition \ref{prop:WF_prop} \eqref{it:G_act} implies that
%\begin{equation} \label{eq:CN}
%\WF(\cF^{*}_W(\eta_{\frac{ 1}{p},\omega}))\subset\bigcup_{x \in Y \times  F} CN_{\pi_2(T) x}^{Y \times  F}.
% \end{equation}
%\RamiF{
%Let $D$ be the union of coordinate hyperplanes in $Y$. By Lemma \ref{lem:exp_comp}, the distribution $\cF^{*}_W(\eta_{\frac{ 1}{p},\omega})$ is smooth on
% $Y_{reg}=Y-D$
%}
%On the other hand, Lemma \ref{lem:exp_comp} says that $\cF^{*}_W(\eta_{\frac{ 1}{p},\omega})$ is regular on
%$Y_0=Y-Y_\infty=Y-p^{-1}(0).$
% This implies, by Proposition \ref{prop:WF_prop},
% that $\WF(\cF^{*}_W(\eta_{\frac{ 1}{p},\omega})) \cap T^*\RamiF{Y_{reg}} \subset \RamiF{Y_{reg}}.$ This, together with (\ref{eq:CN}),
% gives
%$$\WF(\cF^{*}_W(\eta_{\frac{ 1}{p},\omega}))\subset\left ( \bigcup_{x \in \RamiF{D} \times  F} CN_{\pi_2(T) x}^{Y \times  F} \right) \cup \RamiF{Y_{reg}}.$$

%Since $\pi_2(T)$ has finitely many orbits on $\RamiF{D} \times  F$,
% this implies that $\cF^*_W(\eta_{\frac{1}{p},\omega})$ is WF-holonomic.
\end{proof}

%For the proof see \S \ref{ssec:pf_dim_1}.
\subsubsection{\DrinH{Proof} \RamiF{of  {Proposition} \ref{thm:main_c}}}
%In order to deduce  {Proposition} \ref{thm:main_c} from \RamiI{the key lemmas,}%the two lemmas,
%we
\DrinH{Let us} introduce the following \RamiG{\emph{ad hoc}} terminology.
\begin{definition}
A quadruple $(Y,W,\phi,\omega)$ as in Notation \ref{not:eta} is said to be
``good" if  $\RamiT{\hat \eta}_{\phi,\omega} \in \RamiF{\Sc^*}( {Y(F)}\times W^*,
\DrinD{D^{Y(F)\times W^*}_{Y (F)}})$
is  {\RamiJ{WF-holonomic}}.
\end{definition}
\DrinH{Our goal is to show that any quadruple $(Y,W,\phi,\omega)$ satisfying the conditions of Proposition~\ref{thm:main_c} is good. We will need  the following  obvious lemma.}
%\RamiM{Let us now study how this property  depends on $(Y,W,\phi,\omega)$.
%For this, we will need  the following  obvious lemma.}
\begin{lemma} \label{lem:loc_hom}
Let $(Y,W,\phi,\omega)$ be as above.
\RamiR{Let  $e:U \to Y$ be an {\et} map and $f\in O^{\times}(Y)$ be \DrinH{an} invertible  regular function. Then
\begin{enumerate}
 \item \label{lem:loc_hom:1}
$\eta_{\phi,f\omega}= |f| \cdot \eta_{\phi,\omega}.$
 \item \label{lem:loc_hom:2} \DrinH{Let} $\rho_{f}:Y \times W \to Y \times W $ \DrinH{denote} the homothety action as in \DrinH{N}otation \ref{not:F_prop}. Then
$$\eta_{f \phi,\omega}= \rho_{f}(\eta_{ \phi,\omega}).$$
\item \label{lem:loc_hom:3}  \RamiS{Let $e^*(\phi)$ denote the composition $$U \overset{e}{\to} Y \overset{\phi}{\to}\overline W$$

Then}
$$\eta_{e^*(\phi),e^*(\omega)}= (e \times Id_W)^*( \eta_{ \phi,\omega})$$
%\RamiQF{Do you think it would be better to write $e^*(\phi)$ instead of %$\phi \circ e$ and explain (this time) what does it mean?} \DrinH{Yes.}

\end{enumerate}
}
% Let $f_1,f_2 \in O^{\times}(Y)$
%be invertible  {algebraic} functions.
%Then
% {$$\eta_{(f_1 \phi) \circ e,e^*(f_2\omega)}= (e \times Id_W)^*(\rho_{f_1}(|f_2| %\cdot \eta_{ \phi,\omega})) $$}
\end{lemma}

\DrinH{Let us now study how the property of being good  depends on $(Y,W,\phi,\omega)$.}

\begin{proposition}[Locality] \label{prop:loc}
Let $(Y,W,\phi,\omega)$ be as above.
 {
\begin{enumerate}
 \item \label{prop:loc:1} Let $Y=\bigcup U_i$ be a Zariski open cover of $Y$. Suppose that
 the quadruple $(U_i,W,\phi|_{U_i},\omega|_{U_i})$ is good for each $i$ .
Then the quadruple $(Y,W,\phi,\omega)$ is good.
\item \label{prop:loc:2}Let $e:U \to Y$ be an  {\et} map. Suppose that the quadruple $(Y,W,\phi,\omega)$ is good. Then the
 quadruple $(U,W,\RamiS{e^*(\phi)},e^*(\omega))$ is good.
\end{enumerate}
}
\end{proposition}
\begin{proof}
$ $
\begin{enumerate}
 \item By Lemma \ref{lem:loc_hom}\RamiS{\eqref{lem:loc_hom:3}}, we have $\eta_{\phi|_U,\omega|_U}= (  \eta_{ \phi,\omega})|_{U \times W}$.
By Lemma \ref{prop:F_prop}, we get  {$\hat \eta_{\phi|_U,\omega|_U}=  (\hat \eta_{ \phi,\omega})|_{U(F) \times W}$}.
 By Proposition \ref{prop:WF_prop}, this gives
 {$\WF(\hat \eta_{\phi|_U,\omega|_U})= \WF(\hat \eta_{ \phi,\omega})\cap (T^*(U \times W))(F)$}.
This immediately implies the assertion.
\item  {Follows immediately from Lemma \ref{lem:loc_hom}\RamiS{\eqref{lem:loc_hom:3}} and Proposition \ref{prop:WF_Hol}. }
\end{enumerate}
\end{proof}

\begin{proposition}[\RamiG{Homogeneity}] \label{prop:homo}
Let $(Y,W,\phi,\omega)$ be a good quadruple. Let $f_1,f_2 \in O^{\times}(Y)$ be invertible \RamiJ{regular} functions. Then $(Y,W,f_1\phi,f_2 \omega)$ is good.
\end{proposition}
\begin{proof}
By  Lemma \ref{lem:loc_hom}\RamiS{(\ref{lem:loc_hom:1}},\RamiS{\ref{lem:loc_hom:2})}, we have $\eta_{f_1 \phi,f_2 \omega}= (\rho_{f_1}(|f_2| \cdot \eta_{ \phi,\omega})$. By \RamiE{Proposition}  \ref{prop:F_prop2}, we get
%$\cF_W^*(\eta_{f_1 \phi,f_2 \omega})= |f_2| \cdot |f_1|^{2\dim W} \cdot\rho_{f_1^{-1}} (\cF_W^*(\eta_{ \phi,\omega}))$.
\RamiE{$\cF_W^*(\eta_{f_1 \phi,f_2 \omega})= |f_2| \cdot\rho_{f_1^{-1}} (\cF_W^*(\eta_{ \phi,\omega}))$}.
 By Proposition \ref{prop:WF_prop}, this gives $\WF(\cF_W^*(\eta_{f_1 \phi,f_2 \omega}))\subset \rho_{f_1^{ }}^*(\WF(\cF_W^*((  \eta_{ \phi,\omega}))))$ . This immediately implies the assertion.
\end{proof}
 {
\begin{corollary}  \label{cor:homo}
Let $(Y,W,\phi,\omega)$ be a quadruple as above. \DrinH{Assume that $\dim W=1$, so we can interpret $\phi$ as a rational function on $Y$. Then} the property of being good depends only on the divisors of
$\phi$ and $\omega$.
% \DrinH{I would delete the followng:  and not on $\phi$ and $\omega$ themselves.}
%\RamiQF{(erased explanation on divisors)}
%Here by \RamiM{``divisor"} we mean an integral linear combination of irreducible %subvarieties of codimension $1$.  We attach \RamiM{divisors} to  $\phi$ %and $\omega$  by considering \RamiM{their} poles and zeros.
\end{corollary}


%\begin{proposition} \label{prop:bira}
%Let $(Y,W,\phi,\omega)$ be as above. Let $p:\tilde Y \to Y$ be a proper map. Assume that it is a birational equivalence
%Assume also that that $(\tilde Y ,W,\phi \circ p,p^*(\omega))$ is good. Then $(Y,W,\phi,\omega)$ is good.
%\end{proposition}
%\begin{proof}
%It follows from Lemma \ref{lem:loc_hom} and Proposition \ref{prop:WF_Hol}, ??I should change the proposition??
%\end{proof}

We will also need the following standard lemma.
\begin{lemma} \label{lem:mon}
Let $D \subset U$ be an SNC divisor inside a smooth algebraic variety. Let $D_i$ be a collection of divisors in $U$ supported\footnote{\RamiR{Recall that a divisor on a manifold is an integral linear combination of
irreducible subvarieties of codimension $1$. The words ``supported in $D$"
mean that each of the subvarieties is contained in $D$.}} in $D$.
  Then there exist
 a Zariski cover $V_j$ of $U$,  {\et} maps $e_j:V_j \to \A^n$  and divisors $D_{ji}$ in $\A^n$ such that
 $(D_i)|_{V_j}=e_j^*(D_{ji})$ and  \DrinH{each} $D_{ji}$ \DrinH{is} supported \RamiM{in} the union of the coordinate hyperplanes.
\end{lemma}
%\begin{proof}
%Without loss of generality we may assume that $\cF_i$ are trivial.
%\end{proof}
}
\RamiR{
Now we are ready to prove the following particular case of
Proposition \ref{thm:main_c}.
\begin{lemma}\label{lem:sp_case}
Let $(Y,W,\phi,\omega)$  be as in  Proposition~\ref{thm:main_c}. Moreover,
suppose that $W=F$ and
$0\not\in\phi (Y)$. Then $(Y,W,\phi,\omega)$  is good.
\end{lemma}}
\begin{proof}
\RamiR{
The proof is based on Lemma \ref{lem:dim_1}.
We can rewrite $\phi=1/p$, \DrinH{where} $p$ is a regular function on $Y$.} We know that %$\Xi:= U_\infty  \cup Z_1$
 \RamiR{$\Xi:= Y_\infty  \cup Z$} is \RamiM{an} {SNC} divisor in
  \RamiR{$Y$.}%$U$.

Let $D_1$ be the divisor of $p$ and $D_2$ the divisor of %$\omega|_{U}$.
\RamiR{$\omega$}.
By Lemma \ref{lem:mon}, we can find a Zariski cover  $V_j$ of
%$U$
\RamiR{$Y$}, {\et} maps $e_j:V_j \to \A^n$,
and  divisors $D_{\RamiI{j1}},D_{\RamiI{j2}}$ such that
$(D_i)|_{V_j}=e_j^*(D_{ji})$ and the divisors $D_{ji}$ are supported \RamiM{in} the union of the coordinate hyperplanes.
By Proposition \ref{prop:loc}\RamiS{\eqref{prop:loc:1}}, it is enough to show that for any $j$
the quadruple  $(V_j,F,\frac{1}{p|_{V_j}},(\omega|_{V_j}))$ is good.
By Corollary \ref{cor:homo}, we may replace $p|_{V_j}$ and $\omega|_{V_j}$ by any other function and form with the
same divisor.
%\DrinH{Do you really need Proposition \ref{prop:loc} in the next sentence?}
%\RamiQG{I think so, we use proposition \ref{prop:loc}\eqref{prop:loc:2} in order to deduce that $(V_j,F,e_{j}^{*}(\frac{1}{q}),e_{j}^{*}(\eps))$
%is good from the fact that $(\A^n,F,\frac{1}{q},\eps)$ is good.}
 So by Proposition \ref{prop:loc}\RamiS{\eqref{prop:loc:2}}, it is enough to show that
$(\A^n,F,\frac{1}{q},\eps)$ is good %, where
\RamiR{for some regular function $q$  and top differential form $\eps$ on $\A^n$ such that} the divisor of $q$ is
 $D_{\RamiI{j1}}$ and the divisor
of $\eps$ is $D_{\RamiI{j2}}$.
This follows from Lemma  \ref{lem:dim_1}.
\end{proof}

%Now we \RamiR{will deduce {Proposition} \ref{thm:main_c} from this special case
%using Lemma \ref{lem:red_to_dim_1}.}
%%% are ready to prove  {Proposition} \ref{thm:main_c}.
\begin{proof} [Proof of  {Proposition} \ref{thm:main_c}]
$ $
% {Step 1.
Without loss of generality we may assume that $Y$ is irreducible and $\omega \neq 0$.
% {Step 1. Reduction to the case when $Z \cup Y_\infty$ is an SNC divisor, where $Z$ is the zero locus of $\omega$
%and   $Y_\infty:=\phi^{-1}(W_\infty)$.\\
%Let $p: \tilde Y \to Y$ be  a resolution of singularities that strictly resolve $Z \cup Y_\infty$ and such that
% the map  $p^{-1}(U) \to (U)$ is an isomorphism. By Proposition  \ref{prop:bira} it is enough to prove that
%$(\tilde Y ,W,\phi \circ p,p^*(\omega))$ is good. Note that $\tilde Z \cup \tilde Y_\infty=p^{-1}(Z \cup Y_\infty)$
%where $\tilde Z$ is the zero set of $p^*(\omega)$ and $\tilde Y_\infty:=(\phi \circ p)^{-1}(W_\infty)$.
%
%From now we replace $Y$ by $\tilde Y$, $\omega$ by $p^*(\omega)$,$\phi$ by $\phi \circ p$, $Z$ by $\tilde Z$
%and $Y_\infty$ by $\tilde Y_\infty$. Thus we have  $Z \cup Y_\infty$ is an SNC divisor.
%

%Step  {2}. Reduction to the case when $\dim W=1$ \RamiR{and $\phi$ never %vanish.}\\
We have to show that $(Y,W,\phi,\omega)$  is good.
 {We can cover $Y$ by open subsets $U_i$ so that $\phi|_{U_i}=(f_i(x):p_i(x))$,
 where for each $i$, one of the maps $f_i:U_i \to W$ and $p_i:U_i \to F$ never vanishes.}

%Choose coordinates on $W$.
%Let $\{ W_i\}$ be the corresponding standard cover of $\RamiF{\overline{W}}.$ More precisely,
% let $$\tilde{W}_i=\{(x_0,\dots x_n) \in  {W \oplus F}| x_i=1\}  \subset   {W \oplus F},$$
%and let $W_i=pr(\tilde{W}_i) \subset \RamiF{\overline{W}},$ where $pr:(F \oplus W)\setminus\{ 0\} \to \overline{W} $
% is the quotient map.
%Let $U_i:=\phi^{-1} (W_i) \subset Y$.
By Proposition \ref{prop:loc}\RamiS{\eqref{prop:loc:1}},
 it is enough to show that $(U_i,W,\phi|_{U_i},\omega|_{U_i})$ is good.
\begin{itemize}
 \item The case \RamiI{when}  {$p_i$ never vanishes}.\\
By Lemma \ref{lem:exp_comp} it is enough to show that $|\omega|$ is \RamiJ{a WF-holonomic} distribution on $U_i$.
For this it is enough to show that
$(U_{ {i}},F,1,\omega|_{U_ {i}})$ is good.
\RamiR{This follows from }% the previous lemma ()}
\DrinH{Lemma \ref{lem:sp_case}.}
%\RamiR{We will  discuss this case in the next step.}
% This is a special case of the case $\dim W=1$, which we
 %discuss in the next step.
 \item \RamiF{The case  \RamiI{when}  {$f_i$ never vanishes}.}\\
 %we may assume, without loss of generality, that $i=1$.
%Since $pr|_{\tilde{W}_i}:\tilde{W}_i \to {W}_i$ is an isomorphism, we can rewrite $\phi|_{U_{i}}$ as
 %${\phi}|_{U_{i}}=pr \circ \tilde{\phi}$, where $\tilde{\phi}:U_{i} \to F \oplus W$ is given by
%$\tilde{\phi}(x) = (p(x),f(x))$.
 By Lemma \ref{lem:red_to_dim_1}, there exists a submersion
$g:U_{i} \times W \to U_{i} \times F$ such that
 $\RamiT{\hat \eta}_{\phi,\omega}  =g^*({\hat \eta_{\frac{1}{{p_i}},\omega}}).$
So by Proposition \ref{prop:WF_Hol}, it is enough to show that $(U_i,F,\frac{1}{{p_i}},\omega|_{U_i})$ is good.
\RamiR{This again follows from }% the previous lemma ()}
\DrinH{Lemma \ref{lem:sp_case}.}
\end{itemize}
%\\

%The case $i=0$ follows from Lemma \ref{lem:exp_comp} . For the rest of the
% cases,
% (Old Step 3)Step {3}. Reduction \RamiR{of the case when $W=F$ and $\phi$ never vanish} to Lemma \ref{lem:dim_1}.\\
% (Old Step 3)\RamiR{from now on we will asumw that $W=F$ and $\phi$ never vanish. Thus we can reright $\phi=1/p$ when $p$ is a polinomailal on $Y$.}
% (Old Step 3)%Fix $i$ and let $U:=U_i$ {and $p:=p_i$}.
% (Old Step 3)%Let $U_\infty=U\cap Y_\infty$, $Z_1=U\cap Z.$
% (Old Step 3)%and $\overline{Z_1} \subset U$ be the closure of $Z_1$ in $U$.
% (Old Step 3) We know that %$\Xi:= U_\infty  \cup Z_1$
% (Old Step 3) \RamiR{$\Xi:= Y_\infty  \cup Z$} is \RamiM{an} {SNC} divisor in
% (Old Step 3)  \RamiR{Y.}%$U$.
% (Old Step 3)
% (Old Step 3){ Let $D_1$ be the divisor of $p$ and $D_2$ the divisor of %$\omega|_{U}$.
% (Old Step 3)\RamiR{$\omega$}.
% (Old Step 3)By Lemma \ref{lem:mon}, we can find a Zariski cover  $V_j$ of
% (Old Step 3)%$U$
% (Old Step 3)\RamiR{$Y$}, {\et} maps $e_j:V_j \to \A^n$,
% (Old Step 3)and  divisors $D_{\RamiI{j1}},D_{\RamiI{j2}}$ such that
% (Old Step 3)$(D_i)|_{V_j}=e_j^*(D_{ji})$ and the divisors $D_{ji}$ are supported \RamiM{in} the union of the coordinate hyperplanes.
% (Old Step 3)By Proposition \ref{prop:loc}, it is enough to show that for any $j$
% (Old Step 3)the quadruple  $(V_j,F,\frac{1}{p|_{V_j}},(\omega|_{V_j}))$ is good.
% (Old Step 3)By Corollary \ref{cor:homo}, we may replace $p|_{V_j}$ and $\omega|_{V_j}$ by any other function and form with the
% (Old Step 3)same divisor. So by Proposition \ref{prop:loc}, it is enough
% (Old Step 3)  to show that
% (Old Step 3)$(\A^n,F,\frac{1}{q},\eps)$ is good, where \RamiR{for some polinomial $q$  and top diferantila form $\eps$ on $\A^n$ such that}  the divisor of $q$ is $D_{\RamiI{j1}}$ and the divisor
% (Old Step 3)of $\eps$ is $D_{\RamiI{j2}}$.
% (Old Step 3)This follows from Lemma  \ref{lem:dim_1}.
% (Old Step 3)}
\end{proof}

%\subsubsection{Proof of Lemma \ref{lem:red_to_dim_1}} \label{ssec:pf_red_to_dim_1}




%\subsubsection{Proof of Lemma \ref{lem:dim_1}} \label{ssec:pf_dim_1}
%$ $
%{
% We will need the following lemma
% \begin{lemma}
% {Let $Y$ be the affine space with co-ordinates $y_1, \dots, y_n.$ Let $p:Y \to F$ defined by
% %$p(y_1, \dots, y_n)=a \prod y_i^{l_i}$
% $p= \prod_{i=1}^n y_i^{l_i}$
% where  $l_i \in \Z_{\geq 0}$. Let $\omega$ be a top differential form on $Y$ given by $\omega= (\prod_{i=1}^n y_i^{r_i}) d{y_1} \wedge \dots \wedge d{y_n}$ where $r_i \in \Z.$ Suppose that $r_i \geq 0$ whenever $l_i=0.$
%
% Then $\cF_W^{*}(\eta_{\frac{1}{p},\omega})$ is WF-holonomic}
% \end{lemma}

%We are now ready to prove Lemma \ref{lem:dim_1}. Let us remind its formulation.}


%\begin{conjecture}
%\end{conjecture}

%\newpage

\section{Proof of Theorem  \ref{thm:futerO}}\label{sec:ex_boun}
\setcounter{lemma}{0}
%\RamiP{In this section we give an explicit version of Theorem \ref{thm:main_bb}.}\\ \
%RamiS{For this we will need the following notation.}



%\DrinK{In this section we prove Theorem  \ref{thm:futerO}.}

\subsection{A fact from symplectic geometry}
We will need the following lemma, which will be proved in Appendix \ref{app:symp_geom}.


%\DrinK{It would be nice to find a way to bypass Lemma~\ref{l:symplectic_lem} altogether.
%But I don't see how to do it.
%%%I still hope that Lemma~\ref{l:symplectic_lem} is NOT REALLY NECESSARY for us.
%%%A reason for such hope is explained immediately before Corollary~\ref{c:based on symplectic lemma}.
%}

\RamiCPD{
 \begin{lemma}    \label{l:symplectic_lem}
Let $\RamiAJ{W}$ be a finite dimensional vector space over $F$ and $X$ be a manifold. Let $E$ be a vector bundle over $\RamiAJ{Y}$ which is a subbundle of the trivial vector bundle $\RamiAJ{Y} \times \RamiAJ{W}$. Let $E^\bot\subset \RamiAJ{Y} \times \RamiAJ{W}^*$ be its orthogonal complement.
Then $CN_{E^\bot}^{\RamiAJ{Y} \times \RamiAJ{W}^*}=CN_E^{\RamiAJ{Y} \times \RamiAJ{W}}$.

Here
$$CN_E^{\RamiAJ{Y} \times \RamiAJ{W}} \subset T^*(\RamiAJ{Y} \times \RamiAJ{W})=T^*(\RamiAJ{Y}) \times \RamiAJ{W} \times \RamiAJ{W}^*,$$
$$CN_{E^\bot}^{\RamiAJ{Y} \times \RamiAJ{W}^*} \subset T^*(\RamiAJ{Y} \times \RamiAJ{W}^*)=T^*(\RamiAJ{Y}) \times \RamiAJ{W}^* \times \RamiAJ{W},$$
and the symplectic manifolds $T^*(\RamiAJ{Y}) \times \RamiAJ{W} \times \RamiAJ{W}^*$ and $T^*(\RamiAJ{Y}
) \times \RamiAJ{W} \times \RamiAJ{W}^*$ are identified via the map
$$\RamiAJ{W} \times \RamiAJ{W}^*{\buildrel{\sim}\over{\longrightarrow}} \RamiAJ{W}^* \times \RamiAJ{W}, \quad \RamiAJ{(w,\phi)\mapsto (\phi,-w)}.$$
 \end{lemma}
}
%\RamiAF{For a proof, see Appendix \ref{app:symp_geom}.} %\RamiQU{It will be Appendix B.}


\subsection{Some notation}  \label{ss:notation_explicit}
$ $


\RamiAK{
\begin{notn} $ $
\begin{enumerate}
\item
Let $W$ be a vector space. We denote by $\mathbb{P}(W)$  the {projective space whose points are
1-dimensional subspaces} in $W^*$. We denote by $Taut_{\mathbb{P}(W)}$ the tautological  {line} bundle of $\mathbb{P}(W)$ which is {a} subbundle of the {trivial} bundle with fiber $W^*$.
% \RamiQAA{ Which expression is better: ''the constant bundle with fiber %$W^*$" or ``the bundle $\mathbb{P}(W) \times W^*$"?} \DrinK{Trivial bundle.}

\item
\RamiAM{Recall that
\RamiP{$\overline{W}:=\mathbb{P}(W^* \oplus F)$}.
}
\item
Let $W$ be a finite dimensional vector space over $F$ and $X$ be a manifold. Let $E$ be a {vector bundle} over $X$ which is a subbundle of the {trivial} bundle $X \times W$. Then
its orthogonal complement in $X \times W^*$ is denoted by $E^\bot$.

In particular, we denote by $Taut_{\mathbb{P}(W)}^\bot$ the orthogonal complement to $Taut_{\mathbb{P}(W)}\,$, which is a co-dimension $1$ subbundle of the {trivial} bundle $\mathbb{P}(W) \times W$.
\end{enumerate}
\end{notn}
}
%\DrinK{Rami, is it possible to replace the notation $S_f$ by a more ``expressive"
% notation, e.g., by $\Crit_f$ (or
% at least, ${\operatorname{Cr}}_f$ if $\Crit_f$ is too long)?}
\begin{notation}
% \RamiQY{I've chainged ``$S$" to $\Crit$. I cosed one line brack in the %proof of }
  \label{notation:S}
Given a morphism $f:M\to N$ between algebraic manifolds, define $\RamiAJ{\Crit}_f\subset T^*N$ by
$\RamiAJ{\Crit}_f:=f_*(M)$ where $M\subset T^*M$ is the zero section and $f_*$ is as in Definition~\ref{def:dir_and_inv_image}(\ref{def:dir_and_inv_image:3}).
\end{notation}

\begin{rem}   \label{r:S isotropic}
By Lemma~\ref{lem:iso_dir_invs_im}, $\RamiAJ{\Crit}_f$ is isotropic. It is easy to see that if $f:M\to N$ is proper then
the subset $\RamiAJ{\Crit}_f\subset T^*N$ is closed.
\end{rem}

\begin{rem}
The fiber of $\RamiAJ{\Crit}_f$ over $y\in N$ is nonzero if and only if $y$ is a critical value for $f:M\to N$.
\end{rem}


\RamiCPD{
\begin{notation}   \label{n:SNC_notation}
%\DrinK{Should we say that $D$ is an SNC divisor on some algebraic manifold?}
For an SNC divisor $D$ on some algebraic manifold,
let $\hat{D_1}$ denote the disjoint union of the irreducible components of $D$, let $\hat{D_2}$ denote the disjoint union of the pairwise intersections of the irreducible components of $D$, and so on. Let $\hat{D}$ denote the disjoint union of $\hat{D_i}$, $i\ge 1$. Clearly
$\hat D$ is smooth, and if $D$ is projective then so is $\hat{D}$.

%  we let $\nu_D:\hat{D} \to D$ be its normalization. Set
%$D_i=\{z \in D| \#\nu_D^{-1}(z)\geq i\}$, and let $\hat{D_i}$ be the normalization %of $D_i$.
\end{notation}
% \RamiT{(see \S \S\ref{sec:pf_futer} below)}.
% \RamiQH{The deduction is less trivial then I expected. I think it deserves %a subsection.}
}

% \DrinK{Since $D$ is assumed to be SNC (not merely NC) one could change Notation~\ref{n:SNC_notation} as follows:
%
% For an SNC divisor $D$ on some algebraic manifold, let $\hat{D_1}$ denote the disjoint union of the irreducible components of $D$, let $\hat{D_2}$ denote the disjoint union of the pairwise intersections of the irreducible components of $D$, and so on. Let $\hat{D}$ denote the disjoint union of $\hat{D_i}$, $i\ge 1$. Clearly
% $\hat D$ is smooth, and if $D$ is projective then so is $\hat{D}$.
%
% One of the reasons for changing the notation is to have a notation for the  disjoint union of $\hat{D_i}$'s without introducing a new letter. Will this allow us to avoid considering the individual $\hat{D_i}$'s in the rest of the section? (It would be nice to get ride of the numerous symbols $\bigcup$ and $\bigsqcup$ in this section.)
% In Corollary~\ref{cor:exp} I would also prefer to work with the disjoint union of $D'_i$'s and to denote it by
% $\hat{D}'$ rather than $D'$.
% }

\subsection{An explicit upper bound for the wave front}
%\DrinK{
Let $F$ be a local field of characteristic 0. Because of the numerous \RamiAJ{references} to \S\ref{sec:pf_main_bb}, the reader may assume for a while that $F$ is non-Archimedean. However, it will be clear from  \RamiAJ{\S\S}\ref{ssec:arc_pf} that this assumption is not necessary either in  \S\ref{sec:pf_main_bb} or here.

\subsubsection{The goal} \label{sss:thegoal}
Consider the following setting (it is essentially\footnote{%\DrinK
{Only the notation is slightly different: here we denote by $X$ and $\phi$ the objects that were denoted by $\tilde X$ and $\bar \phi \circ \rho$ in the proof of Theorem \ref{t:C}}.} the same as in the proof of Theorem \ref{t:C} given at the end of %Subsection~
\RamiAJ{\S\S}\ref{ss:Reduction to key}).
%}

Let $X,Y$ be  {algebraic} manifolds over $F$ and $W$ a vector space over $F$,  {with}  $\dim W<\infty$.
Let $\phi :X\to Y \times \overline{W}$ be a proper map.  \RamiT{Let $X_0:=\phi^{-1}(Y \times W)$.} Let $\omega$ be a top differential form on $X\RamiT{_0}$.
 Let $Z$ be the \RamiT{closure of the} zero set of $\omega$ \RamiT{in $X$}.
 Let $D:=Z \cup \phi^{-1}((\overline{W}-W)\times Y)$.
Assume that $D$ is an SNC divisor.

%\DrinK{
Then we have the distribution $(\phi|_{X_0})_*(|\omega|))$ on $Y(F)\times W$.
In \S\ref{sec:pf_main_bb} we proved that its partial Fourier transform
$\cF^*(\RamiT{(\phi|_{X_0})}_*(|\omega|)))$ is holonomic, which means that
\begin{equation}   \label{e:property_needed}
WF(\cF^*(\RamiT{(\phi|_{X_0})}_*(|\omega|)))\subset L(F)
\end{equation}
%the wave front of $\cF^*(\RamiT{(\phi|_{X_0})}_*(|\omega|)))$ is contained in the set of $F$-points of
for \emph{some} isotropic algebraic subvariety $L\subset T^*(Y \times  W^{*})$.

Our goal now is to describe a \emph{specific} $L$ with property \eqref{e:property_needed}. The
%format\footnote{\DrinK{The key feature of the description of $L$ is that if the collection $(X,Y,W,\phi,\omega )$
%is defined over a subfield $K\subset F$ then so is $L$.}} of the description given below implies
%Theorem~\ref{thm:futerO}.
definition of $L$ given below is purely algebro-geometric, so the fact that this $L$ satisfies
\eqref{e:property_needed} will imply Theorem~\ref{thm:futerO}.
%}

\subsubsection{Definition of $L$}   \label{sss:Definition of L}
%Let $\nu:\hat{D} \to D$ be the normalization of $D$.
%Let $D_i=\{z \in D| \#\nu^{-1}(z)\geq i\}$ and let $\hat{D_i}$ be the normalization %of $D_i$.
Let \RamiAJ{$\hat D$} be as in Notation~\ref{n:SNC_notation}.
Let \RamiAJ{$\hat D'$} be the union of those components of \RamiAJ{$\hat D$} whose image in \RamiAJ{$ D$} is contained in $\phi^{-1}(\overline{W}-W)$\RamiT{.}
% and  let $\hat{D_i}''$ be the union of the components of $\hat{D_i}$ whose %image in $D_i$ contained in $Z$.

Let $\pi:X\to \overline{W}$ and $\tau:X\to Y$ be the  compositions of $\phi :X\to Y \times \overline{W}$ with the projections $Y \times \overline{W}\to \overline{W}$ and $Y \times \overline{W}\to Y$.
{
Let
$$\pii:\RamiAJ{\hat D'} \to\overline{W}-W= \mathbb{P}(W^*)$$
denote the map induced by $\pi:X\to \overline{W}$.
% Consider $\pi\RamiT{|}_{\RamiAK{\hat D'}}$
 %\RamiQAA{this is not exactly restriction, maybe we should call it $\pi_0$ as in the next corollary? }
% as a map $$\pi\RamiT{|}_{\RamiAJ{\hat D'}}:\RamiAJ{\hat D'} \to\overline{W}-W= \mathbb{P}(W^*)$$
Set
}
 $\RamiAJ{E}:=\pii^{*}(Taut_{\mathbb{P}(W^{*})})$.
 %\pi\RamiT{|}_{\RamiAJ{\hat D'}}^{*}(Taut_{\mathbb{P}(W^{*})})$.
% \DrinK{
%\ If you agree with the notation $\pi_{\infty}$ then please replace $\pi_0$ %by $\pi_{\infty}$ everywhere.
% }
 Recall (see \RamiAK{\S\S\ref{ss:notation_explicit}}) that
$Taut_{\mathbb{P}(W^{*})}\subset \mathbb{P}(W^{*})\times W$; accordingly,
 we \RamiAL{have a map $E \to X \times W$} .
% \DrinK{
% Is $\RamiAJ{E}$ indeed as a sub\RamiT{manifold} of $X \times W$? I doubt %because the map $\hat D'\to X$
% is not injective.
% }
%\DrinK{
Set %$\RamiAL{\tD}$ be the disconnected union
\begin{equation}   \label{e:tDdef}
\RamiAL{\tD:=}(X\times 0) \sqcup\RamiAJ{ (\hat{D} \times 0)} \sqcup{E}\, ;
%\sqcup\coprod D'_i \times 0
\end{equation}
where $\sqcup$ stands for the disjoint union; clearly
%}
$\RamiAL{\tD}$ is a manifold equipped with a natural map
$\RamiAL{\mmup}:\RamiAL{\tD} \to X \times W$.
%\DrinK{Is the notation $\tau'$ below logical? (Should one really think of $\tau'$ as a version of $\tau$?) }
%\RamiQAA{I suggest to change $\tau'$ to $\mu'$.}
%\DrinK{
%Please try to find a new letter for this map (without the symbol "prime").
%For safety, please introduce and use a ``newcommand" for it; then it would %be easy to replace the letter by another one, if necessary.
%}
Let $\RamiAL{\mmu}:\RamiAL{\tD} \to Y \times W$ be the composition $$\RamiAL{\tD}  \overset{\RamiT{\RamiAL{\mmup}}}{\to} X \times W\overset{\tau \times Id}{\to}Y \times  W.$$
Let $\RamiAJ{\Crit}_{\RamiAL{\mmu}}$ be as in Notation  \ref{notation:S}. By Remark~\ref{r:S isotropic},
$\RamiAJ{\Crit}_{\RamiAL{\mmu}}\subset T^*(Y \times  W)$ is an isotropic closed algebraic subvariety.
%\RamiT{Recall that  $\RamiAJ{\Crit}_{\RamiAL{\mmu}}$ is the direct image of the zero section in $T^*(\RamiAL{\tD})$ with respect to $\RamiAL{\mmu}$
%(see Notation \ref{notation:S}).

Finally, define $L\subset T^*(Y \times  W^{*})$ to be the image of $\RamiAJ{\Crit}_{\RamiAL{\mmu}}$ under the symplectic isomorphism $T^*(Y \times  W){\buildrel{\sim}\over{\longrightarrow}} T^*(Y \times  W^*)$ from Lemma \ref{l:symplectic_lem}.
% \RamiQAA{Below are 2 versions of a commutative  diagram that should help to understand the map $\RamiAL{\mmu}$. Do you like any of them? \DrinK{No. They are both scary. And I don't know how to ``read" them.}
% $$\xymatrix
% {      &  Y \times \overline W \  \ar@{->}^{}[rdd] \ar@{->}_{}[ldd]&   \\
%      &   X  \ar@{->}^{\phi}[u] \ar@{->}_{\pi}[rd]  \ar@{->}^{\tau}[ld] &   \\
%    \quad \quad Y \quad \quad\ar@{<-}^{}[d]  &  \hat D\ar@{->}^{}[u] &   \overline  W  \\
%    Y\times W    &  \hat D'\ar@{->}^{}[u] \ar@{->}^{\pi_0 \text{ or } \pi\RamiT{|}_{{\hat D'}} \quad \quad \quad}[r]& \overline  W - W =\mathbb{P}(W^*)\ar@{->}^{}[u]\\       &  E %\simpAr[ru]{\qed}
%    \ar@{}_{\qed}[ru]
%     \ar@{->}_{\RamiAL{\mmu}|_{E}\quad \quad \quad}[lu]\ar@{->}^{}[luu]\ar@{->}^{}[u] \ar@{->}^{}[r]& Taut_{\mathbb{P}(W^{*})} \ar@{->}^{}[u]\\       &  W\ar@{<-}^{}[luu] \ar@{<-}^{}[u] \ar@{<-}^{}[ur]&     \\}
% $$
%
% $$\xymatrix
% {   & &  Y \times \overline W \  \ar@{->}^{}[rddd] \ar@{->}_{}[llddd]&   \\
%   \quad & \quad&X  \ar@{->}_{\pi}[rdd]  \ar@{->}_{\tau}[lldd]  \ar@{->}^{\phi}[u]  & \quad  \\
%     & X \times 0 \ar@{<-}[ru] \ar@{->}_{\quad \quad \quad \quad  \RamiAL{\mmu}|_{X \times 0} }[ldd]&   &   \\
% \quad  \quad  \quad \quad  \quad  \quad Y \quad  \quad  \quad \quad  \quad  \quad \ar@{<-}^{}[d] &  \hat D \times 0\ar@{<-}[r] \ar@{->}_{\quad \quad  \RamiAL{\mmu}|_{\hat D \times 0} }[ld]   &  \hat D\ar@{->}^{}[uu] &   \overline  W  \\
%  Y\times W  &       &  \hat D'\ar@{->}^{}[u] \ar@{->}^{\pi_0 \text{ or } \pi\RamiT{|}_{{\hat D'}} \quad \quad \quad}[r]& \overline
%    W - W    =\mathbb{P}(W^*)\ar@{->}^{}[u]\\
%   &     &  E \ar@{}_{\qed}[ru]%\simpAr[ru]{\qed}
%    \ar@{->}_{\RamiAL{\mmu}|_{E}}[llu]\ar@{->}^{}[lluu]\ar@{->}^{}[u] \ar@{->}^{}[r]& Taut_{\mathbb{P}(W^{*})}    \ar@{->}^{}[u]\\
%    & &  W\ar@{<-}^{}[lluu] \ar@{<-}^{}[u] \ar@{<-}^{}[ur]&     \\}
% $$
%}
\begin{thm} \label{exp}
We keep the notation of \S\ref{sss:thegoal}. Let $L$ be as in \S\ref{sss:Definition of L}.  Then
$WF(\cF^*(\RamiT{(\phi|_{X_0})}_*(|\omega|)))\subset L(F)$.
\end{thm}

% \DrinK{Is it possible to explicitly indicate a dense open subset of $Y \times  W^*$ over which the generalized function $\cF^*(\RamiT{(\phi|_{X_0})}_*(|\omega|))$ is smooth? There is also a similar question about Corollary~\ref{cor:exp} below.}

%\RamiP{The following theorem is an explicit version of Theorem \ref{thm:main_bb}, in particular it implies
%\RamiW{Theorem  \ref{thm:futerO}.}
%
%\begin{thm}\label{exp}
%Let $X,Y$ be  {algebraic} manifolds and $W$ a vector space over $F$,  {with}  $\dim W<\infty$.
%Let $\phi :X\to Y \times \overline{W}$ be a proper map.  \RamiT{Let $X_0:=\phi^{-1}(Y \times W)$.} Let
%$\omega$ be a top differential form on $X\RamiT{_0}$.
% Let $Z$ be the \RamiT{closure of the} zero set of $\omega$ \RamiT{in $X$}.
%Let $D=Z \cup \phi^{-1}((\overline{W}-W)\times Y)$.
%Assume that $D$ is an SNC divisor.
%\DrinK{Then
%$$WF(\cF^*(\RamiT{(\phi|_{X_0})}_*(|\omega|)))\subset L(F),$$
%where $L\subset T^*(Y \times  W^{*})$ is the isotropic subvariety defined below.
%}

%%%%Let $\nu:\hat{D} \to D$ be the normalization of $D$.
%%%%Let $D_i=\{z \in D| \#\nu^{-1}(z)\geq i\}$ and let $\hat{D_i}$ be the normalization %of $D_i$.
%Let $\RamiT{D_i}'$ be the union of those components of $\hat{D_i}$ whose image in $D_i$ is contained in
%$\phi^{-1}(\overline{W}-W)$\RamiT{.}
%%%% and  let $\hat{D_i}''$ be the union of the components of $\hat{D_i}$ whose
%%%%image in $D_i$ contained in $Z$.
%
%Let $\pi:X\to \overline{W}$ and $\tau:X\to Y$ be the  compositions of $\phi :X\to Y \times \overline{W}$ with the
%projections $Y \times \overline{W}\to \overline{W}$ and $Y \times \overline{W}\to Y$.
% Consider $\pi\RamiT{|}_{D'_i}$ as a map $\pi\RamiT{|}_{D'_i}:D'_i \to \mathbb{P}(W^*)$ and consider
%$E_i:=\pi\RamiT{|}_{D'_i}^{*}(Taut_{\mathbb{P}(W^{*})})$ as a sub\RamiT{manifold} of $X \times W$.
%
%Let $\RamiAL{\tD}$ be the disconnected union
%$$(X\times 0) \sqcup\coprod \RamiT{\hat{D}_i \times 0} \sqcup\coprod {E_i}
%%%%\sqcup\coprod D'_i \times 0
%$$
%We have a natural map
%$\RamiT{\mu}:\RamiAL{\tD} \to X \times W$. Let $\RamiAL{\mmu}$ be the composition $$\RamiAL{\tD}  \overset{\RamiT{\mu}}{\to} X
%\times W\overset{\tau \times Id}{\to}Y \times  W.$$
%\DrinK{Let $S_{\RamiAL{\mmu}}\subset T^*(Y \times  W)$ be as in Notation  \ref{notation:S}.}
%%%%\RamiT{Recall that  $S_{\RamiAL{\mmu}}$ is the direct image of the zero section in $T^*(\RamiAL{\tD})$ with respect to
%$\RamiAL{\mmu}$
%%%(see Notation \ref{notation:S}).
%\DrinK{Here I would say that $L\subset T^*(Y \times  W^{*})$ is defined to be the image of the subset
%$S_{\RamiAL{\mmu}}\subset T^*(Y \times  W)$.}



%%%}

%Then $$WF(\cF^*(\RamiT{(\phi|_{X_0})}_*(|\omega|)))\subset \RamiAJ{\Crit}_{\RamiAL{\mmu}}(F)$$
%Note that here we use the
%%%%\RamiT{symplectic}
% identification
%$$T^*(Y \times  W^{*})
%%%%\RamiT{=T^*(Y) \times  T^*(W^{*})=T^*(Y) \times  T^*(W)}
%=T^*(Y)\times  W^{*}\times  W= T^*(Y \times  W),$$
%\RamiAE{as in Lemma \ref{l:symplectic_lem}.}
%%% or Remark \ref{rem:symplectic_rem}.}
%%% \RamiT{as in lemma ??}.
%\end{thm}
%\RamiW{
%\begin{rem}
%Let us briefly explain why Theorem \ref{exp} implies Theorem  \ref{thm:futerO}. %\\
%Let the objects and the morphisms in the following commutative diagram be as in the proof of Theorem
%\ref{t:C} (which is at the end of Subsection~\ref{ss:Reduction to key}):
%  $$
%\xymatrix
%{        X   \ar@{}[r]|-*[@]{\hookrightarrow}   \ar@{->}_{\phi}[d]        & \overline X\ \ar@{->}_{\overline \phi}[d]  &
%\tilde X \ar@{->}_{\rho}[l] \ar@{->}^{\tilde \phi:=\bar \phi \circ \rho}[ld]\\
%         Y\times W \    \ar@{}[r]|-*[@]{\hookrightarrow}   &  Y\times\overline {W}  &. \\ }
%$$
%Now we can take $L:=\RamiAJ{\Crit}_{\RamiAL{\mmu}}$ where $\RamiAL{\mmu}$ is as in Theorem \ref{exp}, when we substitute $\tilde X$ and $%\tilde \phi$ for  $ X$ and $ \phi$.
%\end{rem}
%}
%%%%\RamiQD{Here I probably should explain why it implies theorem \ref{thm:futer}.}

%\newpage

Before proving \RamiW{Theorem  \ref{exp}}, {let us formulate two corollaries in the case that $Y$ is a point. In this case the wave front in question is a subset of $T^*(W^*)=W^*\times W$, and the next corollary gives an upper bound for its intersection with $\RamiAO{(W^*-0)}\times (W-0)$.
}
%observe that in the case when $Y$ is one point we get the following \RamiAK{corollaries}:

\begin{cor}\label{cor:exp}
Let $X$ be  an {algebraic} manifold and $W$ a vector space over $F$,  with  $\dim W<\infty$.
Let $\phi:X\to \overline{W}$ be a proper map.  \RamiT{Let  $X_0:=\phi^{-1}(\RamiAJ{ W})$.} Let $\omega$ be a top differential form on $X\RamiT{_0}$. Let $Z$ be the \RamiT{closure of the} zero set of $\omega$  \RamiT{in $X$}. Let $D=Z \cup \phi^{-1}(\overline{W}-W)$.
Assume that $D$ is an SNC divisor.

%Let $\nu:\hat{D} \to D$ be the normalization of $D$.
%Let $D_i=\{z \in D| \#\nu^{-1}(z)\geq i\}$ and let $\hat{D_i}$ be the normalization %of $D_i$.
Let $\RamiAJ{\hat D}$ be as in Notation~\ref{n:SNC_notation}.
Let $\RamiAJ{\hat D'}$ be the union of those components of $\RamiAJ{\hat D}$ whose image in $\RamiAJ{D}$ is contained in \RamiAJ{$X-X_{0}$}. %Let $\RamiAL{\tD}_{\RamiT{0}}$ be the disconnected union $\RamiAL{\tD}_{\RamiT{0}}=\coprod {\RamiT{D_i}'}$.
Let %$\RamiAL{\pii}$
$\RamiAL{\pii}:\RamiAJ{\hat D'}\to \overline{W}-W=\mathbb{P}(W^*)$ be the natural map.
{Define $\mathrm{PCrit}_{{\pii}}\subset\mathbb{P}(W^*)\times\mathbb{P}(W)$ to be the set of pairs $(z,H)$, where $z\in \mathbb{P}(W^*)$ and
$H\subset \mathbb{P}(W^*)$ is a projective hyperplane\footnote{Recall that a point of $\mathbb{P}(W)$ is the same as a hyperplane $H\subset\mathbb{P}(W^*)$.} containing $z$ such that $\pii :\hat D'\to \mathbb{P}(W^*)$ is not transversal\footnote{By definition, non-transversality of $\pii :\hat D'\to \mathbb{P}(W^*)$ to $H$ at $x\in \pii^{-1}(H)$
means that the image of $d_x\pii :T_x\hat D'\to T_{\pii (x)}\mathbb{P}(W^*)$ is contained in $T_{\pii (x)}H$.
}
to $H$ at some point of  $\pii^{-1}(z)$. Let $L'\subset (W^*-0)\times (W-0)$ denote the preimage of $\mathrm{PCrit}_{{\pii}}$ with respect to the map
$$(W^*-0)\times (W-0)\to\mathbb{P}(W)\times\mathbb{P}(W^*)=\mathbb{P}(W^*)\times\mathbb{P}(W).$$
Then
\begin{enumerate}
\item[(a)] \RamiAJ{$L' \subset (W^*-0) \times(W-0)$ is  an  isotropic closed algebraic subvariety.}
\item[(b)]
$WF(\RamiT{\cF^*((\phi|_{X_0})_*(|\omega|))})\cap ((W^*-0) \times(W-0)) \subset \RamiT{L'(F)}.$

%$$WF(\RamiT{\cF^*(\phi_*(|\omega|))})\cap(W^*-0) \times (W-0) \subset S(F)
%\cap(W^*-0) \times (W-0)$$
%\RamiQH{We can earaze ``$\cap(W^*-0) \times (W-0)$" from the r.h.s. Do we
%want to? }
\end{enumerate}
}
\end{cor}



{
\begin{rem}   \label{r:PCrit}
The set $\mathrm{PCrit}_{{\pii}}\subset\mathbb{P}(W^*)\times\mathbb{P}(W)$ introduced above is the ``projectivization" of the set $\Crit_{\pii}\subset T^*(\mathbb{P}(W^*))$ from
Notation~\ref{notation:S}. More precisely, $\mathrm{PCrit}_{{\pii}}$ canonically identifies with the quotient
of $\Crit_{\pii}-\mbox{\{zero section\}}$ by the action of $\mathbb{G}_m\,$.
\end{rem}
}

% Old version follows:
%
% Let $\RamiAJ{\Crit}_{\RamiAL{\pii}}\subset T^*(\mathbb{P}(W^*))$ be the isotropic subvariety defined in
% Notation~\ref{notation:S}.  \RamiAO{Let $PT^*(\mathbb{P}(W^*))$ be the projectivization  of $T^*(\mathbb{P}(W^*))$ and  $\mathrm{PCrit}_{{\pii}}$ be the image of ${\Crit}_{{\pii}}$ there. Note that for any $\ell \in \mathbb{P}(W^*)$ we have $T_{\ell}(\mathbb{P}(W^*))=\Hom(\ell,W/\ell)$. Thus $PT^{*}_{\ell}(\mathbb{P}(W^*))=(W/\ell)^*=\ell^\bot.$ This gives us an embedding $PT^*(\mathbb{P}(W^*)) \subset \mathbb{P}(W^*) \times \mathbb{P}(W)$. We also have a natural map}
% %$T^*(\mathbb{P}(W^*)) \subset \mathbb{P}(W^*) \times W^*$
% %\DrinK{I don't think so: if $l\subset W$ is a 1-dimensional subspace then %the cotangent space to $\mathbb{P}(W^*)$ at $l$ equals $\Hom (W/l,l)$, which %is
% %a subspace in $W^*\otimes l$ but not in $W^*$.} and a natural \RamiQH{} %map
%
% $$\RamiAO{(W^*-0)} \times(W-0) {\buildrel{\sim}\over{\longrightarrow}}
% (W-0) \times \RamiAO{(W^*-0)} \to \mathbb{P}(W^*) \times \RamiAO{\mathbb{P}(W)},$$  \RamiAE{where the first arrow is as in Lemma \ref{l:symplectic_lem}.}
% Let $L$ be the inverse image of $\RamiAO{\mathrm{PCrit}}_{\RamiAL{\pii}}$ in $W^* \times(W-0)$.
% %\DrinK{Is it obvious that this $L$ is isotropic? (For the $L$ from \S\ref{sss:Definition %of L} this was indeed obvious?) Maybe this should be one of the statements %of the Corollary?
% %}
%
% Then
% \begin{enumerate}
% \item \RamiAJ{$L \subset \RamiAO{(W^*-0)} \times(W-0)$ is  an  isotropic subset.}
% \item
% $WF(\RamiT{\cF^*((\phi|_{X_0})_*(|\omega|))})\cap (\RamiAO{(W^*-0)} \times (W-0)) \subset \RamiT{L(F)}.$
%
% %$$WF(\RamiT{\cF^*(\phi_*(|\omega|))})\cap(W^*-0) \times (W-0) \subset S(F)
% %\cap(W^*-0) \times (W-0)$$
% %\RamiQH{We can earaze ``$\cap(W^*-0) \times (W-0)$" from the r.h.s. Do we
% %want to? }
%\end{enumerate}
%End of old version.

% \RamiQAC{The formulation is long, I think it is possible to make it shorter by just describing the map $\mu|_{E}$ in this case and defining $L:=\Crit \mu|_{E}$. But then it will be less close to your original note and less explicit. What do you think?
%
%  I finally understand correctly the sentence  ``Note that $T^{*}\mathbb{P}_\infty$ is a quotient of a submanifold of $(W\setminus\{0\} ) \times W^*$  by an action of $\mathbb{G}_m$" from page 1 of the file ``Wave\_front\_c .pdf''. So another option is to describe the subset of $(W\setminus\{0\} ) \times W^*$ and the quotient map to $T^{*}\mathbb{P}_\infty$. It will be slightly closer to your original note, but I think it will be longer and not more explicit.
% }
%\RamiQD{Should I write a proof why the corollary follows from the theorem?}
%\RamiT{

{
\begin{proof}
Let $p:W-0\to \mathbb{P}(W^{*})$ be the canonical map. We have an isotropic closed algebraic subvariety
\begin{equation}    \label{e:p*Crit}
p^*(\Crit_{\pii})\subset T^*(W-0)=(W-0)\times W^*=W^*\times (W-0).
\end{equation}
By Remark~\ref{r:PCrit} and the definition of $L'$, we have
\begin{equation}   \label{e:formula for L'}
L'=p^*(\Crit_{\pii})\cap ((W^*-0)\times (W-0)),
\end{equation}
which proves statement (a).

Let $\RamiAJ{\tD}$ and $\RamiAL{\mmu}{:\tD\to Y\times W} = W$ be as in \S\ref{sss:Definition of L}.
By  Theorem \ref{exp},
\begin{equation}   \label{eq:2WF_bnd}
WF(\RamiT{\cF^*((\phi|_{X_0})_*(|\omega|))})\cap (W^* \times (W-0))
\subset \Crit_{\mmu} (F)\cap ((W-0) \times W^*)
\end{equation}
(here we  identify $W \times W^*$ with $W^* \times W$, just as in formula \eqref{e:p*Crit}).

Our $\tD$ was defined by formula~\eqref{e:tDdef} to be a disjoint union of three sets. It is clear that
$\mmu^{-1}(W-0)$ is contained in the third one, denoted by $E$. Moreover, one has a Cartesian square
\[
\xymatrix
{       \mu^{-1}(W-0) \ar@{->}^{}[r] \ar@{->}^{}[d]& W-0
 \ar@{->}^{p}[d]  \\
        \hat D' \ar@{->}^{\pii}[r]                                 & \mathbb{P}(W^*)  \\
}
\]
So the r.h.s. of  \eqref{eq:2WF_bnd} equals $p^*(\Crit_{\pii})$. Thus we see that
\[
WF(\RamiT{\cF^*((\phi|_{X_0})_*(|\omega|))})\cap ((W^*-0) \times (W-0))\subset p^*(\Crit_{\pii})
\cap ((W^*-0) \times (W-0)).
\]
Combining this with \eqref{e:formula for L'}, we get statement (b).
\end{proof}
}

%\begin{proof}
%Let $E$, $\RamiAJ{\tD}$, and $\RamiAL{\mmu}{:\tD\to Y\times W} = \RamiAP{W}$ be as in
%\S\ref{sss:Definition of L}.
%%%%Let $\RamiAL{\tD}'=\coprod {E_i} \subset \tD$.
%{Recall that $E\subset\tD$, see formula~\eqref{e:tDdef}.}
%  Clearly
%  \begin{equation}   \label{e:Crit=Crit}
%  \RamiAJ{\Crit}_{\RamiAL{\mmu}}\cap \RamiAP{((W-0) \times W^*)} =
% \RamiAJ{\Crit}_{\RamiAL{\mmu}|_{\RamiAJ{E}}}\cap\RamiAP{((W-0) \times W^*)}.
%  \end{equation}
%  Thus, by Theorem \ref{exp}, we know that
%  \begin{equation}   \label{eq:WF_bnd}
%WF(\RamiT{\cF^*((\phi|_{X_0})_*(|\omega|))})\cap (W^* \times (W-0))
%\subset  \RamiAJ{\Crit}_{\RamiAL{\mmu}|_{\RamiAJ{E}}}(F)\cap \RamiAP{((W-0) \times W^*),}
% \end{equation}
%\RamiAP{(note that here we identify $W \times W^*$ with $W^* \times W$).
%
 %So it is left to show that ${\Crit}_{{\mmu}|_{{E}}}\cap ((W^*-0) \times (W-0))\subset L'$.
% \DrinK{Probably this inclusion is, in fact, an equality?}
% Let $(v,w) \in {\Crit}_{{\mmu}|_{{E}}}\cap ((W-0) \times (W^*-0))$. Let $z\in \mathbb{P}(W^{*})$ be the line
%spanned by $v$ and $H \subset \mathbb{P}(W^*)$ be the projective hyperplane orthogonal to $w$. We need
%to show that $(z,H)\in\mathrm{PCrit}_{{\pii}}$. We know that there exists \DrinK{$(x,u) \in E$ such that
%$\mu (x,u)=v$ and
% $\Im d_{(x,u)} \mu \subset w^{\bot}$, where $w^{\bot}$ is the orthogonal complement of $w$ in $W$.}
%%% $(x,u) \in \mu|_E^{-1}(v) \subset  E$
%%% such that $d_{(x,u)}^{*} \mu(w)=0$.
%Clearly $x\in \pi_{\DrinK{\infty}}^{-1}(z)$. So it is left to show that $z\in H$ and
%$\Im d_{x}{\pi_\infty} \subset T_zH$. This follows from the fact that $\Im d_{(x,u)} \mu \subset w^{\bot}$.
%}


% \RamiQH{We can erase ``$\cap W^* \times (W-0)$" from the r.h.s. Do we want to? }
% Now all it is left to show is that $\RamiAJ{\Crit}_{\RamiAL{\mmu}|_{\RamiAJ{E}}}\cap (W^* \times (W-0))=L$. For this, note that $\RamiAJ{E}$ is a line bundle  over $\RamiAJ{\hat D'}$. The assertion follows now from the fact that $ \RamiAJ{\hat D'}=\RamiAL{\pii^*}( Taut_{\mathbb{P}(W^{*})})$ and from the fact that the map $\RamiAL{\mmu}$ is induced from $\RamiAL{\pii}$ in the obvious way.
%\end{proof}
%}



\begin{cor}   \label{c:explcit_open}
Let $X,W,\phi,\omega,{\hat D'}$ and $\RamiAL{\pii}$ be as in  Corollary \ref{cor:exp}.
Define $U$ to be the set of all $\ell \in W^*-0$ such that the map $\RamiAL{\pii}:{\hat D'}\to  \mathbb{P}(W^*)$ is transversal to the hyperplane $H_{\ell}\subset  \mathbb{P}(W^*)$ corresponding to $\ell$.
%$H_{\ell}$, where $H_{\ell}\subset  \mathbb{P}(W^*)$ is the hyperplane corresponding to $\ell$.
Then $\cF^*((\phi|_{X_0})_*(|\omega|))|_{U(F)}$ is smooth.

%Any $\ell \in W^*-0$  defines a %projective
%hyperplane  $H_{\ell}:= \mathbb{P}((\ell^{\bot} )^*) \subset  \mathbb{P}(W^*).$

%% Let $X$ be  an {algebraic} manifold and $W$ a vector space over $F$,  with  $\dim W<\infty$.
%% Let $\phi:X\to \overline{W}$ be a proper map.  \RamiT{Let  $X_0:=\phi^{-1}(\RamiAJ{ W})$.}
%%Let $\omega$ be a top differential form on $X\RamiT{_0}$. Let $Z$ be the \RamiT{closure of the} zero
%%set of $\omega$  \RamiT{in $X$}. Let $D=Z \cup \phi^{-1}(\overline{W}-W)$.
%% Assume that $D$ is an SNC divisor.
%%
%% Let $\RamiAJ{\hat D}$ be as in Notation~\ref{n:SNC_notation}.
%% Let $\RamiAJ{\hat D'}$ be the union of those components of $\RamiAJ{\hat D}$ whose image in
%%$\RamiAJ{ D}$ is contained in \RamiAJ{$X-X_{0}$}. %Let $\tD_{\RamiT{0}}$ be the disconnected union $\tD_{\RamiT{0}}=\coprod {\RamiT{D_i}'}$.
%%  Let $\pi_{\RamiT{0}}$ be the natural map
%%$\pi_{\RamiT{0}}:\RamiAJ{\hat D'}\to \overline{W}-W=\mathbb{P}(W^*)$.

%%Let $G:=\pi_0^*(Taut_{\mathbb{P}(W^{*})}^{\bot})$. We have a natural map
%% $\tilde\pi_0:G\to W^*$. Let $U\subset W^*$ be the set of regular values %of this map.

  %Let
 %$$U:=\{\ell \in W^*-0| \text{ the map $\pi_0$ is transversal to } H_{\ell}\}.$$
 % Or more explicitly
 %$$U:=\{\ell \in W^*-0| \forall x \in {\pi_0}^{-1}(H_{\ell}), \text{ we have } \Im(d_x\RamiAL{\pii})+  T_{\RamiAL{\pii}(x)} H_{\ell} =
  %T_{\pi_0(x)} \mathbb {P}(W^*)\}.$$
\end{cor}

%Recall that transversality in the definition of $U$ means that
%$\Im(d_x\RamiAL{\pii})+  T_{\RamiAL{\pii}(x)} H_{\ell} = T_{\RamiAL{\pii}(x)} \mathbb {P}(W^*)$ for all
%$x \in \RamiAL{\pii^{-1}}(H_{\ell})$. \RamiQAD{I think the last sentence is unnecessary now.}
It is clear that $U$ is Zariski open in $W^*- 0$. %$W^*-\{ 0\}$.

%\RamiQAD{ In the proof we use only formula \eqref{eq:WF_bnd} from the last corollary. This formula almost
%did not change and the proof still works. I do not see an obvious way to make a shorter proof in view of the
%new formulation of the last corollary, so I suggest to leave it as it is.}


\begin{proof}
{
By Corollary~\ref{cor:exp}, $\cF^*((\phi|_{X_0})_*(|\omega|))$  is smooth on the open subset
\begin{equation} \label{e:open_of_smoothness}
(W^*-0)-q^{-1}(p(\mathrm{PCrit}_{{\pii}}))\subset W^*-0,
\end{equation}
where $q:W^*-0\to  \mathbb {P}(W)$ and $p:\mathbb {P}(W^*) \times \mathbb {P}(W) \to \mathbb {P}(W)$ are the projections and $\mathrm{PCrit}_{{\pii}}\subset\mathbb{P}(W^*)\times\mathbb{P}(W)$ is as in Corollary~\ref{cor:exp}.
The subset \eqref{e:open_of_smoothness} clearly equals $U$.
}
\end{proof}
%
%\RamiAL{We keep the notations of the proof of Corollary \ref{cor:exp}.
%
%Let $p:W^* \times W \to W^*$ be the projection.
%By \eqref{eq:WF_bnd}, we know that $\cF^*((\phi|_{X_0})_*(|\omega|))$  is smooth outside
%$Z:= p(\Crit_{\RamiAL{\mmu}|_{\RamiAJ{E}}} \cap W^* \times (W-0))$.
%So it is enough to show that $Z \cap U=\emptyset.$  Let $\ell \in U.$ By the definition of $U,$ for any
%$x \in {\mmu|_E}^{-1}(\ell ^{\bot} )-0$ we have $\Im(d_x \tilde\mmu)+  \ell ^{\bot}  =  W.$ Therefore
%$d_x^{*} \mmu(\ell ) \neq 0.$ Thus  $(\mmu(x),\ell )\notin \Crit_{\mmu}$. This implies that $x \notin Z.$
%}

%??To be edited??
%Let $\RamiAL{\tilde \pi_\infty}:=\RamiAL{\mmu}|_{E}:E\to W$, where $\RamiAL{\mmu}$ %and $E$ are as in  \RamiAL{\S\S}\S\ref{sss:Definition of L}.
% .  be the natural map.
% As in Corollary  \ref{cor:exp}, we have
%$$\RamiAJ{\Crit}_{\RamiAL{\mmu}}\cap (W^* \times (W-0)) = \RamiAJ{\Crit}_{\tilde %\pi_0}\cap(W^* \times (W-0)).$$
%  Note that it is equal to $\RamiAL{\mmu}|_E$ where $\RamiAL{\mmu}$ is as %in  \S\ref{sss:Definition of L}. We have:
%\DrinK{Can we just refer here to formula \eqref{e:Crit=Crit} instead of %rewriting it?}

% \begin{multline*}
% U:=\{\ell  \in W^*-0| \forall x \in {\tilde \pi_0}^{-1}(\ell ^{\bot} )-0 \text{, we have } \Im(d_x \tilde\pi_0)+  \ell ^{\bot}  =  W\}=\\=\{\ell \in W^*-0| \forall x \in {\tilde \pi_0}^{-1}(\ell ^{\bot} )-0, \text{ we have } d_x^{*} \tilde\pi_0(\ell ) \neq 0\}=\\=\{\ell \in W^*-0| \forall x \in {\tilde \pi_0}^{-1}(\ell ^{\bot} )-0, \text{ we have } (\tilde\pi_0( x),\ell )\notin \Crit_{\tilde\pi_0}\}=\\=\{\ell  \in W^*-0| \forall y \in W-0,  \text{ we have } ( y,\ell )\notin \Crit_{\tilde\pi_0}\}=\\=\{\ell  \in W^*-0| \forall y \in W-0,  \text{ we have } ( y,\ell )\notin \Crit_{\RamiAL{\mmu}}\}.
%  \end{multline*}
%
% \DrinK{Is it possible to write the above proof in words? The above sequence of equalities is hard to read.}
%
% The assertion follows now from \RamiW{Theorem  \ref{exp}}.% and the fact that.



% \RamiQZ{
% This is essentially the same statement as you wrote in the file ``Kazhdan-Hrushovski\_via\_resolution2.pdf"
%  (when the former is interpreted correctly).
%
%  In order to make this corollary follow immediately from Theorem \ref{exp},
%   we need to give an alternative (equivalent) description of $L$, where
%    the bundle $E:=\pi\RamiT{|}_{\RamiAJ{\hat D'}}^{*}(Taut_{\mathbb{P}(W^{*})})$
%     is replaced with  $G:=\pi\RamiT{|}_{\RamiAJ{\hat D'}}^{*}(Taut_{\mathbb{P}(W^{*})}^{{\bot}}).$
%      In fact, our proof will be shorter if we will use this alternative
%       definitions.
%
% I suggest giving both descriptions of $L$ and stating that they are equivalent. %Then this corollary will follow immediately.
%}


\subsection{Proof of Theorem~\ref{exp}}
%In order to prove theorem \ref{exp}, we
We will proceed in stages analogous to the stages of the proof of  theorem \ref{thm:main_bb}.

\begin{lemma}\label{lem:exp_bn}
Let $Y$ be an algebraic manifold.
Let $p :Y\to F$ be a regular function such that
 $p \neq 0$ on a dense subset \RamiT{$Y_0$}. Let $\omega$ be a \RamiT{rational} top differential form \RamiT{on $Y$ which is regular on $Y_0$}. Let $D$ be the union of the zero sets of $\omega$ and of $p$. Assume that $D$ is an SNC divisor. {Just as in Notation~\ref{not:eta}, let
%Considering $\frac{1}{p}$ as a map $Y \to \mathbb{P}^1=\bar{\A}^1$, we  get
$\eta_{\frac{1}{p},\omega}\in\Sc^*(Y(F)\times F)$ denote the pushforward of $|\omega |$ with respect to
the map $Y_0\to Y\times F$ given by $y\mapsto (y,p(y)^{-1})$.
Let
${\hat \eta}_{\frac{1}{p},\omega}\in {\Sc^*}(Y(F)\times F, {D^{Y(F)\times F}_{Y (F)}})$ be its partial Fourier transform.
}

Let $\RamiAJ{\hat D}$ be as in Notation~\ref{n:SNC_notation}.
\RamiT{Let $\RamiAJ{\hat D'}$ be the union of those components of $\RamiAJ{\hat D}$ whose image in $\RamiAJ{ D}$ is contained in} {the zero set of $p$.}
%$\phi^{-1}(\overline{W}-W)$.
%\DrinK{What is $\phi$, $W$, $\overline{W}$? What is $\phi^{-1}(\overline{W}-W)$ below?
%Probably $\phi=1/p$, $(\overline{W}-W)=\{\infty\}$, so
%$\phi^{-1}(\overline{W}-W)$ is the set of zeros of $p$??? }
%Let $\nu:\hat{D} \to D$ be the normalization of $D$. Let $D_i=\{z \in D| \#\nu^{-1}(z)=i\}$. %Let $Z_0=Y-Z$.

Let $\RamiAL{\widetilde H}:= (\RamiAJ{Y}\times F)  \sqcup  \RamiT{(\RamiAJ{\hat D}\times F)} \sqcup
 \RamiT{(\RamiAJ{\hat D}' \times 0)}$. We have a natural map $\RamiT{\nu}:\RamiT{\RamiAL{\widetilde H}} \to Y \times F$.

Then
$$WF({\hat \eta}_{\frac{1}{p},\omega})\subset \RamiAJ{\Crit}_{\RamiT{\nu}}(F)$$

%\RamiT{=(Y\times F)\cup \bigcup CN_{\RamiT{\hat{D}_i \times F} }^{Y\times %F} \cup \bigcup CN_{D'_i \times 0}^{Y\times F}(F)}

\end{lemma}
\RamiT{
\begin{proof}
Just as in the proof of lemma \ref{lem:sp_case}, it suffices to consider the case where
%We follow the proof of lemma \ref{lem:sp_case}. Namely, as in Corollary \ref{cor:homo}, the  set
%$WF({\hat \eta}_{\frac{1}{p},\omega})$ depends only on the divisors of $p$ and $\omega$ and as in
%Lemma \ref{prop:loc}\eqref{prop:loc:2}, the statement is local in the \et topology.
%
%OOPS! In one direction, the statement is Zariski-local, in the other direction it is stale local, just as in
%Lemma \ref{prop:loc}\eqref{prop:loc:2}
%
%Thus we can assume that
$Y$ is an affine space and $p$ and $\omega$ are given by monomials. In this case the statement follows from
Lemma \ref{lem:equv}
and Proposition \ref{prop:WF_prop}\eqref{it:G_act}. \end{proof}
}
%\RamiQD{This lemma is proved the same way as Lemma \ref{lem:dim_1} and step %3 of the proof of Proposition \ref{thm:main_c}. Is it enough to say this %or should I give a full proof?}

%\begin{notation}
%Let $V$ be a finite dimensional vector space over $F$. Denote by $\mathbb{P}(V)$ the space of hyperplanes in $V$. Let $Taut_{\mathbb{P}(V)}$ be the tautological bundle of $\mathbb{P}(V)$  (under this realization). More explicitly, $$Taut_{\mathbb{P}(V)}
%:=\{(H,v)|H\subset V \text{ is a hyperplane,  and } v \in H\}.$$ Note that $\overline{V}-V\cong \mathbb{P}(V^*)$.
%\end{notation}

\begin{prop}\label{exp_1}
Let $Y$ be an  {algebraic} manifold and $W$ a vector space over $F$,  {with}  $\dim W<\infty$.
Let $\phi :Y\to \overline{W}$ be a regular map. \RamiT{Let $Y_0:=\phi^{-1}( W)$.}  Let $\omega$ \RamiT be a {rational} top differential form \RamiT{on $Y$ which is regular on $Y_0$}. Let $Z$ be the zero set of $\omega$. Let $D=Z \cup \phi^{-1}(\overline{W}-W)$. Assume that $D$ is an SNC divisor.
%Recall that we have $\eta_{\phi,\omega}\in\Sc^*(Y {(F)}\times W)$ and
%${\hat \eta}_{\phi,\omega}\in {\Sc^*}(Y {(F)}\times W^{*},{D^{Y(F)\times W^{*}}_{Y (F)}})$.
%\DrinK
{Just as in Notation~\ref{not:eta}, let
$\eta_{\phi,\omega}\in\Sc^*(Y {(F)}\times W)$ denote the pushforward of $|\omega |$ with respect to
the map $Y_0\to Y\times W$ given by $y\mapsto (y,\phi (y))$.
Let
${\hat \eta}_{\phi,\omega}\in {\Sc^*}(Y {(F)}\times W^{*},{D^{Y(F)\times W^{*}}_{Y (F)}})$ be its partial Fourier transform.
}


%Let $\nu:\hat{D} \to D$ be the normalization of $D$. Let $D_i=\{z \in D| \#\nu^{-1}(z)=i\}$. Decompose $D_i=D_i' \cup D_i''$, where $D_i' = D_i \cap \phi^{-1}(\overline{W}-W)$ and $D_i = D_i'' \cap Z$.
Let $\RamiAJ{\hat D}$ be as in Notation~\ref{n:SNC_notation}.
Let $\RamiAJ{\hat D'}$ be the union of those components of ${\RamiAJ{\hat D}}$ whose image in $\RamiAJ{ D}$ is contained in
$\phi^{-1}(\overline{W}-W)$\RamiT{.}
%and  let $\hat{D_i}''$ be the union of the components of $\hat{D_i}$ whose
%image in $D_i$ contained in $Z$



Consider $\phi_{\RamiAJ{\hat D'}}$ as a map \RamiAJ{$\phi_{\hat D'}:\hat D' \to \mathbb{P}(W^*)$} and  \RamiAJ{$G:=\phi^{*}_{\hat D'}(Taut_{\mathbb{P}(W^{*})}^\bot)$} as a subvariety of $Y \times W^*$. Let
$\RamiAL{\widetilde H}:= (Y\times W^{*})\sqcup  \RamiT{(\RamiAJ{\hat D} \times W^{*})} \sqcup G
%\sqcup \coprod D'\RamiT{'}_i \times W^{*}
$.
We have a natural map $\RamiT{\nu}:\RamiAL{\widetilde H} \to Y \times W^{*}$.


%Let $Z_0=Y-Z$.

Then
$WF({\hat \eta}_{\phi,\omega})\subset \RamiAJ{\Crit}_{\RamiT{\nu}}(F)$.
%\RamiT{=(Y\times W^{*})\cup \bigcup CN_{\hat{D}_i \times W^{*}}^{Y\times %W^{*}} \cup \bigcup CN_{G_{i}}^{Y\times W^{*}}(F)}
\end{prop}

\RamiP{
\begin{proof}
{We follow the proof of Proposition  \ref{thm:main_c}. The claim is local, so we can reduce the problem to the case when $\phi=(f:p)$ where $f:Y \to W$ and $p:Y\to F$ are regular functions and one of them never vanishes. Let us analyze the two cases:
\begin{itemize}
 \item The case when  $p$ never vanishes.\\
By Lemma \ref{lem:exp_comp}.
$$WF({\hat \eta}_{\phi,\omega}) =WF(f_{\phi} \cdot  {pr_{W}}^{*}(|\omega|))=WF({pr_{W}}^{*}(|\omega|))={pr_{\RamiS{W}}}^{*}(WF(|\omega|)).$$
 The assertion follows now from Lemma \ref{lem:exp_bn} (after noticing that in this case $D_i'$ and $G_i$ are empty).

 \item \RamiF{The case  \RamiI{when}  {$f$ never vanishes}.}\\
 %we may assume, without loss of generality, that $i=1$.
%Since $pr|_{\tilde{W}_i}:\tilde{W}_i \to {W}_i$ is an isomorphism, we can rewrite $\phi|_{U_{i}}$ as
 %${\phi}|_{U_{i}}=pr \circ \tilde{\phi}$, where $\tilde{\phi}:U_{i} \to F \oplus W$ is given by
%$\tilde{\phi}(x) = (p(x),f(x))$.
 By Lemma \ref{lem:red_to_dim_1}, we have a submersion
$g:Y \times W^{{*}} \to Y \times F$ such that
 $\hat \eta_{\phi,\omega}  =g^*({\hat \eta_{\frac{1}{{p}},\omega}}).$
% \DrinK{I replaced here $W$ by $W^*$. Am I right?}
So by Proposition \ref{prop:WF_prop2} \eqref{prop:WF_prop2:2}, $WF(\hat \eta_{\phi,\omega})\subset g^*(WF({\hat \eta_{\frac{1}{{p}},\omega}}))$.
Let $\RamiAL{\widetilde H}^1$ and $\nu^1:\RamiAL{\widetilde H}^1 \to Y \times F$ be the variety $\RamiAL{\widetilde H}$ and map $\nu$ from Lemma \ref{lem:exp_bn}.  By Lemma \ref{lem:exp_bn}, it is enough to show that $\RamiAJ{\Crit}_\nu= g^*(\RamiAJ{\Crit}_{\nu^1})$.
Let \RamiAJ{$\hat{D}_{j} ,\hat D'_{j}\subset Y$ and $  G_{j}\subset Y \times W^*$} be the components of \RamiAJ{$\hat{D},\hat D'$ and $G$}.   Note that
$$\RamiAJ{\Crit}_{\nu^1}
=(Y\times F)\cup \bigcup CN_{\RamiT{\hat{D}_{\RamiAJ{j}}\ \times F} }^{Y\times F} \cup \bigcup CN_{\RamiAJ{\hat D'_{j}} \times 0}^{Y\times F}$$
and
\begin{equation}   \label{e:S_nu}
\RamiAJ{\Crit}_{\nu}
=(Y\times W^{*})\cup \bigcup CN_{\hat{D}_{\RamiAJ{j}} \times W^{*}}^{Y\times W^{*}} \cup \bigcup CN_{G_{\RamiAJ{j}}}^{Y\times W^{*}}
\end{equation}



The assertion follows now from the fact that $G_{\RamiAJ{j}}=g^{-1}(\RamiAJ{\hat D_{j}} \times 0) $.
%this followos from lemma ?? inview of the following cortizian squaer:
%$$
%\xymatrix
%{        \tilde H   \ar@{->}_{\nu}[r]   \ar@{->}_{\phi}[d]        &  Y\times %W^{*}\ \ar@{->}_{g}[d] \\
%         \tilde H_1 \    \ar@{->}_{\nu_1}[r]   &  Y\times F  \\ }
%$$
%So, by Lemma ?? it is enough to show that
%\begin{multline*}
%(Y\times W^{*})\cup \bigcup CN_{D_i\times 0}^{Y\times W^{*}} \cup \bigcup %CN_{G_{i}}^{Y\times W^{*}} \cup \bigcup CN_{D''_i \times W^{*}}^{Y\times %W^{*}}=\\=g^{*}((Y(F)\times F)\cup \bigcup CN_{D_i \times 0}^{Y\times F} %\cup \bigcup CN_{D_i\times F}^{Y\times F}(F))
%\end{multline*}

\end{itemize}
}
%\RamiQD{I probably need to give a proof here}
\end{proof}

%Using Lemma \ref{l:symplectic_lem} one obtains the following corollary:
}

%\DrinK{You have proved the above Proposition by showing that $g^*(S_{\nu^1})=S_\nu$. In the next
%Corollary you use Lemma \ref{l:symplectic_lem} to show that $\RamiAJ{\Crit}_{\nu}= \rho(S_{\RamiT{\pi}})$. I wonder if it is
%possible to prove the equality $g^*(S_{\nu^1})=\rho(S_{\RamiT{\pi}})$ directly without using
%Lemma \ref{l:symplectic_lem}.
%
%One could try as follows. Hopefully, $g^*(S_{\nu^1})=S_{\lambda}$, where
%$$\lambda :(Y \times W^*)\times_{Y \times F}\tilde H^1\to Y \times W^*$$
%is obtained from $\nu^1:\tilde H^1 \to Y \times F$  by base change with respect to the submersion
%$g:Y \times W^* \to Y \times F$. If so then it remains to show that $S_{\lambda}=\rho(S_{\RamiT{\pi}})$, where
%$\pi:\tD \to Y \times W$ is defined in the next Corollary. But now I don't see why this should be true.
%}

\begin{cor}   \label{c:based on symplectic lemma}
In the notations of Proposition \ref{exp_1},
let $\RamiAJ{E}:=\phi^{*}_{\RamiAJ{\hat D'}}(Taut_{\mathbb{P}(W^{*})})$ as
 a subvariety of $Y \times W$.
Let $\RamiAL{\tD}:= (Y\times 0)\sqcup  \RamiAJ{(\hat D \times 0)  \coprod E}\,$.
 %\sqcup \coprod D'_i \times 0$.
We have a natural map $\RamiAL{\mmu}:\RamiAL{\tD} \to Y \times W$. Let
$\rho:T^*(Y\times W) {\buildrel{\sim}\over{\longrightarrow}} T^*(Y\times
 W^*)$
be the standard identification (\RamiAE{as in Lemma \ref{l:symplectic_lem}}).



%Let $Z_0=Y-Z$.

Then
$WF({\hat \eta}_{\phi,\omega})\subset \rho(\RamiAJ{\Crit}_{\RamiAL{\mmu}})(F)$.
%Let $Z_0=Y-Z$.

%Then
%$$WF({\hat \eta}_{\phi,\omega})\subset (Y(F)\times W^{*})\cup \bigcup CN_{D_i(F) %\times W}^{Y(F)\times W^{}} \cup \bigcup CN_{E_i(F)}^{Y(F)\times W^{}}\cup %\bigcup CN_{D'_i(F) \times 0}^{Y(F)\times W^{}}$$
\end{cor}

\begin{proof}
{By Proposition~\ref{exp_1}, it suffices to}
\RamiAI{
show that $\RamiAJ{\Crit}_{\nu}
= \rho(\RamiAJ{\Crit}_{\RamiAL{\mmu}})$. Combining formula~\eqref{e:S_nu} with
%As before, $$S_{\nu}=(Y\times W^{*})\cup \bigcup CN_{\hat{D}_{ij} \times W^{*}}^{Y\times W^{*}}
% \cup \bigcup CN_{G_{ij}}^{Y\times W^{*}}.$$ By
Lemma \ref{l:symplectic_lem}, we get
$$\RamiAJ{\Crit}_{\nu}
%(Y\times W^{*})\cup \bigcup CN_{\hat{D}_{ij} \times W^{*}}^{Y\times W^{*}} \cup
% \bigcup CN_{G_{ij}}^{Y\times W^*}
=\rho\left( (Y\times 0)\cup \RamiAJ{\bigcup CN_{\hat{D}_{j} \times0}^{Y\times W} \cup \bigcup CN_{G_{j}^{\bot}}^{Y\times W}}\ \right).$$
%\RamiQX{Do you prefer 3 short equalities here instead of the one long equality above?}
Clearly, $\RamiAJ{E=G^{\bot}\subset \hat D \times W.}$ Thus
$ G_{\RamiAJ{j}}^{\bot}$ are the components of $\RamiAJ{E}$. Therefore,
$(Y\times 0)\cup \ \bigcup CN_{\hat{D}_{\RamiAJ{j}} \times0}^{Y\times W} \cup \bigcup CN_{G_{\RamiAJ{j}}^{\bot}}^{Y\times W} = \RamiAJ{\Crit}_{\RamiAL{\mmu}}\,$.
}
\end{proof}

From Corollary~\ref{c:based on symplectic lemma} one deduces the following statement (\RamiT{using Proposition \ref{prop:WF_prop2}\eqref{prop:WF_prop2:3} in the same way as in the proof of Theorem \ref{t:C}}):
\RamiT{
\begin{cor}
In the situation of Theorem \ref{exp}, we have
$$WF(\cF^*((\phi|_{X_0})_*(|\omega|)))\subset (\tau \times \id_{W^*})_{*}(\rho(\RamiAJ{\Crit}_\RamiAL{\mmup}))(F)$$
where $\RamiAL{\mmup}:\RamiAL{\tD} \to X \times W$ and $\tau:X\to Y$ are as in \S\ref{sss:Definition of L}\ and $\rho:T^*(Y\times W) {\buildrel{\sim}\over{\longrightarrow}} T^*(Y\times W^*)$ is the standard identification (\RamiAE{as in Lemma \ref{l:symplectic_lem}}). \qed
\end{cor}
}

% (Long formulation)\begin{cor}
% (Long formulation)Let $X,Y$ be  {algebraic} manifolds and $W$  a vector space over $F$,  {with}  $\dim W<\infty$.
% (Long formulation)Let $\phi :X\to Y \times \overline{W}$ be a proper map.  Let $\omega$ be a top differential form on $X$. Let $Z$ be the zero set of $\omega$. Let $D=Z \cup \phi^{-1}(Y \times (\overline{W}-W))$.
% (Long formulation)Assume that $D$ is an SNC divisor.
% (Long formulation)
% (Long formulation)
% (Long formulation)
% (Long formulation)%Let $\nu:\hat{D} \to D$ be the normalization of $D$. Let $D_i=\{z \in D| \#\nu^{-1}(z)=i\}$.
% (Long formulation)%Decompose $D_i=D_i' \cup D_i''$, where $D_i' = D_i \cap  \pi^{-1}(\overline{W}-W)$ %and $D_i'' = D_i \cap Z$.
% (Long formulation)Let $\RamiT{D_i}'$ be the union of the components of $\hat{D_i}$ whose image in $D_i$ contained in $\phi^{-1}(Y \times\overline{W}-W)$\RamiT{.}
% (Long formulation)
% (Long formulation)
% (Long formulation)Let $\pi:X\to \overline{W}$ be the composition of $\phi$ with the projection to $ \overline{W}$ and $\tau:X\to Y$ be the composition of $\phi$ with the projection to $ Y$.
% (Long formulation) Consider $\pi_{D'_i}$ as a map $\pi_{D'_i}:D'_i \to \mathbb{P}(W^*)$ and consider $E_i:=\pi^{*}_{D'_i}(Taut_{\mathbb{P}(W^{*})})$ as a subspace of $X \times W$.
% (Long formulation)
% (Long formulation)Let $\tD$ be the disconnected union
% (Long formulation)$$(X\times 0) \sqcup\coprod {D_i \times W} \sqcup\coprod {E_i}\sqcup\coprod D'_i \times 0$$
% (Long formulation)We have a natural map $\mu:\tD \to X \times W^*$.
% (Long formulation)%Let $\RamiAL{\mmu}$ be the composition $$\tD  \overset{i}{\to} X \times W^*\overset{\tau %\times Id}{\to}Y \times  W^*.$$
% (Long formulation)%Consider $\tD \subset T^* \tD$ as the zero section.
% (Long formulation)
% (Long formulation)%Let $\pi:X\to \overline{W}$ be the composition of $\phi$ with the projection %to $ \overline{W}$ and $\tau:X\to Y$ be the composition of $\phi$ with the %projection to $ Y$.
% (Long formulation)%Let $\RamiAL{\mmu}:=\tau\times Id:X\times W^* \to Y\times W^*$
% (Long formulation)
% (Long formulation)Then $$WF(\cF^*(\phi_*(|\omega|)))\subset (\tau \times \id_{W^*})_{*}(\rho(S_\mu))(F)$$
% (Long formulation)
% (Long formulation)where $\rho:T^*(Y\times W) \to T^*(Y\times W^*)$ be the standard identification as before.
% (Long formulation)%Then $$WF(\phi_*(|\omega|)})\subset \RamiAL{\mmu}_{*}\left ((X\times W)\cup \bigcup %CN_{D_i \times 0}^{X\times W} \cup \bigcup CN_{E_i}^{X\times W}\cup \bigcup %CN_{D'_i \times W}^{X\times W}\right )$$
%\end{cor}

Theorem \ref{exp} follows from this corollary in view of the following lemma:


\begin{lemma}
Let $X,Y$ be manifolds and $V$ be a vector space. Let $A \subset T^*(X \times V) = T^*(X \times V^*)$  be a subset. Let $\phi:X \to Y$ be a map. Then $(\phi \times Id_V)_*(A)=(\phi \times Id_{V^*})_*(A)$.

Note that in the left hand side $A$ is considered as a subset in $T^*(X \times V)$ and in the right hand side $A$ is considered as a subset in $T^*(X \times V^*)$. The equality is under the standard identification $T^*(Y \times V) = T^*(Y \times V^*)$.
\end{lemma}
%\RamiQD{I think that the lemma should stay here since it too specific for % \S\ref{sec:WF-Holonomic} and \S\ref{sec:prel}, however if we put an appropriate %remark after Definition \ref{def:dir_and_inv_image}, the proof becomes shorter.}
\begin{proof}
%For the proof, let us first explain the procedure of direct image in terms %of symplectic geometry. Namely, let $\pi: M \to N$ be a map of manifolds.
%It gives rise to a correspondence $\Lambda_\pi \subset T^*(M) \times T^*(N)$ %by $\Lambda_\pi=\{((x,v),(y,w))|y=\pi(x), v=d\pi^*(w) \}$. Now let $S$ and %$T$ be
% a symplectic manifold and $\Lambda \subset S \times T$ be a  correspondence.
% For a subset $Z \subset S$, we set $\Lambda(Z)=\{y\in T|\exists x\in Z %\text{ such that }(x,y)\in \Lambda\}$.
%For a subset $Z \subset T^*(M)$, we have $\pi_*(Z)=\Lambda_\pi(Z)$.

The \RamiAE{lemma  follows from Remark \ref{rem:simp_gem} and} the equality
$$\Lambda_{\phi \times Id_V}=\Lambda_{\phi \times Id_{V^*}}\, ,$$
where $\Lambda_{\phi \times Id_V}$ and $\Lambda_{\phi \times Id_{V^*}}$ have the same meaning as in
Remark \ref{rem:simp_gem}.
\end{proof}
%}
%\RamiT{

%\newpage


\section{Proof of Theorem \ref{thm:futer} \RamiAF{in the non-Archimedean case}}\label{sec:pf_futer}
{In this section we deduce Theorem \ref{thm:futer} from Theorem \ref{thm:futerO} assuming that the local field $F$ is non-Archimidean. A slight modification of the same argument allows to prove Theorem~\ref{thm:futer} in the Archimedean case as well, see \S\ref{ssec:arc_pf} below.}

\medskip

$ $
%\RamiW{Theorem \ref{thm:futer} follows from Theorem \ref{thm:futerO} using %the following standard lemma:}
%\RamiQT{The following  text (until Lemma \ref{lem:geom_1}) is probably obsolete.\\ %}
%
\RamiAbs{
\RamiY{We will need the following lemma:}
\begin{lemma}\label{lem:part_Fo}
Let $X$ be an analytic manifold, let $f$ be an analytic function on it and $\xi$ be a distribution on it.
\RamiAA{Let $i:X\to X\times F$ be defined by $i(x):=(x,f(x))$.}
%Let $i:X \to X \times F$
% be the graph of $f$.
 Let $\eta:=\RamiW{\cF\RamiAA{^*}_F(i_*\xi)}$. Then
 \begin{enumerate}
 \item \label{lem:part_Fo:1}  $WF(\eta) \subset  T^*(X) \times F \subset T^*(X \times F)$
 \item \label{lem:part_Fo:2} $j^*(\eta)=(\psi \circ f)\cdot\xi $, where $j:X \to  X \times F$ is given by $j(x)=(x,1)$. \RamiQO{Maybe it should be $-1$, I need to check. } Note that $j^*(\eta)$ is defined by virtue of \eqref{lem:part_Fo:1} and Proposition~\ref{prop:WF_prop3}.
 \end{enumerate}
  \end{lemma}

  \begin{proof}
  $ $
 \begin{enumerate}
 \item Follows from the definition of the wave front set.
 \item By continuity, it \RamiAA{is} enough to consider the case when $\xi$ is a Schwartz \RamiAA{function}. In this case, the assertion is obvious.
 \end{enumerate}
  \end{proof}


\RamiY{
  \begin{cor}\label{cor:part_Fo}
Let $X,Y$ be analytic manifolds, let $f$ be an analytic function on $X$ and $\xi$ be a distribution on $X$. Let $\phi:X \to Z$ be a proper map. Let
 $\phi':=\phi\times f:X \to Z \times F$
 Let $\eta:=\cF_F(\phi'_*(\xi))$. Then
 \begin{enumerate}
 \item \label{cor:part_Fo:1}  $WF(\eta) \subset  T^*(Z) \times F \subset T^*(Z \times F)$
 \item \label{cor:part_Fo:2} $\RamiAA{k}^*(\eta)= \RamiAA{\phi_{*}}((\psi \circ f)\cdot\xi)$, where $\RamiAA{k}:Z \to  \RamiAA{Z} \times F$ is given by $\RamiAA{k}(\RamiAA{z})=(\RamiAA{z},1)$. Again $j^*(\eta)$ is defined by virtue of \eqref{lem:part_Fo:1} and Proposition~\ref{prop:WF_prop3}.
 \end{enumerate}
  \end{cor}
    \RamiAA{
  \begin{proof}
  Let $i:X\to X\times F$ and $j:X\to X \times F$ be as in Lemma \ref{lem:part_Fo}. Then
$\eta:=\cF^{\RamiAA{*}}_F(\phi'_*\xi )$ is equal to the direct image of $\RamiAL{\mmup}:=\cF^{\RamiAA{*}}_F(i_*\xi)$ with
respect to the map $\alpha: X\times F\to Z\times F$ induced by $\phi :X\to Z$. By lemma \ref{lem:part_Fo} we have $WF(\mu) \subset  T^*(X) \times F \subset T^*(X \times F)$ and $(\psi \circ f)\cdot\xi =j^{*}(\mu)$. By Proposition \ref{prop:WF_prop4} this implies  both assertions.
  \end{proof}
  }
%  }

\RamiQO{
Here is an alternative proof of Corollary \ref{cor:part_Fo}, which does not use Lemma \ref{lem:part_Fo} and Proposition \ref{prop:WF_prop4}. First we need the following lemma:


\begin{lemma}\label{lem:part_Fo2}
Let $X$ be an analytic manifold. Let  $\xi$ be a distribution on $X \times F$ such that $p|_{\supp(\xi)}$ is proper, where $p:X \times F$ is the projection.\\ Let $\eta:=\cF_F(\xi)$. Then
 \begin{enumerate}
 \item \label{lem:part_Fo2:1}  $WF(\eta) \subset  T^*(X) \times F \subset T^*(X \times F)$
 \item \label{lem:part_Fo2:2} $j^*(\eta)=p_{*}((\psi \circ pr_{F})\cdot\xi)$, where $j:X \to  X \times F$ is given by $j(x)=(x,1)$ and $pr_{F}:X\times F\to F$ is the projection.  Note that $j^*(\eta)$ is defined by virtue of \eqref{lem:part_Fo:1} and Proposition~\ref{prop:WF_prop3}.
 \end{enumerate}
  \end{lemma}

  \begin{proof}
  $ $
 \begin{enumerate}
 \item Follows from the definition of the wave front set.
 \item By continuity, it is enough to consider the case when $\xi$ is a Schwartz function. In this case, the assertion is obvious.
 \end{enumerate}
  \end{proof}


  \begin{cor}\label{cor:part_Fo2}
Let $X,Y$ be analytic manifolds, let $f$ be an analytic function on $X$ and $\xi$ be a distribution on $X$. Let $\phi:X \to Z$ be a proper map. Let
 $\phi':=\phi\times f:X \to Z \times F$
 Let $\eta:=\cF_F(\phi'_*(\xi))$. Then
 \begin{enumerate}
 \item \label{cor:part_Fo2:1}  $WF(\eta) \subset  T^*(Z) \times F \subset T^*(Z \times F)$
 \item \label{cor:part_Fo2:2} ${k}^*(\eta)= {\phi_{*}}((\psi \circ f)\cdot\xi )$, where ${k}:Z \to  {Z} \times F$ is given by ${k}({z})=({z},1)$. Again $j^*(\eta)$ is defined by virtue of \eqref{lem:part_Fo:1} and
 Proposition~\ref{prop:WF_prop3}.
 \end{enumerate}
  \end{cor}
  \begin{proof}
  Part \eqref{cor:part_Fo2:1} obviously follows from Lemma \ref{lem:part_Fo2}. Part \eqref{cor:part_Fo2:2} follows from Lemma \ref{lem:part_Fo2} and the following equality:
  ${\phi_{*}}((\psi \circ f)\cdot\xi )=p_{*}(\phi'_{*}((\psi \circ f)\cdot\xi ))=p_{*}( (\psi \circ pr_{F})\cdot \phi'_{*}{\xi} )$.
   \end{proof}

  If I don't find a reference for Proposition \ref{prop:WF_prop4}, I prefer this proof. If I do, then I don't know. What do you prefer?
 }

 %By Proposition \ref{prop:WF_prop2}\eqref{prop:WF_prop2:3} this implies % \eqref{cor:part_Fo:1}.
% Now
%apply Lemma \ref{lem:part_Fo} (and something else?).
}
}
\RamiAbs{
\RamiAC{We will %also
need the following geometric lemmas:
\RamiQT{You have proposed a different, more general, exposition of those lemmas, but I still need to think it through. So I leave the lemmas here for the time being.}
\begin{lemma}\label{lem:geom_1}
Let $Y$ be an algebraic manifold and  $W$ a finite dimensional $F$ vector space. Let $X=Y \times W \times F$. Let $L$ be a conic (locally closed) isotropic smooth subvariety of $T^*(X)$, which is stable under
 homotheties of $ W \times F$. Let $Z:=Y \times W \times \{1\}$.

Then
\begin{enumerate}
\item \label{lem:geom_1:1} $L \cap CN_Z^{X} \subset X$
\item \label{lem:geom_1:2} $L$ intersects transversally  $T^*(X)|_Z\subset T^*(X)$
\end{enumerate}
\end{lemma}
\begin{proof}
$ $
\begin{enumerate}
\item A general point in $CN_Z^{X}$ looks like $x=(y,w,1,0,0,\alpha)$. Assume $x \in L$. We have to show that $\alpha =0$. Assume the contrary.  Since $L$ is conic, we have:
\begin{equation}\label{pf_lem:geom_1:1}
(0,0,0,0,0,1) \in T_x L \subset T_x (T^*(X))=T_y Y \times W  \times F \times  T^{*}_y Y\times W^*  \times F.
\end{equation}
Since $L$ is stable with respect to homothety by $ W \times F$, we have:
\begin{equation} \label{pf_lem:geom_1:2}
(0,w,1,0,0,0) \in T_x L.
\end{equation}
This contradicts the fact that $L$ is isotropic.
\item  Follows immediately from formula \eqref{pf_lem:geom_1:2} above.
\end{enumerate}
\end{proof}
\begin{lemma}\label{lem:geom_2}
Let $X$ be an algebraic manifold and $Z \subset X$ be an algebraic submanifold. Let $L$ be a conic (locally closed) isotropic smooth subvariety of $T^*(X)$. Assume that
\begin{enumerate}
\item  $L \cap CN_Z^{X} \subset X$,
\item  $L$ intersect transversally  $T^*(X)|_Z\subset T^*(X)$.
\end{enumerate}
Then $i^{*}(L) \subset T^*(Z)$ is isotropic, where $i:Z \hookrightarrow X$ is the embedding, and  $i^{*}(L) $ has the same meaning as
%is as defined
in Definition \ref{def:dir_and_inv_image}\eqref{def:dir_and_inv_image:2}.
\end{lemma}
\RamiQR{Maybe we can put the proof in Appendix B.}
\begin{proof}
Let $L':= L \cap T^*(X)|_Z$ and $L'':=i^{*}(L)$. Without loss of generality we may assume that the map $L' \to L''$ is a submersion. The statement now reduces to the case when $X,Z,L$ are linear spaces, which is obvious.
\end{proof}

\begin{corollary} \label{cor:geom}
Let everything be as in
Lemma  \ref{lem:geom_1}, but without the assumption that $L$ is smooth.
Let $i:Y \times W \to Y \times W \times F$ be given by $i(y,w)=(y,w,1)$.
 Then

\begin{enumerate}
\item \label{cor:geom:1} $L \cap N_i \subset X.$
\item \label{cor:geom:2} $i^*(L) \subset T^*(Y \times W)$ is isotropic.
\end{enumerate}

\end{corollary}
\begin{proof}
We can stratify $L$ into smooth submanifolds, and then prove for each one separately using lemmas \ref{lem:geom_1} and \ref{lem:geom_2}.
\end{proof}

}
     Now we are ready to prove Theorem \ref{thm:futer}.
}%
%\begin{proof}[Proof of Theorem \ref{thm:futer} (in the non-Archimedean case).]
\DrinI{In Theorem \ref{thm:futer} we are given $\phi :=X \to Y\times W$ and $p:X\to K$. Let
  $\phi'=\phi\times p :X \to Y\times W \times K$. The idea is to apply Theorem \ref{thm:futerO} to $W\times K$ instead of $W$ and $\phi':X\to Y\times W \times K$ instead of $\phi :=X \to Y\times W$. Let
 $L' \subset T^*(Y\times W^{*} \times K)$ be the isotropic subvariety provided by Theorem \ref{thm:futerO} in this situation (in particular, $L'$ is stable under the homotheties of $W^*$). We can also assume that $L'$ is conic (otherwise replace $L'$ by its biggest conic subvariety).
%\RamiQS{Do you mean that the subvariety provided by Theorem \ref{thm:futerO} is not conic? Why?}.
Consider the embedding $ j:Y\times W^*=Y\times W^*\times\{ 1\}\hookrightarrow Y\times W^*\times F$. %\RamiQS{I still need to check the sign.}
%$j:Y\times W^{*} \to Y\times W^{*} \times K$  is given by $j(y,w)=(y,w,1)$,
\RamiAE{Define $L$ to be the Zarizki closure\footnote{\RamiAE{In fact, $j^*(L')$ is closed, but this is not essential to us.}} of $j^*(L')$,}
%and set $L=j^*(L')$,
 where $j^*$ has the same meaning as in Definition \ref{def:dir_and_inv_image}\eqref{def:dir_and_inv_image:2}.}
% where
% } \RamiAB{$j^*$ has the same meaning as %is as defined
% in Definition \ref{def:dir_and_inv_image}\eqref{def:dir_and_inv_image:2}.
%}
%It is easy to see that $L$ is isotropic.
%Indeed, let $(y,w,\alpha,\beta)\in L.$
\RamiAC{By \RamiAF{Lemma \ref{lem:iso_dir_invs_im} {and Remark~\ref{r:easy_facts}},}
%Corollary \ref{cor:geom}\eqref{cor:geom:2},
 $L$ is isotropic.}
%\RamiQP{I need to explain why. I'll try to do it  tomorrow}.

%\DrinJ{I would rather formulate somewhere a \emph{general} statement saying %that any pushforward and any pullback preserves the class of isotropic (constructible %in the sense of algebraic geometry) subsets. I think this is true and follows %from the interpretation of direct and inverse images in terms of symplectic %geometry
%(which should probably be formulated as a separate remark or lemma\footnote{Currently %this statement for direct images is inside the proof of a certain lemma} %and the fact that Lagrangian correspondences between symplectic varieties %preserve the class of isotropic subsets.
%RamiQS{Lemma \ref{lem:geom_1} is such a general statement, but it has additional %assumptions. Do you mean that we do not need these assumptions?} \DrinJ{Yes, %this is what I mean.}
%}
%\DrinJ{Is it true that $L$ is closed (if $L'$ is assumed closed)?
%Maybe yes, but I am afraid that the proof relies on the fact that $L'$ has
%zero intersection with the normal cone. The latter fact will be checked
%only at the end of the proof.}
%\RamiQS{I think that you are right. Anyhow
%it is not really important to us. Do you want to define $L$ to be the
%closure and give a footnote that in fact it is closed?  }  \DrinJ{Yes.}
% \DrinJ{Where did we define and use $j^*(L')$? Why is $j^*(L')$ Lagrangian?}

 \DrinI{Let us show that $L$ has the property required in Theorem~\ref{thm:futer}.
  Let $F$ be a local field equipped with an embedding $K\hookrightarrow F$. Set $W_F:=W\otimes_KF$. The problem is to show that the wave front of the distribution
\begin{equation}  \label{e:ditsribution_in_question}
\RamiAB{\mu:=}\cF^*_{W_F}((\phi_F)_*((\psi \circ p_F) \cdot |\omega_F|))
\RamiAD{=\cF^*_{W_F}((\phi_F)_*(p_{F}^{*}(\psi ) \cdot |\omega_F|))}
\end{equation}
is contained in $L(F)$. By the definition of $L'$, the wave front of the distribution
\begin{equation}    \label{e:ditsribution_we_control}
\RamiAB{\mu':=}\cF^*_{W _F\times F}((\phi'_F)_*(|\omega_F|))
\end{equation}
is contained in $L'(F)$.





% The remaining argument consists of two steps.
% %Now it remains to do two things.
% First, one has to show that the restriction of the distribution
% \eqref{e:ditsribution_we_control} to the hypersurface\footnote{By the way, it is not OK to denote the embedding
% $Y\times W^*=Y\times W^*\times\{ 1\}\hookrightarrow Y\times W^*\times F$ by $j$ if you use the symbol $j$ for the embedding $Y\times W=Y\times W\times\{ 1\}\hookrightarrow Y\times W\times F$.}
% \begin{equation}   \label{e:the_hypersurface}
% Y\times W^*=Y\times W^*\times\{ 1\}\subset Y\times W^*\times F
% \end{equation}
%  makes sense and is equal to the distribution \eqref{e:ditsribution_in_question}. You did this using the theory of wave fronts, but one can do this directly.
% Note that the restriction of the distribution \eqref{e:ditsribution_we_control} to the hypersurface \eqref{e:the_hypersurface} makes sense because the distribution
% \eqref{e:ditsribution_we_control} is, in fact, a \emph{continuous} map from $F$ to the space of distributions on
% $Y\times W^*$.
%
% The second step is to bound the wave front of the restriction of \eqref{e:ditsribution_we_control}
% to the hypersurface \eqref{e:the_hypersurface} in terms of the wave front of \eqref{e:ditsribution_we_control} itself.
% Here I see a problem but also a way out.
%
% First, the problem. You want to apply Proposition~\ref{prop:WF_prop3}. But then you have to check that the wave front of \eqref{e:ditsribution_we_control} is in ``good position" with respect to the hypersurface \eqref{e:the_hypersurface}. You checked a similar property \emph{before} doing Fourier transform with respect to $W$. But I cannot see how this allows you to conclude anything about the wave front \emph{after} Fourier transform with respect to $W$ (because I don't know a direct relation between the wave front before and after Fourier transform).
%
% But hopefully there is a way out. We know that $L'$ is isotropic and also \emph{stable under a certain action of the multiplicative group}, see property (i) from Theorem~\ref{thm:futerO}. I hope that these two properties imply
% ``good position" with respect to the hypersurface \eqref{e:the_hypersurface}.
%
% %By Corollary \ref{cor:part_Fo},
% %$(\phi_F)_*((\psi \circ p_F) \cdot |\omega_F|)=j^*(\cF^*_F(\phi'_F)_*(|\omega_F|))$. So by
% %Propositions~\ref{prop:WF_prop5} and \ref{prop:F_prop}\eqref{prop:F_prop:4}, the ditsribution
% %
% %$\cF^*_W((\phi_F)_*((\psi \circ p_F) \cdot |\omega_F|))=\cF^*_W(j^*(\cF^*_F(\phi'_F)_*(|\
%
 }

%  Let $\phi'=\phi\times p :X \to Y\times W \times K$.
% Let $L' \subset T^*(Y\times W^{*} \times K)$ \RamiAA{be} as in Theorem \ref{thm:futerO} (where we substitute $W \times \RamiZ{K}$ and $\phi'$ for $W$ and $\phi$). Let $L=j^*(L')$ where $j:Y\times W \to Y\times W \times K$  \RamiAA{is given by $j(y,w)=(y,w,1)$}. It is easy to see
% %(using Lemma \ref{lem:Lag_dir_im})
% that $L$ is isotropic.

\DrinI{First, let us show that
\begin{equation} \label{e:mu_and_eta}
\mu =\RamiAD{\mu'|_{Y(F)\times \RamiAE{W_F^{*}} \times \{1\}}}%j^*(\mu' ),%\quad j:Y\times W^*=Y\times W^*\times\{ 1\}\hookrightarrow Y\times W^*\times F,
\end{equation}
where %$j:Y\times W^* \to Y\times W \times K$  is given by $j(y,w)=(y,w,1)$ and
the equality \eqref{e:mu_and_eta} is understood in the sense of
Definition\RamiAD{ \ref{def:cont_dep}. }%/Convention X \footnote{\DrinJ{I %suggest to formulate it somewhere before Proposition~\ref{prop:WF_prop3}.
%According to this Definition (or Convention) the equality \eqref{e:mu_and_eta}  just means that for any test function $f$ on $X(F)\times W_F$ the integral of $\mu'$ against $f$ is a \emph{continuous} (rather than generalized) function of $t\in F$ and its value at $t=1$ equals the integral of $\mu$ against $f$.}}.
\DrinI{To this end, for each $t\in F$ consider the distribution
\[
\mu_t:=\cF^*_{W_F}((\phi_F)_*(p_{F}^{*}(\psi_{t})  \cdot |\omega_F|)),
\] where $\psi_{t}$ is the additive character of $F$ defined by $\psi_t(x)=\psi(tx)$. Note that $\mu_1=\mu$, so
\eqref{e:mu_and_eta} follows from the next lemma. }
\begin{lem}\label{lem:mu_t}
$ $
\begin{enumerate}
\item \label{lem:mu_t:1} $\{\mu_t\}_{t\in F}$ is a continuous family of distributions\footnote{{As usual, the space of distributions on $Y(F)\times W_F^*\,$ is equipped with the weak topology.}} on $Y(F)\times W_F^*\,$.

\item \label{lem:mu_t:2}
The distribution on $Y(F)\times W_F^* \times F$ corresponding to the family $\{\mu_t\}$ equals $\mu'$; \RamiAE{that is, for any $f\in \Sc(Y(F)\times W^{*}_F ,\C_{Y(F)} \boxtimes D_{W^{*}_F})$ and $g\in \Sc(F)$, we have
\begin{equation}
\langle \mu',f \boxtimes g\rangle=\int_{t\in F}\mu_t (f)g(t) dt.
\end{equation}
 }
\end{enumerate}
\end{lem}
%\RamiQT{I slightly changed the formulation of the lemma, is it OK?} \DrinK{Yes.}


\begin{proof}
Statement \eqref{lem:mu_t:1} is clear.
%\footnote{\DrinJ{I think it is indeed clear in the non-Archimedean case.}}
\RamiAE{Let us } prove \eqref{lem:mu_t:2}. \RamiAE{We have:
%we have to check that
%for any
\RamiAD{
% $f\in \Sc(Y(F)\times W^{*}_F ,??)$ and $g\in \Sc(F)$.
%
% \DrinJ{Are you sure that $f\in \Sc(Y(F)\times W_F )$?  Is $f$ a function on $Y(F)\times W_F $ or on  $Y(F)\times W^*_F$? Is there a locally constant sheaf involved?}
%
% \DrinJ{Is is possible to use the scalar product notation in the proof below? (More precisely, I suggest to write
% the ``scalar product of a test function and a distribution" rather than the ``value of a distribution on a test function" because this better agrees with our intuition.) Another suggestion: instead of writing a very very long chain of equalities, compute the l.h.s. and the r.h.s, then explain why they are equal.}
\begin{multline*}
 \left \langle \mu',f \boxtimes g\right\rangle  =\left \langle (\phi'_F)_*(|\omega_F|),\cF_{W_F} (f) \boxtimes \cF_{F} (g)\right\rangle=\left \langle |\omega_F|,(\phi'_F)^*(\cF_{W_F} (f) \boxtimes \cF_{F} (g))\right\rangle=\\
=\left \langle |\omega_F|,(\phi_F)^*(\cF_{W_F} (f)) \cdot p_{F}^*(\cF_{F} (g))\right\rangle=\left \langle |\omega_F|,(\phi_F)^*(\cF_{W_F} (f)) \cdot  p_{F}^*\left(\int_{t\in F}\psi_t \cdot g(t)dt\right)\right\rangle
\end{multline*}
On the other hand
\begin{multline*}
 \int_{t\in F} \left \langle\mu_t ,f \right\rangle g(t) dt
 =\int_{t\in F}\left \langle(\phi_F)_*(|\omega_F| \cdot p_{F}^*(\psi_t)),\cF_{W_F} (f)  \right\rangle g(t) dt=\\= \int_{t\in F}\left \langle(|\omega_F| \cdot p_{F}^*(\psi_t),(\phi_F)^*(\cF_{W_F}
(f))\right\rangle g(t)  dt =  \int_{t\in F} \left\langle |\omega_F|,(\phi_F)^*(\cF_{W_F} (f)) \cdot p_{F}^*(\psi_t)\right\rangle g(t)  dt
\end{multline*}
So it remains to prove that
$$\left \langle |\omega_F|,(\phi_F)^*(\cF_{W_F} (f)) \cdot  p_{F}^*\left(\int_{t\in F}\psi_t \cdot g(t)dt\right)\right\rangle=  \int_{t\in F} \left\langle |\omega_F|,(\phi_F)^*(\cF_{W_F} (f)) \cdot p_{F}^*(\psi_t)\right\rangle g(t)  dt$$
This follows from the fact that, for each particular $f$ and $g$, the integral can be replaced by a finite sum.
}
%\RamiQT{In the Archimedean case, the explanation is slightly different.}\\  %\RamiQW{In the proof, of the Archimedean case we have written that this\\ % part is analogous to the non-Archimedean case. I think that this is good\\ % enough.} \DrinK{I agree.}
% \begin{multline*}
%  \left \langle \mu',f \boxtimes g\right\rangle  =\left \langle (\phi'_F)_*(|\omega_F|),\cF_{W_F} (f) \boxtimes \cF_{F} (g)\right\rangle=\left \langle |\omega_F|,(\phi'_F)^*(\cF_{W_F} (f) \boxtimes \cF_{F} (g))\right\rangle=\\
% =\left \langle |\omega_F|,(\phi_F)^*(\cF_{W_F} (f)) \cdot p_{F}^*(\cF_{F} (g))\right\rangle=\left \langle |\omega_F|,(\phi_F)^*(\cF_{W_F} (f)) \cdot  p_{F}^*\left(\int_{t\in F}\psi_t \cdot g(t)dt\right)\right\rangle
%  =\\=
%  \int_{t\in F} \left\langle |\omega_F|,(\phi_F)^*(\cF_{W_F} (f)) \cdot p_{F}^*(\psi_t)\right\rangle g(t)  dt= \int_{t\in F}\left \langle(|\omega_F| \cdot p_{F}^*(\psi_t),(\phi_F)^*(\cF_{W_F} (f))\right\rangle g(t)  dt=\\= \int_{t\in F}\left \langle(\phi_F)_*(|\omega_F| \cdot p_{F}^*(\psi_t)),\cF_{W_F} (f)  \right\rangle g(t) dt=
% %\int_{t\in F}(\phi_F)_*(\cF_{W_F} (|\omega_F| \cdot p_{F}^*(\psi_t) ))(f  )g(t) dt=\\=
%  \int_{t\in F} \left \langle\mu_t ,f \right\rangle g(t) dt
%\end{multline*}
%
% \RamiQS{Do you think I need to explain the second line in the equality in more detail?}
% \DrinJ{I don't understand the question. Which equality are you asking about and what kind of explanation do you have in mind? }\RamiQT{The equalities
% \begin{multline*}
% |\omega_F|\left((\phi_F)^*(\cF_{W_F} (f)) \cdot p_{F}^*(\cF_{F} (g))\right)=|\omega_F|\left((\phi_F)^*(\cF_{W_F} (f)) \cdot  p_{F}^*\left(\int_{t\in F}\psi_t \cdot g(t)dt\right)\right)=\\=\int_{t\in F}|\omega_F|\left((\phi_F)^*(\cF_{W_F} (f)) \cdot p_{F}^*(\psi_t)\right)g(t) dt
% \end{multline*}
% and more explicitly:
% should I explain the meaning of $$\int_{t\in F}\psi_t \cdot g(t)dt$$
% }
%This implies that $\mu_t=\mu'|_{Y(F)\times W_F \times \{t\}}$
}
\end{proof}

%\RamiQS{Do you like the argument above more then the argument below?}
%
% \DrinJ{I cannot answer this question until you edit the proof of the lemma.
% On the other hand, here is a question.
% In the ``argument below" it is clearly said that in order to prove \eqref{e:mu_and_eta} we prove \eqref{e:xi_and_xi'} (this is something like a lemma). Are you doing more or less the same in the above computation?
%  If yes then is it possible to make it clear? }\RamiQT{The above computation is a direct proof of \eqref{e:mu_and_eta}. Under this approach I think that it is not longer then the proof of \eqref{e:xi_and_xi'}.}


%\RamiQS{I did not edit the rest of the proof.}

% \RamiAC{By Proposition~\ref{prop:F_prop}\eqref{prop:F_prop:4}, w}e have
% %\[
% %\mu' =\cF^*_W(\xi')), \quad \mu' =\cF^*_W(\xi')),
% %\]
% \RamiAC{
% \[
% \mu =\cF^*_{W_{F}}(\xi), \quad \mu' =\cF^*_{W_{F}}(\xi'),
% \]
% }
% where
% \[
% \xi:=(\phi_F)_*((\psi \circ p_F) \cdot |\omega_F|) ,\quad \xi':=\cF^*_F((\phi'_F)_*(|\omega_F|)),
% \]
% So Proposition \ref{prop:WF_prop5} tells us that in order to prove \eqref{e:mu_and_eta}, it suffices to show that
%
% \begin{equation} \label{e:xi_and_xi'}
% \xi =k^*(\xi' ),%\quad j:Y\times W^*=Y\times W^*\times\{ 1\}\hookrightarrow Y\times W^*\times F,
% \end{equation}
% where $k:Y\times W \to Y\times W \times K$  is given by $k(y,w)=(y,w,1)$.
%  }
%
%  \DrinJ{Rami, formula \eqref{e:xi_and_xi'} indeed follows from Corollary~\ref{cor:part_Fo}, but is it possible to write a \emph{direct} proof of \eqref{e:xi_and_xi'}, without using wave fronts?
% Just write the formulas for the scalar product of $\xi'$ and $\xi$ against a test function on
%  $Y(F)\times W_F$ (skip the minor details of the computation) and say that the scalar product for $\xi'$ depends continuously on the parameter $t\in F$ and after setting $t=1$ becomes equal to the scalar product for $\xi$.
 }

\DrinI{Thus we have proved
\RamiAE{\eqref{e:mu_and_eta}.}
%\eqref{e:xi_and_xi'}.
By assumption, the wave front of $\mu'$ is contained in $L'(F)$.
So by {Corollary~\ref{c:Treves}}, to prove that the wave front of $\mu$ is contained in $L(F):=(\overline{j^*L'})(F)$, it suffices to check that $L'(F)$ satisfies the condition of
{Proposition~\ref{prop:Treves}}.
%Proposition~\ref{?} \DrinJ{(By this I mean an elementary variant of
%Proposition~\ref{prop:WF_prop3})}.
 In other words, we have to check that if $z\in \RamiAE{j}(Y\times \RamiAE{W^{*}})$ and
$\xi\in T^*_z(Y\times \RamiAE{W^{*}}\times K)$ are such that $(z,\xi )\in L'$ and $\xi$ is conormal to $k(Y\times W)$ then $\xi =0$.
Recall that $L'$  is assumed to be conic and stable under the action of the \RamiAH{multiplicative} group ${\mathbb G}_m$ on $Y\times \RamiAE{W^{*}}\times K$ that comes from  homotheties of $\RamiAE{W^{*}}\times K$; in other words, $L$ is stable under
${\mathbb G}_m\times {\mathbb G}_m\,$. So the tangent space to the $({\mathbb G}_m\times {\mathbb G}_m)$-orbit of $(z,\xi )$ has to be isotropic. This means that $\xi$ vanishes on the tangent space to the ${\mathbb G}_m$-orbit
of $z$. On the other hand, $\xi$ is assumed to be conormal to $\RamiAE{j}(Y\times \RamiAE{W^{*}})$. So $\xi=0$.
}
%\end{proof}
%Now
%let us show that $L$ satisfies condition \eqref{thm:futer:2} \RamiQO{Should %be (ii)}.
%choose an embedding $K \hookrightarrow F$.


%Set $\xi':=\cF^*_F((\phi'_F)_*(|\omega_F|))$. Then $\mu' =\cF^*_W(\xi'))$.
%On the other hand, by Corollary \ref{cor:part_Fo}, we have
%$\mu=\cF^*_{W_F}(k^*(\xi))$,
%where $k:Y\times W \to Y\times W \times K$  is given by $k(y,w)=(y,w,1)$.  So \eqref{e:mu_and_eta} follows from
%Proposition \ref{prop:WF_prop5}.



%\RamiAB{
%Now
%let us show that $L$ satisfies condition \eqref{thm:futer:2} \RamiQO{Should %be (ii)}.
%choose an embedding $K \hookrightarrow F$.

%Let $\xi:=\cF^*_F((\phi'_F)_*(|\omega_F|)).$
%By Corollary \ref{cor:part_Fo}, we have
%$$(\phi_F)_*((\psi \circ p_F) \cdot |\omega_F|)=k^*(\xi),$$
%where $k:Y\times W \to Y\times W \times K$  is given by $k(y,w)=(y,w,1)$,  so $$\mu=\cF^*_{W_F}((\phi_F)_*((\psi %\circ p_F) \cdot |\omega_F|))=\cF^*_{W_F}(k^*(\xi)).$$
%By Proposition~\ref{prop:F_prop}\eqref{prop:F_prop:4},
%$$\cF^*_{W_F}(\xi)=\eta \in \G_{L'}(Y_{F} \times W_F \times F,D_{Y_{F}}\boxtimes \C_{W_F \times F}).$$
%\RamiAC{By Corollary \ref{cor:geom}\eqref{cor:geom:1}, }
%%%%It is easy to see that
% $${L'} \cap N_j\subset Y_{F} \times W^{*}_F \times F.$$
%%%\RamiQP{I need to explain why. I'll try to do it  tomorrow.}
% Therefore by  Proposition \ref{prop:WF_prop5},
% $$\mu=\cF^*_W(k^*(\xi))=j^* (\cF^*_W(\xi))=j^{*}(\eta).$$
%%%% $\cF^*_W(j^*(\cF^*_F(\phi'_F)_*(|\omega_F|)))=j^* (\cF^*_W(\cF^*_F((\phi'_F)_*(|\omega_F|)))$
%%%% and by
%%%%Proposition~\ref{prop:F_prop}\eqref{prop:F_prop:4}, $j^* (\cF^*_W(\cF^*_F((\phi'_F)_*(|\omega_F|)))=j^*(\cF^*_{W %\times F}((\phi'_F)_*(|\omega_F|)))$.
%The assertion now follows  from Proposition~\ref{prop:WF_prop3}.
%}
%\end{proof}

% ?
%
% The fact that $L$ satisfies condition \eqref{thm:futer:2} \RamiQH{Something
%  is wrong with the reference. It should be (ii). Maybe it is the tex compiler
%   problem. I'll try to fix it. In the worst case, I'll do it manually.}
%    follows now from Corollary \ref{cor:part_Fo}.\\       \RamiQH{Strictly
%     speaking, we also use here the fact that iterative Fourier transform
%      is equal to a total Fourier transform,
%    but I think it is too trivial to mention.}\RamiQN{ There are other rather
%     trivial things that we use here. Do you want me to add more details
%      here?}    \end{proof}



%      Now we are ready to prove Theorem \ref{thm:futer}.
%      \begin{proof}[Proof of Theorem \ref{thm:futer} (in the non-Archimedean
%       case).]
%      The proof follows the proof of Theorem \ref{t:C}, with additional
%       considerations regarding the additive character $\theta$ which are
%        based on Lemma \ref{lem:part_Fo}. Without loss of generality, we
%         may assume that $\theta=\psi$. Let $\phi'=\phi\times p :X \to Y\times
%          W \times F$.
%
%    Let the objects and the morphisms in the following commutative diagram
%     be analogous to those in the proof of Theorem \ref{t:C}.
%      $$
%    \xymatrix
%    {        X   \ar@{}[r]|-*[@]{\hookrightarrow}   \ar@{->}_{\phi'}[d]
%            & \overline X\ \ar@{->}_{\overline \phi'}[d]  & \tilde X \ar@{->}_{\rho}[l]
%             \ar@{->}^{\tilde \phi':=\bar \phi' \circ \rho}[ld]\\
%             Y\times W\times F \    \ar@{}[r]|-*[@]{\hookrightarrow}   &
%               Y\times\overline {W\times F}  & \\ }
%    $$
%
%    Let $\tD $ and  $\RamiAL{\mmu}: \tD \to Y\times {W^{*}\times F}$ be
%     as in Theorem \ref{exp} when we substitute $\tilde X,\tilde \phi$ and
%      $W \times  F$  by  $ X,\phi$ and $W$. Let $S=j^*(S_{\RamiAL{\mmu}})$ where
%       $j:Y\times W \to Y\times W \times F$ is as in Lemma \ref{lem:part_Fo}.
%        It is easy to see (using Lemma \ref{lem:Lag_dir_im}) that $S$ is
%         isotropic and conic. Thus by Lemma \ref{lem:isotr_discrip} we can
%          find  a Lagrangian conic variety  $L \supset S$. The fact that
%           $L$ satisfies \ref{thm:futer}(ii)%\eqref{thm:futer:2} \RamiQT{I
%            made the reference manually, for the meanwhel.} follows now
%             from Theorem \ref{exp} and Lemma \ref{lem:part_Fo}\eqref{lem:part_Fo:2}.
%    %\   Let $\overline X,\tilde X,\rho: \tilde X \to \overline X, \phi$
%    \RamiQH{Strictly speaking, we also use here the fact that iterative
%     Fourier transform is equal to a total Fourier transform,
%    but I think it is too trivial to mention.}
%    \end{proof}
%

\section{\RamiT{The Archimedean case}} \label{sec:arc}
\RamiT{
%In this section we briefly repeat the results of the paper for the Archimedean case.
In \S\S \ref{ssec:arc_dist} we recall the terminology relevant for the Archimedean case (in particular, the notion of partially Schwartz distribution).
In \S\S \ref{ssec:arc_pf} we
% give an exact formulation of our result in the Archimedean case and
explain what should be added to the proof from \S\ref{sec:pf_main_bb}
 of non-Archimedean case
to make it valid in the Archimedean case \DrinI{(essentially, the only new ingredient is the elementary
Lemma~\ref{lem:dim_1_arch}).}

Throughout the section $F$ is an Archimedean field (i.e. $F$ is $\R$ or $\C$). Recall that we equip
$F$ with the  \emph{normalized} absolute value, which in case of $F=\C$ is the \emph{square} of the classical one.

\subsection{Distributions in the Archimedean case} \label{ssec:arc_dist}
Let $M$ be a smooth (real) manifold. Recall that the space $C^\infty_c(M)$ of test functions on $M$ is the space of smooth compactly supported functions endowed with the \RamiU{standard}
%natural direct limit of \Fre spaces
topology (\RamiU{recall that in this topology, a sequence} converges if and only if it has a compact joint support and converges uniformly with all its derivatives). Recall also that the space of distributions $C^{-\infty}(M,D_M)$ on $M$ is defined to be the dual of $C^\infty_c(M)$. Similarly, for any  \DrinI{smooth vector} bundle we can consider its smooth compactly supported sections and generalized sections.

\RamiAG{We will use the same notations as in \S\S\ref{ssec:dist} but we will replace $\Sc$ with $C^\infty_c$ and $\G$ with $C^{-\infty}$. The reason is that $\Sc$ and $\G$ stands for Schwartz, and in the non-Archimedean case Schwartz functions are just smooth compactly supported functions and Schwartz distributions are just distributions, unlike the Archimedean case.
%The statements of \S\S\ref{ssec:dist}
%that precede \S\S\S\ref{sssec:ft}

% \RamiQV{Do you think that it is a good idea to put the beginning of  \S\S\ref{ssec:dist} is a separate subsubsection (with some name, e.g. ``generalities") so we can refer to it more easily here?}
% \DrinK{Yes, this would be good.}
\RamiAM{The content of \S\S\S\ref{sssec:fun_sp} and \S\S\S\ref{sssec:op_dist}}
hold\RamiAM{s} for the Archimedean case, with the obvious modifications (e.g. $l$-spaces are replaced with smooth manifolds and locally constant sheaves are replaced with  \DrinI{smooth vector} bundles). The statements of  \S\S\S\ref{sssec:wfs} hold with minor modifications.
\RamiAH{In particular,
%Definition \ref{def:cont_dep} should be changed in the following way.% for the Archemadian case:
{the role of Definition \ref{def:cont_dep} is played by the following one.}

\begin{definition}\label{def:cont_dep_arch}
Let $\xi\in C^{-\infty}(X \times Y)$ be a generalized function on a product
of analytic manifolds. We will say that $\xi$ depends continuously on $Y$ if

(i) for any $f \in C^{\infty}_c(X,D_X)$, the generalized function $\xi_f \in
C^{-\infty}(Y) $ given by
$\xi_f(g)=\xi(f \boxtimes g)$ is continuous;

(ii) for any $y \in Y$, the
functional $f \mapsto \xi_f(y)$ is continuous.

In this case we \DrinI{define} $\xi|_{X \times \{y\}} \in  C^{-\infty}(X
\times \{y\})$   by  $\xi|_{X \times \{y\}}(f):=\xi_f(y)$.
\end{definition}

{
\begin{rem}   \label{r:closed graph}
Using the closed graph theorem one can show that (i) implies (ii) (and moreover, (i) implies continuity of the map $C^{\infty}_c(X,D_X)\to C(Y)$ given by $f\mapsto\xi_f$). We will not need this fact.
\end{rem}
}

}
\RamiAH{
 We present the rest of the content of \S\S\S\ref{sssec:wfs},} with more details, for both  the Archimedean and the non-Archimedean case, in Appendix \ref{app:WF}.}


In order to discuss partial Fourier transform \RamiAG{as in  \S\S\S\ref{sssec:ft}} we will need to discuss test functions which are partially Schwartz.
\begin{definition}
Let $M$ be a smooth manifold and $V$ be a real vector space.
\begin{enumerate}
\item We define the space \RamiU{$C_{c}^{\infty,V}(M \times V)$} of partially Schwartz (along $V$) test functions on $M \times V$ to be the space of all smooth functions $f$ on $M \times V$ such that $\supp(f) \subset \RamiU{K} \times V$ for some compact $\RamiU{K \subset M}$, and for any polynomial differential operator $D$ on $V$ \RamiU{and any smooth differential operator $D'$ on $M$}, the function $\RamiU{D'}Df$ is bounded.
%\DrinI{Do we require boundedness of $D_1D_2f$ if $D_1$ is a differential %operator on $M$ and $D_2$ is a polynomial differential operator on $V$?}
\item
\RamiU{We define a topology on this space in the following way. For any compact $K \subset M$, we let $C^{\infty,V}_K(M \times V)$ be the subspace of $C_{c}^{\infty,V}(M \times V)$ that consists of all functions %which are
supported on $K \times V$. We define the topology on $C^{\infty,V}_K(M \times V)$ by the semi-norms $f \mapsto |D'Df|$, where $D'$ and $D$ are as above. We define the topology on  $C_{c}^{\infty,V}(M \times V)$ to be the direct limit topology.}

%\RamiQI{So if you approve let's use this definition .}

%\DrinJ{I approve it (at least for now). If problems appear we can try to %use something like Definition 6.3.3 on p. 144 of \cite{Hor}.}


\item \DrinI{Let $\xi$ be a distribution  on $M \times V$, i.e., a continuous linear functional on
$C^\infty_c(M\times V)$.
We say that  $\xi$ is \emph{partially Schwartz} along $V$ if this functional} can be continuously extended to the space of partially Schwartz test functions.
\item We can clearly extend the above definition to generalized sections of bundles of the type $E \boxtimes D_V$, where $E$ is a bundle on $M$.
\item We say that a generalized section of $E \boxtimes \C_V$, (where $\C_V$ is the constant bundle on $V$) is partially Schwartz if it becomes so after multiplication by a Haar measure on $V$.
\item The space of Schwartz generalized sections will be denoted by $C^{-\infty,V}(\dots)$.
\end{enumerate}
\end{definition}

\DrinI{Now the results of  \S\S\S\ref{sssec:ft} (with natural modifications) are valid (with the same standard proofs) for distributions which are partially Schwartz along the relevant vector space. Here is the precise formulation, whose only new ingredient is the fact that the operations performed on distributions preserve the partially Schwartz property.}

%Now the results of  \S\S\S\ref{sssec:ft} (with natural modifications) are valid for distributions which are partially
%Schwartz along the relevant vector space with the same standard proofs. Let us repeat the formulation %explicitly\RamiV{, the only thing that we are actually adding is the fact that the operations we perform on
%distributions in \S\S\S\ref{sssec:ft} preserve the partially Schwartz property.}
%(for the convenience of the reader the modifications are in bold phase). %\DrinJ{Using so much bold face is not a good idea. I also wonder whether %formulating so many obvious statements is a good idea. How do you think, %is it possible to just give one or maybe two examples (e.g., to formulate %only statement (6) below)?}
\begin{proposition} \label{prop:F_prop_arch}
Let $W,L$ be a real vector spaces and $X$ be a smooth manifold.
Let $\xi \in C^{-\infty}(X \times W,D_{X \times W})$, which is Schwartz along $W$.

\begin{enumerate}
\item
Let $U \subset X$ be an open set. Then %\textbf
{$\xi_{U\times W}$ is partially Schwartz along $W$ and} $\cF_{W}^{{*}}(\xi)|_{U\times W}=\cF_{W}^{{*}}(\xi|_{U\times W})$.
\item %\textbf
{Let $\xi' \in C^{-\infty}(X \times W,D_{X \times W})$ and Let $X=\bigcup U_i$ be an open cover of $X$. Assume that $\xi'|_{U\times W}$ is partially Schwartz along $W$. Then $\xi'$ is  partially Schwartz along $W$.}
\item Let $f\in C^\infty(X)$ be a smooth function. Then %\textbf
{$f\xi$ is partially  Schwartz along $W$ and} $\cF_{W}^{{*}}(f\xi)=f\cF_{W}^{{*}}(\xi).$
\item Let $p:X \to Y$ be a proper map of smooth manifolds. Then %\textbf
{$p_*\xi$ is partially  Schwartz along $W$ and} $\cF_{W}^{{*}}(p_*\xi)=p_*\cF_{W}^{{*}}(\xi).$
\item Let $\nu$ be as  in Lemma \ref{prop:F_prop2} and  $\rho_\nu$  be as in Notation \ref{not:F_prop}
%Let $C^{-\infti,W}(X \times W^*,D_X\boxtimes \C_W)$ be the space of sections which are Schwartz along $W$.
Then %\textbf
{the vertical arrows in the following diagram preserve the space of partially Schwartz functions and} it is commutative.

\begin{equation*}
\xymatrix
{        C^{-\infty,W^*}(X \times W^*)  \ar@{<-}^{\cF^*_{W}\quad}[r] \ar@{<-}^{(\rho_{{\nu^t}})^*}[d]& C^{-\infty,W}(X \times W,\C _X\boxtimes D_{W}) \ar@{<-}^{(\rho_{\nu})_* }[d]  \\
         C^{-\infty,L^*}(X \times L^*)  \ar@{<-}^{\cF^*_L\quad}[r]                                 & C^{-\infty,L}(X \times L,\C _X\boxtimes D_{L})
}
\end{equation*}
\end{enumerate}
\end{proposition}
% \RamiQM{The rest of the subsection should be changed.}
% The Archimedean analog of Lemma \ref{lem:part_Fo} also holds. But in order for its formulation to make sense we need the following obvious lemma:
% \begin{lemma}\label{lem:part_Fo_def_arch}
% Let $X$ be a \DrinI{smooth} manifold, let $f:X\to F$ be a  \DrinI{smooth} function and
% $\xi$ be a distribution on $X$. Let $i:X \to X \times F$ be the graph of $f$.
% Then the distribution $i_*(\xi)$ is Schwartz along $F$.
%  % \RamiQH{We do not actually need ``analytic'' here,
%  %I just put it to be consistent with the non-Archimedean case}
%   \end{lemma}
%
%
\subsection{\DrinI{On the proofs of} the main results in the Archimedean case} \label{ssec:arc_pf}
%The \DrinJ{ proof of Theorem~\ref{thm:main_C_Arch} is parallel to the proof}
$ $
\RamiY{The proof of Theorem \ref{thm:main_C_Arch}  follows the same lines as the proof of Theorem \ref{thm:main_bb}, but in each step we have to check that the distributions we consider are partially Schwartz along the relevant vector space. In other words we should prove part\RamiAE{s} \RamiAG{\eqref{thm:main_C_Arch:1} and \eqref{thm:main_C_Arch:2}} of Theorem \ref{thm:main_C_Arch}
%\eqref{thm:main_C_Arch:1} and \eqref{thm:main_C_Arch:2}
% \RamiQT{I made the reference manually, for the time being.}
 %\RamiQM{Should be (i) and (ii)}
together.
The reduction to Lemma \ref{lem:dim_1}
      is the same as  in  Theorem \ref{thm:main_bb}, but in Lemma  \ref{lem:dim_1} itself we need to be more careful. Namely, we have to precede it with the following lemma:}

%    Let us repeat the formulation in a way which is more appropriate for the Archimedean case (again we use a bold phase for the natural modifications):
%
%    \begin{thm}\label{thm:exp_arch}
%    Let $X,Y$ be {algebraic} manifolds and $W$ a vector space over $F$,  {with}  $\dim W<\infty$.
%    Let $\phi :X\to Y \times \overline{W}$ be a proper map. \RamiT{Let $X_0:=\phi^{-1}(Y \times W)$.} Let $\omega$ be a top differential form on $X\RamiT{_0}$.
%     Let $Z$ be the \RamiT{closure of the} zero set of $\omega$  \RamiT{in $X$}. Let $D=Z \cup \phi^{-1}((\overline{W}-W)\times Y)$.
%    Assume that $D$ is an SNC divisor.
%
%    Let $\RamiT{D_i}'$ be the union of the components of $\hat{D_i}$ whose image in $D_i$ contained in $\phi^{-1}(\overline{W}-W)$\RamiT{.}
%
%    Let $\pi:X\to \overline{W}$ be the composition of $\phi$ with the projection to $ \overline{W}$ and $\tau:X\to Y$ be the composition of $\phi$ with the projection to $ Y$.
%     Consider $\pi_{D'_i}$ as a map $\pi_{D'_i}:D'_i \to \mathbb{P}(W^*)$ and consider $E_i:=\pi^{*}_{D'_i}(Taut_{\mathbb{P}(W^{*})})$ as a subspace of $X \times W$.
%
%    Let $\tD$ be the disconnected union
%    $$(X\times 0) \sqcup\coprod \RamiT{\hat{D}_i \times 0} \sqcup\coprod {E_i}
%    %\sqcup\coprod D'_i \times 0
%    $$
%    We have a natural map
%    $\RamiT{\mu}:\tD \to X \times W$. Let $\RamiAL{\mmu}$ be the composition $$\RamiAL{\tD}  \overset{\RamiT{\mu}}{\to} X \times W\overset{\tau \times Id}{\to}Y \times  W.$$
%    %\RamiT{Recall that  $S_{\RamiAL{\mmu}}$ is the direct image of the zero section %in $T^*(\tD)$ with respect to $\RamiAL{\mmu}$ (see Notation \ref{def:dir_and_inv_image}\eqref{def:dir_and_inv_image:4}).
%    %}
%    Then \textbf{the distribution $(\phi|_{X_0})_*(|\omega|)$ is partially Schwartz along $W$ and}
%
%    $$WF(\cF^*(\RamiT{(\phi|_{X_0})}_*(|\omega|)))\subset S_{\RamiAL{\mmu}}(F)$$
%
%    %Note that here we use the
%    %%\RamiT{simplectic}
%    % identification
%    %$$T^*(Y \times  W^{*})
%    %%\RamiT{=T^*(Y) \times  T^*(W^{*})=T^*(Y) \times  T^*(W)}
%    %=T^*(Y)\times  W^{*}\times  W= T^*(Y \times  W).$$
%    %% \RamiT{as in lemma ??}.
%    \end{thm}
%
%    \begin{theorem} \label{thm:futer_arch}
%    Let $K$ be a characteristic $0$ field and $W$ a finite-dimensional $K$-vector space. Let $X,Y$ be smooth algebraic manifolds over $K$. Let $\phi: X \to Y \times W$ be a proper map,
%     $\omega$ a \RamiR{regular} top differential form on $X$, and $p$ be a regular function on $X$.
%
%    Then \textbf{the distribution $(\phi_F)_*(\theta \circ p_F \cdot |\omega_F|)$ is partially Schwartz along $W$, and}
%     there exists a Lagrangian algebraic subvariety $L \subset T^*(Y\times W^*)$ such that
%    \begin{enumerate}
%    \item[(i)]   $L$ is stable with respect to the action of the multiplicative group on $W$;
%    \item[(ii)]   \RamiM{F}or any embedding of $K$ into any local field $F$ (Archimedean or not), and any additive character
%     $\theta$ of $F$, we have
%     $$WF(\cF^*((\phi_F)_*(\theta \circ p_F \cdot |\omega_F|)))\subset L(F).$$
%     \end{enumerate}

%Here $L(F)\subset T^*(Y\times W^*)(F)$ is the set of $F$-points of $L$, $\omega_F$ and $p_F$ are obtained from
%$\omega$ and $p$ by extension of scalars from $K$ to $F$, and $|\omega_F|$ is the corresponding measure on $X(F)$.
%\end{theorem}
%    The proof of Theorem \ref{thm:futer_arch} works smoothly as in the non-Archimedean case. In the proof of Theorem \ref{thm:exp_arch}, the reduction to Lemma \ref{lem:dim_1}
%     %\RamiQH{(in fact it is not exactly Lemma \ref{lem:dim_1}, but rather its explicit counterpart, which is part of
%     %Lemma \ref{lem:exp_bn}. I think it is better not to confuse the reader with those details)}
%     also works smoothly, but in Lemma  \ref{lem:dim_1} itself we need to be more careful. Namely, we have to precede it with the following lemma:

 \begin{lemma}  \label{lem:dim_1_arch}
Let $Y$ be the affine space with coordinates $y_1, \dots, y_n.$ Let $p:Y \to F$ be defined by
$p= \prod_{i=1}^n y_i^{l_i}$,
where
$l_i \in \Z_{\geq 0}$. Let $\omega$ be the top differential form on $Y$
given by $\omega= (\prod_{i=1}^n y_i^{r_i}) d{y_1} \wedge \dots \wedge d{y_n}$, where $r_i \in \Z$.
 Suppose $r_i \geq 0$ whenever $l_i=0$,
so $\omega$ is regular on the set $Y_0:=\{ y\in Y | p(y)\neq 0\}$.
Define $i:Y_0\hookrightarrow Y \times F$ by $i(y):=(y,p(y)^{-1})$.
 %the graph of $1/p$.
 Then the distribution $i_*(|\omega|)$ is Schwartz, \RamiU{and in particular it is partially Schwartz} along~$F$.
\end{lemma}

%\DrinI{In the proof below are you proving that $i_*(|\omega|)$ is Schwartz %along $F$ or that
%it is Schwartz ``in all directions"? (So far, this is just a  question, %no action is required).}

%\DrinJ{Below is a slight modification of your proof. Could you please check
%it?}

\begin{proof}
Consider the scalar product of $i_*(|\omega|)$ against $f \in C_c^\infty(Y \times F)$. It suffices to get for it an estimate of the form
\begin{equation}   \label{e:form_of_estiamate}
|i_*(|\omega|)(f)|\le \sup_{(y,x)\in Y\times F} |u(y,x)f(y,x)|,
\end{equation}
where $u$ is some polynomial on $Y\times F$.

For brevity, write $y$ instead of $(y_1,\dots,y_n)$ and $y^r$ instead of $\prod\limits_{i=1}^n y_i^{r_i}$.
Set
$$s(y):= \prod\limits_{i=1}^n (1+|y_i^{2}|).$$
We have
\begin{multline*}
|i_*(|\omega|)(f)|=|\int_{Y_0} y^{r} f(y,p(y)^{-1})dy |\leq\\
\leq \int_{Y_0} |y^{r}| \cdot|f(y,p(y)^{-1})|dy\leq C\cdot \sup_{y\in Y_0} s(y)\cdot |y^r|\cdot |f(y,p(y)^{-1})|,
\end{multline*}
where $C:= \int\limits_{Y}  s(y)^{-1}dy$.


The conditions on $l_i$ and $r_i$ imply that for $N$ big enough the function $q(y):=p(y)^N\cdot y^r$ is a polynomial. We have
%Now set $q(y):=p(y)^N\cdot y^r$, where $N:=\max\limits_i (-r_{i})$. Then $q$ is a polynomial
%(because $r_i \geq 0$ whenever $l_i=0$). So
\begin{multline*}
 \sup_{y\in Y_0} s(y)\cdot |y^r|\cdot |f(y,p(y)^{-1})|=\sup_{y\in Y_0} s(y)\cdot |q(y)|\cdot |p(y)|^{-N}\cdot |f(y,p(y)^{-1})|\le\\
 \le
\sup_{(y,x)\in Y\times F} s(y)\cdot |q(y)|\cdot |x^N|\cdot |f(y,x)|.
\end{multline*}
Thus we get an estimate of the form \eqref{e:form_of_estiamate}.
\end{proof}
}
\RamiY{Theorems \ref{thm:futerO} and \ref{exp} and Corollary \ref{cor:exp} are also proven in the same way as in the non-Archimedean case. So we are left with Lemma \ref{lem:Sch2} and Theorem  \ref{thm:futer}.}



\RamiAG{In fact, we will need a slightly stronger version of Lemma \ref{lem:Sch2}.
For its formulation we will need the following notion.\footnote{{In connection with
Definition~\ref{d:strictly continuous}, see Definition~\ref{def:cont_dep_arch} and Remark~\ref{r:closed graph}.}}
\begin{defn}   \label{d:strictly continuous}
Let $V$ be a real vector space and $X,Y$ be  smooth manifolds.
We call a family of generalized functions $\xi_t \in C^{-\infty,V}(Y \times V)$ parameterized by $t \in X$ strictly continuous if it gives rise to a continuous map $C_c^{\infty,V}(Y \times V,D_{Y \times V}) \to C(X)$\RamiAH{, where the topology on $C(X)$ is the open compact one}.
\end{defn}
The following lemma is a  stronger version of Lemma \ref{lem:Sch2}:
\begin{lem} \label{lem:Sch3}
%In the notations of  Lemma \ref{lem:Sch2},
%%%let $F$ be an Archimedean local field, with an embedding $K \hookrightarrow %F$.  Then
%the family of distributions
%$\xi_t:= (\phi_F)_*((\psi_t \circ p_F) \cdot |\omega_F|)\in C^{-\infty,W_F}(Y(F) \times W_F,D_{Y(F) \times W_F})$
%(where $\psi_{t}$ is the additive character of $F$ defined by $\psi_t(x)=\psi(tx)$) is strictly continuous.
{
In the situation of  Lemma \ref{lem:Sch2}, set
$$\xi_t:= (\phi_F)_*((\psi_t \circ p_F) \cdot |\omega_F|)\in C^{-\infty}(Y(F) \times W_F,D_{Y(F) \times W_F}),$$
where $t\in F$ and $\psi_{t}$ is the additive character of $F$ defined by $\psi_t(x)=\psi(tx)$. Then each $\xi_t$
is partially Schwartz along $W_F$ and the family of distributions $\xi_t$, $t\in F$, is  strictly continuous.
}
\end{lem}
}


\RamiAG{In order to prove this lemma} we will need the following \RamiAG{one}:
\begin{lem}\label{lem:dir_is_Sch}
Let $Y$ be \RamiAG{an algebraic} manifold and let $V_1,V_2$ be finite dimensional $F$-vector spaces. Choose a  \RamiAG{Haar measure} on $V_2$.  Let $Z\subset Y \times V_1\times V_2$ be an algebraic subvariety such that the projection of $Z$ \RamiAG{to} $Y \times V_1$ is proper (and hence finite). Let $\xi$ be a distribution on $Y(F) \times V_1\times V_2$ which is Schwartz along $V_1\times V_2$ and supported on $Z(F)$. Let $p: Y \times V_1\times V_2 \to  Y \times V_1$ be the projection. Then $p_*(\xi)$ is Schwartz along $V_1$.

\RamiAG{
Moreover, if $\xi_t \in C^{-\infty,V_1\times V_2}(Y(F) \times V_1\times V_2)$ is a strictly continuous family of distributions which are supported on $Z(F)$, then $p_*(\xi_{t}) \in C^{-\infty,V_1}(Y(F) \times V_1)$ is a strictly continuous family of distributions.
}
\end{lem}
\RamiAG{
For the proof we will need the following lemma:
\begin{lem}\label{lem:ex_poly}
Let $Y$ be an affine algebraic manifold and $V$ be a finite dimensional $F$-vector space. Let $Z\subset Y \times V$ be an algebraic subvariety such that the projection of $Z$ to $Y$ is proper (and hence finite). Then there exists a real polynomial $p$ on $Y$ and a norm $\| \cdot \|$ on $V$ such that for any $(y,v) \in Z(F)$, we have {$\max (\| v \|,1) \le p(y)$.}

% \DrinK{
% Rami, I am requiring $p(y)\ge 1$ because otherwise I don't see why the operator $pr^*_p$  from the
% proof of Lemma \ref{lem:dir_is_Sch} can be continuously extended to a map $C_c^{\infty,V_1}(X \times V_1)\to C_c^{\infty,V_1\times V_2}(X \times V_1 \times V_2)$. (Indeed, after differentiating the function $f(y,v_1,v_2)=\phi(\| v_2 \|/p(y,v_1))$ you get $p$ in the denominator.) I hope that once we require $p(y)\ge 1$ then
% $pr^*_p$ can be continuously extend. Please doublecheck this.
%
% }
\end{lem}

\begin{proof}
Let $z_i$ be the coordinates on $V$. Since the projection of $Z$ to $Y$ is  finite, we can find polynomials $\{a_{ij}\}_{j=1\dots N_i}$ on $Y$ such that $(z_i)^{N_i+1}+\sum_{j=1\dots N_i} a_{ij}(y)(z_i)^j=0$ for all $(y,z)\in Z(F)$. This easily implies the assertion.
%\RamiQV{To  be  added}
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem:dir_is_Sch}]
%For simplisity fix polinomial masures on $Y,V_1$...
{We can assume that $Y$ is affine.}
By Lemma \ref{lem:ex_poly}, we can find a real polynomial $p$ on $Y \times V_1$ such that for any $(y,v_1,v_2)\in Z(F)$, we have %$\| v_2 \| < p(y,v_1)$.
{$\max (\| v \|,1) \le p(y)$.} Let $\phi$ be a smooth  function on $\R$ such that $\phi([-1,1])=1$ and $\phi(\R-[-2,2])=0$. Let $f\in C^\infty(Y \times V_1\times V_2)$ be defined by
$f(y,v_1,v_2)=\phi(\| v_2 \|/p(y,v_1)).$ Let $$pr:X \times V_1 \times V_2 \to X \times V_1$$ be the projection. Define $$pr^*_p:C_c^{\infty}(X \times V_1,D_{X \times V_1})\to C_c^{\infty}(X \times V_1 \times V_2,D_{X \times V_1 \times V_2})$$ by $pr^*_p(g)=pr^*(g)\cdot f$. It is easy to see that $pr^*_p$  can be continuously extended to a map $$C_c^{\infty,V_1}(X \times V_1)\to C_c^{\infty,V_1\times V_2}(X \times V_1 \times V_2)$$   and that for any $g\in C_c^{\infty}(X \times V_1)$ and $\xi \in C_c^{-\infty}(X \times V_1 \times V_2)$, we have:
$$\langle \xi, pr^*_p(f)\rangle=\langle pr_*(\xi) ,f \rangle.$$ This proves the assertion.
%\RamiQV{I need to take care about masures}
\end{proof}



Now we can deduce Lemma \ref{lem:Sch3} from Lemma \ref{lem:dir_is_Sch} and Theorem  \ref{thm:main_C_Arch}\eqref{thm:main_C_Arch:1}.
\begin{proof}[Proof of Lemma \ref{lem:Sch3}]
%\RamiQV{To  be  added}
Let   $\phi'=\phi\times p :X \to Y\times W \times K$ and $\xi':=(\phi'_F)_*(|\omega_F|)$. By Theorem \ref{thm:main_C_Arch}\eqref{thm:main_C_Arch:1} the distribution $\xi'$ is partially Schwartz with respect to $W \times F$. For any $t \in F$, let $\xi'_t :=\xi' \cdot 1_{Y(F) \times W_F} \boxtimes \psi_t$. It is easy to see that $\xi'_t$ is a strictly continuous family of partially Schwartz distributions and $\pr_*(\xi'_t)=\xi_t$. Lemma \ref{lem:dir_is_Sch} now implies the assertion.
\end{proof}

%\DrinK{
Now let us prove Lemma \ref{lem:mu_t}  in the Archimedean case. The distributions $\mu_t$ and $\mu'$ from Lemma~\ref{lem:mu_t} can be written as
\[
\mu_t=\cF_{W_F}^*(\xi_t) ,\quad \mu'=\cF_{W_F}^*(\eta ) ,
\]
where
\begin{equation}  %\label{e:ditsribution_in_question1}
\xi_t:=(\phi_F)_*((\psi_t \circ p_F) \cdot |\omega_F|), \quad t\in F,
\end{equation}
\begin{equation}    %\label{e:ditsribution_we_control1}
\eta:=\cF_F^*((\phi'_F)_*(|\omega_F|)),
\end{equation}
and $\phi':X \to Y\times W \times K$ is defined by $\phi'=\phi\times p$.
By Lemma~\ref{lem:Sch3}, each $\xi_t$ is partially Schwartz along $W_F$ and the family of distributions
$\{\xi_t\}$ is  strictly continuous. So each $\mu_t$ is a well-defined distribution and the family $\{\mu_t\}$ is  continuous. This proves Lemma~\ref{lem:mu_t}(1). It is easy to check that the distribution on
$Y(F)\times W_F \times F$ corresponding to the family $\{\xi_t\}$ equals $\eta$. By strict continuity of
$\{\xi_t\}$, this implies Lemma~\ref{lem:mu_t}(2), which says that the distribution on
$Y(F)\times W^*_F \times F$ corresponding to the family $\{\mu_t\}$ equals $\mu'$.

Theorem \ref{thm:futer} is deduced from Lemma~\ref{lem:mu_t} just as in the non-Archimedean case.
%}





%The deduction of Theorem \ref{thm:futer} from Lemma \ref{lem:mu_t} is the same as in the non-Archimedean
%case. % In order to proof Lemma ?? we will need the following notion
%For the proof of the Archemedian counterpart of  Lemma 5.1.1  we will first
%prove the following lemma.
%%\[
%%\mu_t:=\cF^*_{W_F}((\phi_F)_*(p_{F}^{*}(\psi_{t})  \cdot |\omega_F|)),
%%\] where $\psi_{t}$ is the additive character of $F$ defined by $\psi_t(x)=\psi(tx)$. %Note that $\mu_1=\mu$, %so
%\begin{lem}\label{lem:xi_t}
%$ $
%Let
%\begin{equation}  \label{e:ditsribution_in_question1}
%\xi_t:=(\phi_F)_*((\psi_t \circ p_F) \cdot |\omega_F|),\quad t\in F
%\end{equation}
%and
%\begin{equation}    \label{e:ditsribution_we_control1}
%\eta:=\cF_F^*((\phi'_F)_*(|\omega_F|)),
%\end{equation}
%where $\phi':X \to Y\times W \times K$ is defined by $\phi'=\phi\times p$.
%Then
%%\begin{enumerate}
%%\item \label{lem:xi_t:1} $\{\xi_t\}_{t\in F}$ is a (weakly)%strictly
%%continuous family of distributions on $Y(F)\times W_F$.
%\item \label{lem:xi_t:2}
%the distribution on $Y(F)\times W_F \times F$ corresponding to the family $\{\xi_t\}$ equals $\eta$; that is, for %any $f\in C^{\infty}_c(Y(F)\times W_F , D_{Y(F)\times W_F})$
%%\RamiQV{I need o take care about masures}
%and $g\in C^{\infty}_c(F)$, we have
%\begin{equation}\label{eq:arch_fam}
%\langle \eta,f \boxtimes g\rangle=\int_{t\in F}\xi_t (f)g(t) dt.
%\end{equation}
%%\end{enumerate}
%\end{lem}
%The proof is analogous to the proof of Lemma  \ref{lem:mu_t}  in the non-Archimedean case.
%\begin{cor}
%Equality \eqref{eq:arch_fam} holds for any $f\in C^{\infty,W_F}_c(Y(F)\times W_F)$
%%\RamiQV{I need to take care about masures}
%and $g\in \Sc(F)$ (in particular, the right hand side is defined).
%\end{cor}
%\begin{proof}
%$ $
%\begin{enumerate}[{Case} 1,]
%\item $g$ is compactly supported:\\
%follows from Lemma \ref{lem:xi_t} since both sides of \eqref{eq:arch_fam} depend continuously on $f \in
%C^{\infty,W_F}_c(Y(F)\times W_F)$ (the left hand side
%%\RamiQV{I need o take care about masures}
%by definition and the right hand side by strict continuity, which was proved in Lemma~\ref{lem:Sch3}).
%\item the general case:\\
%follows from the previous case.
%\end{enumerate}
%\end{proof}
%
%The Archimedean case of Lemma \ref{lem:mu_t} follows immediately from the last corollary.
}



%\RamiQN{I think I can give a simple proof of this lemma and I can reduce %everything else to it. I do not see an easier way. What do you think about %this approach?}
%\RamiQM{Here I need to think more.}
\appendix
\section{The wave front set} \label{app:WF}
\setcounter{lemma}{0}
%\RamiQH{I need to re-write this section uniformly, so I think we should %use the general distribution notations rather than Schwartz.}
In this section we give an overview of the theory of the wave front set \RamiAH{as developed in \cite{Hor} for the Archimedean case and in \cite{Hef} for the non-Archimedean case.

We will discuss these two cases simultaneously. We will discuss the wave front set of general distributions which are functionals on smooth compactly supported functions. We will use the notations $C^{-\infty}$ and $C_c^{\infty}$ for the spaces of generalized functions and test functions as in \S\S\ref{ssec:arc_dist}. Note that in the non-Archimedean case, there is no difference between Schwartz functions and smooth compactly supported functions, and between general distributions and Schwartz distributions.
}

We explain here the results that  we quote in \S\ref{sssec:wfs}. We give an explicit  reference for some of them and provide proofs for the others.

\begin{defn}\label{def:wf}$ $
\begin{enumerate}
\item
Let $V$ be an $F$-vector space, \RamiAH{with} $\dim V<\infty$.
Let $f \in C^{\infty}(V^*)$ and $w_0 \in V^*$. We  say that $f$ \emph{vanishes asymptotically in the direction of} $w_0$
%\footnote{\RamiQC{I do not really like this name. Do you have better suggestion?}}
if there exists $\rho \in \RamiAH{C^\infty_c}(V^*)$ with $\rho(w_0) \neq 0$ such that the function $\phi \in C^\infty(V^* \times F)$ defined by $\phi(w,\lambda):=f(\lambda w) \cdot \rho(w)$ is a Schwartz function.

\item
Let $U \subset V$ be an open set and $\nu \in \RamiAH{C^{-\infty}(U,D_U)}$. Let $x_0 \in U$ and $w_0 \in V^*$.
We say that $\nu$ is \emph{smooth at} $(x_0,w_0)$ if there exists a compactly supported \RamiAH{non-negative} function $\rho \in \RamiAH{C^\infty_c} (V)$ with $\rho(x_0)\neq 0$ such that $\cF^*(\rho \cdot \nu)$ vanishes asymptotically in the direction of
$w_0$.
\item
The complement in $T^*U$
of the set of smooth pairs $(x_0, w_0)$ of $\nu$ is called the
\emph{wave front set of} $\nu$ and denoted by $WF(\nu)$.
\end{enumerate}
\end{defn}
% \begin{definition}
% %\begin{itemize}
%  %\item  \RamiF{We say that a distribution $\xi$ on a linear space $W$ converges to $0$ along $v \in W$  if there ??
%  %\item
% Let $U \subset F^n$ be an open subset and $\xi \in \Sc^*(U)$ be a
% distribution. We say that $\xi$ is smooth at $(x_0, v_0) \in T^*U$
% if there are open neighborhoods $A$ of $x_0$ and $B$ of $v_0$ such
% that for any $\phi \in \Sc(A)$ there is an
% $N_\phi \in \R_{>0}$
% %$N_\phi > 0$
% such that
% for any
% $\lambda \in F$ satisfying
% $|\lambda| > N_\phi$
% %$\lambda > N_\phi$
% we have
% ${\cF^*((\phi \xi))}|_{\lambda B} = 0$. The complement in $T^*U$
% of the set of smooth pairs $(x_0, v_0)$ of $\xi$ is called the
% wave front set of $\xi$ and denoted by $WF(\xi)$.
% %\end{itemize}
% \end{definition}

%\begin{remark}$ $
%\begin{enumerate}
%\item
%\RamiN{In the non-Archimedean case this notion is equivalent to the one in} \cite{Hef} with two
%differences:
%
%\begin{enumerate}
%\item
%The notion in \cite{Hef} is more general and depends on the choice of
%some open subgroup $\Lambda \subset F^{\times}$ (in our case $\Lambda = F^{\times}$).
%
%\item
% In \cite{Hef} the wave front set of \RamiN{$\nu$} is defined to
%be a subset of $T^*U -(U \times \{ 0\})$ (in our notation this subset is
%$WF(\RamiN{\nu}) -(U \times \{0\})$.
%\end{enumerate}
%\RamiN{\item In the Archimedean case this notion is equivalent to the one in \cite{Hor} with
%analogous differences (when $\Lambda$ is fixed to be $\R_{> 0}$). Namely, let $WF_H$ be the
%the wave front set defined by L.~H\"ormander \cite[Definition 8.1.2]{Hor}, then we have
%\begin{equation}\label{eq:WF_relation}
%WF(\nu)-(U \times \{0\})= F^{\times}\cdot WF_H(\nu)
%\end{equation}
%\item Let us briefly explain why %our definition is
%\eqref{eq:WF_relation} holds. The main problem is that %the approach in \cite{Hor}
%H\"ormander does not require the function $\phi$ from Definition \ref{def:wf}(1) to be Schwarz
%but only requires that it rapidly decay at $\infty$ (without condition on derivatives). However this
%is overcome using two facts:
%\begin{enumerate}
%\item Let $D$ be a differential operator with constant coefficients. Then
%$D \cF^*( \rho \cdot \nu )=\cF^*(p\cdot \rho \cdot \nu )$, where $p$  is the polynomial
%corresponding to $D$.
%
%\item Lemma 8.1.1 of \cite{Hor} which states that if $\cF^*(\nu )$ vanishes asymptotically in the
%direction of $v$ then  so does $\cF^*(\rho \cdot \nu )$
%for any smooth compactly supported function $\rho$.
%\end{enumerate}
%\item The relation with the definition of \cite{Hef} can be also easily explained using the non-
%Archimedean analog of Lemma 8.1.1 of \cite{Hor}
%}
%\end{enumerate}
%\end{remark}

\DrinF{\begin{remark}
Let $WF_H (\nu )$ denote the wave front set defined by L.~H\"ormander \cite[Definition 8.1.2]{Hor} for $F=\R$ and by D.~Heifetz \cite{Hef} for non-\RamiP{Archimedean} fields $F$. Let us explain the relation between
$WF_H (\nu )$ and $WF (\nu )$. First of all, $WF_H (\nu )$ is a subset of $T^*U -(U \times \{ 0\})$
stable under multiplication by $\lambda\in\Lambda$, where $\Lambda \subset F^{\times}$ is
some open subgroup (the definition of $WF_H$ from \cite{Hef} explicitly depends on a choice of
$\Lambda$, H\"ormander always takes $F=\R_{> 0}$)\RamiO{, However it is not necessarily stable under multiplication by
 $F^{\times}$}. Second,
\begin{equation}\label{eq:WF_relation}
WF(\nu)-(U \times \{0\})= F^{\times}\cdot WF_H(\nu).
\end{equation}
To prove \eqref{eq:WF_relation} for $F=\R$, one needs the following observation.
In Definition \ref{def:wf}(2) we require not only the function $\cF^*( \rho \cdot \nu )$ to rapidly decay at $\infty$ but also the same property for $D \cF^*( \rho \cdot \nu )$, where $D$ is any differential operator with constant coefficients. However, it suffices to require the rapid decay of
$\cF^*( \rho \cdot \nu )$ (as in \cite[Definition 8.1.2]{Hor}): the rest follows from
\cite[Lemma 8.1.1]{Hor} combined with the formula $D \cF^*( \rho \cdot \nu )=\cF^*(p\cdot \rho \cdot \nu )$, where $p$  is the polynomial corresponding to $D$.
\end{remark}
}
%\RamiQC{I'll need to edit the rest of this subsection accordingly.}


%\begin{remark}
%This notion appears in \cite{Hef} with two differences.

%1) The notion in \cite{Hef} is more general and depends on some
%subgroup $\Lambda \subset F$, in our case $\Lambda = F$.

%2) The notion in  \cite{Hef} defines the wave front set of $\xi$ to
%be a subset in $T^*U -U \times 0$. In our notation this subset will be
%$WF(\xi) -U \times 0$.
%\end{remark}

The following lemma is trivial.
{
\begin{lemma} \label{lem:WF_prop_aff}

Proposition \ref{prop:WF_prop} \RamiAH{\eqref{prop:WF_prop:1}-\eqref{prop:WF_prop:4}}
%(1-4)
holds for the case \RamiAH{when} $X\RamiAH{\subset}F^n$ \RamiAH{is an open set} and $\xi \in \RamiAH{C^{-\infty}(X,D_X)}$. Namely:
\begin{enumerate}
\item $P_{T^*(X)}(WF(\xi))=WF(\xi) \cap (X) =\Supp(\xi).$
\item If $WF(\xi) \subset X$ {if and only if }$\xi$ \RamiE{is smooth}.
\item Let $U \subset X$ be an open set. Then $WF(\xi|_U) = WF(\xi) \cap T^*(U).$
\item Let $\xi' \in \RamiAH{C^{-\infty}(X,D_X)}$ and $f,f' \subset C^\infty(X)$. %be locally constant functions.
Then
$$WF(f \xi+f' \xi') \subset WF(\xi) \cup
WF(\xi').$$
\end{enumerate}

\end{lemma}
}

%   \begin{lemma}
%   Let $U \subset F^n$ be an open subset and $\xi \in \Sc^*(U)$ be a
%   distribution. Then $WF(\xi)$ is closed, invariant with respect to the
%   homothety $(x,v) \mapsto (x,\lambda v)$ and
%   $$P_{T^*U}(WF(\xi))=WF(\xi) \cap (U \times 0) =\Supp(\xi).$$
%   \end{lemma}
%   \begin{lemma}
%   Let $V \subset U \subset F^n$ be open subsets and $\xi \in
%   \Sc^*(U)$. Then $WF(\xi|_V) = WF(\xi) \cap T^{*}V.$
%   \end{lemma}
%
%   \begin{lemma}
%   Let $U \subset F^n$ be an open subset, $\xi_1,\xi_2 \in \Sc^*(X)$
%   be distributions and $f_1,f_2 \in C^\infty(X)$ be locally constant functions. Then $WF(f_1 \xi_1+f_2 \xi_2) \subset WF(\xi_1) \cup
%   WF(\xi_2)$.
%   \end{lemma}

\begin{corollary}
For any locally constant sheaf \RamiAH{(or, in the Archimedean case, a vector bundle)}  $E$ on \RamiAH{$X$,} we can define the wave
front set of any element in $\RamiAH{C^{-\infty}}(\RamiAH{X},E)$.
{Moreover, the last lemma (Lemma \ref{lem:WF_prop_aff}) will hold in this case, too.}
\end{corollary}
%We will use the following statement from \cite{Hef}, see Theorem
%2.8.

\begin{proposition}%[\RamiAH{see \cite[Theorem ??]{Hor} for the Archimedean case and  \cite[Theorem 2.8.]{Hef} for the non-Archimedean case}]
[\RamiAH{see \cite[Theorem 8.2.4]{Hor} and  \cite[Theorem 2.8.]{Hef}}]
\label{submrtion}
Let $U \subset F^m$ and $V \subset F^n$ be open subsets, and
suppose that  $f: U \to V$ is an analytic submersion. Then for any
$\xi \in \G(V)$, we have $WF(f^*(\xi)) \subset f^*(WF(\xi))$.
\end{proposition}

\begin{corollary} \label{iso}
Let $V, U \subset F^n$ be open subsets and $f: V \to U$ be an
analytic isomorphism. Then for any $\xi \in \G(V)$, we have
$WF(f^*(\xi)) = f^*(WF(\xi))$.
\end{corollary}

\begin{corollary}\label{cor:bundle}
Let $X$ be an analytic manifold, $E$ be a locally constant sheaf \RamiAH{(or, in the Archimedean case, a vector bundle)}
on $X$ . We can define the wave front set of any element in
$\Sc^*(X,E)$ and $\G(X,E)$. Moreover, {Lemma \ref{lem:WF_prop_aff} and Proposition \ref{submrtion} hold for this case.}
%Proposition \ref{submrtion}
%holds for submersions between analytic manifolds.
\end{corollary}

% \begin{notation}
% Let \RamiA{$X,Y$ be manifolds} and let $p:X \to Y.$
% % be a
% %analytic map of analytic manifolds.
% Let $C \in T^*(X)$ be a (locally closed) submanifold.
% \RamiA{Let $dp: p^*(T(Y)) \to T(X)$ be the differential and  $dp^*: (T^*(X)) \to p^*(T^*(Y))$ be the dual map. Denote $p_*(C)=p({dp^*}^{-1}(C)) \subset T^*(Y).$
% Alternatively, in the analytic case one can define it on the set-theoretical
% level by $$\{(y,v) \in T^*(Y)|\exists x \in p^{-1}(y) \text{ with } (x,(d_xp)^*(v)) \in C \}$$
% %Denote by $p_*(C) \subset T^*(Y)$ the following set: $$\{(y,v) \in T^*(Y)|\exists %x \in p^{-1}(y) \text{ with } (x,(d_x(p))^*(v)) \in C \}$$
% }
% \end{notation}
%

\begin{proposition}%[\cite{Aiz}, Theorem 4.1.5] %\label{}
Proposition \ref{prop:WF_prop} \RamiAH{\eqref{it:G_act}}  holds.

\RamiAH{Namely,
  let} $G$ be an analytic group acting on an analytic manifold $X$ and a locally constant sheaf \RamiAH{(or, in the Archimedean case, a vector bundle)}   $E$ over it. Suppose $\xi \in \RamiAH{C^{-\infty}}(X,E)$ is
$G$-{invariant}. Then $$ WF(\xi) \subset \{(x,v) \in T^*X(F)|v(\g x)=0\}$$
where $\g$ is the Lie algebra of $G$.
\end{proposition}
\RamiAH{
\begin{proof}
In the non-Archimedean case, this is Theorem 4.1.5  of \cite{Aiz}.
In the  Archimedean case, the same proof works.
\end{proof}
}
The following  proposition is {essentially} proved in \cite{Gab} for the Archimedean case.
 We include its proof
%\RamiAH{for the non-Archimedean case}
here for completeness.
\begin{proposition} \label{prop:wf_push}
Let $p:X \to Y$ be {an analytic map}
%submersion
and \RamiL{$E$} be a locally constant sheaf \RamiAH{(or, in the Archimedean case, a vector bundle)}  over \RamiK{$Y$}.
 Let $\xi \in \RamiAH{C^{-\infty}(X,p^{\RamiAN{!}}(E))}$\RamiAN{, where $p^{!}$ is the pullback twisted by relative densities (see \S\S\S\ref{sssec:op_dist}\eqref{it:dist_notstd2})}. %\DrinK{What does %the question mark mean here?} \RamiQAB{It is the operation that you should %do with bundles to define pushforward. We introduce it in  \ref{sssec:op_dist}\eqref{it:dist_notstd2}. %I used this notations in other papers before, but it is not standard. I %do not know what is the standard notation and whether there is any. Both %\cite{Hor} and \cite{tre} do not discuss bundles. I think this can be thought %of as an analytic version of $!$, where we replace the top differential %form by densities. Can you suggest a better notation? I asked about it in %the previous versions.}
% be a distribution.
Assume $p|_{\supp (\xi)}$ is proper.
Then
% $$\WF(p_*(\xi)) \subset \{(y,v) \in T^*(Y)|\exist x \in p^{-1}(y) \text{ %with } (x,(d_x(p))^*(v)) \in \WF(\xi) \}$$
 $$\WF(p_*(\xi)) \subset p_*(\WF(\xi)) $$
\end{proposition}
\RamiAH{
For the proof we will need the following lemma:
\begin{lemma}\label{lem:WF_mon}
Let $V$ be an $F$-vector space, with $\dim V<\infty$. Let $\xi \in C^{-\infty}(V,D_V)$ be a compactly supported distribution. Assume $\cF(\xi)$ vanishes asymptotically along $v \in V^*$. Then for any $\rho \in C_c^{\infty}(V)$, the function $\cF(\rho \cdot \xi)$ vanishes asymptotically along $v$.
\end{lemma}
\begin{proof}
In the Archemedian case, this is  Lemma 8.1.1 from \cite{Hor}. In the non-Archemedian case, it is obvious.
\end{proof}
}

\begin{proof}[\RamiAH{Proof of Proposition \ref{prop:wf_push}}]
$ $
\begin{enumerate}[{Case} 1.]
 \item  $p$ is a submersion.\\
Without loss of generality, we may assume \RamiAH{that $E=D_Y$,} %is trivial
$Y=F^k$ and $X=Y \times D$, where $D \subset F^n$ is a standard
\RamiAH{open}
 poly-disk, \RamiAH{$\supp(\xi) \subset  Y \times D'$ where  $D' \subset D$ is a closed poly-disk} and $p$ is the projection. Let $y \in Y$. Let $v \in T^*_y{Y}$ be such that for any $x \in D$, we have $((y,x),(d_{(y,x)}(p))^*(v)) \notin \WF(\xi)$. We have to show that $p_*(\xi)$ is smooth at $(x,v)$.

By the definition of $WF(\xi)$, we can find for any $x \in \RamiAH{D}$  \RamiAH{a non-negative function $f_x\in C_c^\infty(X)$ such that $f_x((y,x))\neq 0$ and $\cF(f_x \cdot \xi)$ vanishing asymptotically in the direction of $(0,v)$.
%Using the partition of unity,
{So} we can construct a non-negative function $f\in C_c^\infty(X)$ such that $f$ does not vanish on
{$p^{-1}(y)\cap (Y\times D')$} and $\cF(f \cdot \xi)$ vanishes asymptotically in the direction of $(0,v)$. By Lemma \ref{lem:WF_mon}, we may assume that {$f$ has the form $f(y',x')=g(y')$ for
%any $(y',x') \in Y \times D=X$ and some non-negative
some $g \in C_c^\infty(Y)$}. This implies that $\cF(g \cdot p_*(\xi))$ vanishes asymptotically in the direction of $v$.
%Withou loss of generality we may asume that $f_x$ is the characteristic function of the product $A_x \times A'_x$ where $x \in A_x \subset D$, $y \in A'_x \subset Y$ are nigberhouds
}



 %neighborhoods $x \in A_x \subset D$, $y \in A'_x \subset Y$, $0 \in B_x \subset T^*_x(D)$ and $v \in B'_x \subset
% T^*_y(Y)$ such that \RamiM{the following condition holds:}
%
%for any $\phi \in \Sc(A_x \times A'_x)$ there exists $N_{\phi,x}$ such that
%for any
%$\lambda \in F$ satisfying
%$|\lambda| > N_{\phi,x}$\RamiM{,}
%we have
%${\cF^*(\phi \xi))}|_{\lambda (B_x\times B'_x)} = 0$.

%Let $U_i$ be a finite, open, compact refinement of the cover $A_x$\RamiM{,} i.e. \RamiM{a} finite, open, compact cover of $D$ such that for any $i$ there exists $x_i$ such that $U_i \subset A_{x_i}$. Let $A =\bigcap A'_{x_i}$ and $B=\bigcap B'_{x_i}$. It is enough to prove that for any $\phi \in \Sc(A)$, there exists $N_{\phi}$ such that
%for any
%$\lambda \in F$ satisfying
%$|\lambda| > N_{\phi}$
%we have
%${\cF(\phi (p_*\xi))}|_{\lambda B} = 0$.


%Let ${\alpha}_i \in \Sc(U_{i})$ be a partition of unity for  the cover $U_i$. Consider ${\alpha}_i \boxtimes \phi \in %\Sc(A_{x_i} \times A'_{x_i}).$ Let $N_\phi=max_i N_{{\alpha}_i \boxtimes \phi,x_i}$. Let $\lambda \in F$ satisfying
%$|\lambda| > N_{\phi}$. Now,
%\begin{multline*}
%\Fou{(\phi (p_*\xi))}|_{\lambda B}=\Fou( (p_*\sum ({\alpha}_i \boxtimes \phi) \xi))|_{\lambda B}=
%\\= \sum \Fou( (p_* ({\alpha}_i \boxtimes \phi) \xi))|_{\lambda B}= \sum (\Fou(({\alpha}_i \boxtimes \phi) \xi)|%_{0\times T^*_y(Y) })|_{\lambda B}=0
%\end{multline*}


\item  $p$ is a closed embedding.\\
Without loss of generality, we may \RamiM{assume  $E$} is trivial,
 $X=D^k$ \RamiM{and} $Y =X \times D^n$, where $D \subset F$ is a  disk and $p$ is the standard embedding.
In this case the assertion is obvious.


\item  The general case.\\
It follows from the previous cases by decomposing $p=pr \circ \gamma$, where $\gamma:X \to X \times Y$ is the graph embedding
and $pr:X \times Y \to Y$ is the projection.


\end{enumerate}
\end{proof}
\subsection{\RamiAM{Pullback of distributions}}
\RamiAM{In order to discuss pullback of
distributions under general maps,}%
%Now we would like to discuss Proposition \ref{prop:WF_prop3}. For this,
\RamiAH{ we need to define the topology on the space $C_\Gamma^{-\infty}(X,E)$ of generalized sections whose wave front set is included in $\Gamma \subset T^*(X)$.
\begin{defn}
We will define the topology in terms of converging sequences rather than open sets, but one can easily modify this definition in order to get an actual definition of topology.
% \RamiQW{\cite{Hor} does the same thing even without saying this. So I think %it is  OK}. \DrinK{Yes, it is OK.}
 Let us first define some auxiliary topologies on some related spaces.
\begin{enumerate}
 \item In the Archimedean case, the space of Schwartz functions on a vector space $V$ is equipped with a well known \Fre topology. In the non-Archimedean case, we say that a sequence of Schwartz functions  converges if all its elements are in the same finite dimensional vector space and it converges there.
\item We say that a sequence of functions $f_i$ in the space $C^{\infty}_{v_0}(V)$ of smooth functions on $V$ which vanishes asymptotically along ${v_0}$ converges if there exists a  $\rho \in C^\infty_c(V)$ with $\rho(v_0) \neq 0$ such that the sequence of functions $\phi_i \in C^\infty(V \times F)$ defined by $\phi(v,\lambda):=f_i(\lambda v) \cdot \rho(w)$ converges in the $\Sc(V)$.
\item We say that a sequence of distributions $\xi_i \in C^{-\infty}_\Gamma(V,D_v)$  converges if it weakly converges and for any $(x,w)\notin \Gamma,$ there exists a function $\rho \in C^\infty_c(V)$ with $\rho(x) \neq 0$ such that the sequence $\cF(\rho \cdot \xi_i) \in C^{\infty}_{w}(V^*)$ converges.
\item This easily defines  a topology on $C^{-\infty}_\Gamma(X,E)$, for any analytic variety $X$ and a locally constant sheaf (or in the Archimedean case, a vector bundle) on $X$.
\end{enumerate}
\end{defn}


\begin{proposition}[{\cite[Theorem 8.2.4.]{Hor} and  \cite[Theorem 2.8.]{Hef}}]
\label{prop:2WF_prop3}
%We have the following generalization of Proposition \ref{prop:WF_prop2} \eqref{prop:WF_prop2:2}.
Let $p: Y \to X$ be an analytic map of analytic manifolds, and let
$$N_p= \{(x,v)\in T^{*}X| x = p(y)  \text{ and } d_{\RamiAB{y}}^* p(v)=0 \text{ for some } y\in Y \}.$$
Let $E$ be a locally
constant sheaf (or, in the Archimedean case, a vector bundle) on $X$.
Let $\Gamma \subset T^{*}X$ be a conic closed subset such that $\Gamma \cap N_p \subset X$.

%Let  $$\G_p(X,E)=\{\xi \in \G_p(X,E)|WF(\xi) \cap N_p \subset X \}.$$
Then the  map $p^{*}:C^\infty(X,E)\to C^\infty(Y,p^*(E))$
 has a unique continuous extension
%\RamiQO{Here I need to clarify about the topology}
to a
map
$p^{*}:C^{-\infty}_\Gamma(X,E)\to C^{-\infty}(Y,p^*(E))$.
 Moreover,
for any $\xi \in C^{-\infty}_\Gamma(X,E)$, we have:
$WF(p^*(\xi)) \subset p^*(WF(\xi))$.
\end{proposition}
}


\begin{rem}\label{rem:iver_im}
Here is an explicit procedure to compute $p^*(\xi)$: we may assume that $X$ is a vector space and $E$ is trivial. Let $f_1\in C^{\infty}_c(X)$ and $f_2\in C^{\infty}_c(X,D_X)$ such that $f_1(0)=1$ and $\int f_2=1$. Let $\lambda_{i} \in F$ be a sequence that converges to infinity. Let $\xi_i:= \rho_{\lambda_{i}}(f_{1}) \cdot (\rho_{\lambda_{i}^{-1}}(f_{2}) * \xi)$, where $\rho_{\lambda_{i}}$ is the homothety as defined in Notation \ref{not:F_prop}. Note that $\xi_i$ are smooth and compactly supported functions. Now, $p^*(\xi)$ is the weak limit of $p^*(\xi_i)$.
\end{rem}

%\DrinK
{
Now we can prove Proposition~\ref{prop:Treves}. First, let us recall its formulation.
Let $$\xi\in \G(X \times Y)$$ be a generalized function on a product of analytic manifolds. Assume that
$\xi$ depends continuously on $Y$, so for each $y\in Y$ we have the generalized function $\xi|_{X \times \{y\}}$
from Definition~\ref{def:cont_dep}. Assume also that $WF(\xi) \cap CN_{X \times \{y\}}^{X \times Y}\subset X \times Y$, so for each $y\in Y$ we have the pullback $j_y^*(\xi )$ in the sense of
Proposition~\ref{prop:2WF_prop3}, where
$j_y:X \times \{y\}\hookrightarrow X \times Y$ is the embedding. Proposition~\ref{prop:Treves} says that in this situation
$$j_y^*(\xi)=\xi|_{X \times \{y\}}\, .$$
To prove this equality, it suffices to compute $j_y^*(\xi)$ using Remark~\ref{rem:iver_im} and choosing $f_1, f_2\in C^{\infty}_c(X\times Y)$ to be compatible with the product structure on $X \times Y$.
}



%Proposition \ref{prop:Treves} follows from Remark \ref{rem:iver_im}, when we choose $f_1$ and $f_2$ to be %compatible with the product $X \times Y$.

\section{Symplectic geometry of the co-tangent bundle} \label{app:symp_geom}
% \label
%\oldWFHol
%\newWFHol
In this section we \RamiAH{provide a proof of the facts from the symplectic geometry of the co-tangent bundle that we used in \S\S\ref{sec:lag}}

\subsection{\RamiAH{Images of isotropic {subsets}}}
\RamiAH{
We will prove Lemma \ref{lem:iso_dir_invs_im} using  Remark \ref{rem:simp_gem}. For this, we will need the following notion:

\begin{defn}
Let $M$, $N$ be symplectic manifolds. A \hrefHid{http://ncatlab.org/nlab/show/Lagrangian+correspondence}{Lagrangian correspondence}  (or, respectively, isotropic correspondence) between them is a subvariety $L \subset M \times N$ which is Lagrangian (or, respectively, isotropic) with respect to the symplectic form $\omega_M \boxtimes (-\omega_N)$.
\end{defn}

Lemma \ref{lem:iso_dir_invs_im} follows now from the next one:
\begin{lemma}\label{lem:lag_cor}
$ $
 \begin{enumerate}
  \item \label{lem:lag_cor:1} Let $\phi:X_1 \to X_2$ be a morphism of manifolds. Then the correspondence $\Lambda_\phi \subset T^*(X_1) \times T^*(X_2)$ described in Remark \ref{rem:simp_gem} is Lagrangian.
  \item \label{lem:lag_cor:2} Let $M$, $N$ be algebraic symplectic manifolds. Let $L \subset M \times N$ be an isotropic  correspondence  between them. Let $I \subset M$  be an isotropic {constructible subset}. Then the  {constructible subset} $L(I) \subset N$  is also isotropic.
 \end{enumerate}
\end{lemma}

{Both statements are well known. For the second one, see, e.g.,
\cite[Prop.~ 2.7.51]{CG} or \cite[Lemma~1]{G}. To prove the first one, note that the symplectic form on
$T^*(X_i)$ is the differential of the canonical 1-form $\eta_i$ on $T^*(X_i)$ and that the pullbacks of $\eta_1$  and $\eta_2$ to $\Lambda_\phi \subset T^*(X_1) \times T^*(X_2)$ are equal to each other.}

%\begin{proof}
%  Part  \eqref{lem:lag_cor:1} is obvious.
%Let us prove Part  \eqref{lem:lag_cor:2}. Let  $p$ and $q$ be the projections from $L$ to $M$ and $N$
%(correspondingly).
%\begin{enumerate}[{Case} 1:]
%\item $M,N$ are linear spaces, $L,I$ are linear subspaces and $p(L)=I$.\\
%Obvious.
%\item $L,I$ are smooth, %$L \subset I \times N$
% $p$ and $q$ are submersions to their images (which  are also smooth) and $p(L)=I$.\\
%Follows immediately from the previous case.
%\item %The case when
%$p(L)=I$.\\
%%%%Let $p,q$ be the projections from $L$ to $M$ and $N$ corespondingly.
%Let us prove the statement by induction on $\dim(L),\dim(I),\dim(p(L))$ and $\dim(q(L))$.
%
%%%% Let $I':=I \cup p(L)$ and $L':=L \cup p^{-1}(I)$.
%Let $L'$ be the smooth locus of $L$ and $L_0:=L-L'$. Clearly $L(I)=L'(I) \cup L_0(I)$. By Proposition
%\ref{prop:sub_iso}, the subvarieties  $L_0$ and $L'$ are isotropic. Thus, by  the induction hypothesis,
%$L_0(I)$ is isotropic. So it is left to show that $L'(I)$ is isotropic.
%
%Let $I'$ be the set of  smooth points of $I$ which are regular values of the map $p|_{L'}$ considered as a map
%$L' \to I$. By the (algebraic) Sard's lemma, $I'$ is Zariski dense in $I$. Let $L''= p^{-1}(I') \cap L$. Again, using
%Proposition \ref{prop:sub_iso} and the induction hypothesis, it is enough to show that  $L''(I')$ is isotropic.
%
%Let $S$ be the set of (smooth) regular values of the map $q|_{L''}$ considered as a map $L'' \to q(L'')$. Let
%$L''':=q^{-1}(S) \cap L''$. Again, it is enough to show that  $L'''(I')$ is isotropic. This follows from the previous
%case.
%\item The general case.\\
% Let $I':=I \cap p(L)$ and $L':=L \cap p^{-1}(I)$. Now the assertion follows from the previous case and
%Proposition \ref{prop:sub_iso}.
% \end{enumerate}
%   \end{proof}
}


% deduce \RamiI{the statements listed in \S\ref{sec:WF-Holonomic}}
%Proposition \ref{prop:WF_Hol}
%from the results described above.
%
%Proposition  \ref{prop:WF_Hol} (1) follows from Corollary \ref{cor:bundle} and the following obvious lemma.
%\begin{proposition}
%Let $X$ be an analytic variety. Let $\xi \in \Sc^*(X)$ be a WF-Holonomic %distribution.
%Let $p: Y \to X$ be an analytic submersion. Then $p^*(\xi)$ is WF-Holonomic.
%\end{proposition}

%\begin{lemma}
%Let $p:Y \to X$ be an
%analytic submersion  of smooth analytic varieties, and let $A \subset X$  be a smooth analytic  subvariety.
%Then $p^*(CN_A^X)=CN_{p^{-1}(A)}^Y$. \qed
% \end{lemma}


%{The following proposition clearly implies  Proposition \ref{prop:WF_Hol} %(2)}
%\begin{proposition} %[{* Proposition ?? (1)}]%\label{prop:hol_push}

%Let $p:X \to Y$ be  \oldWFHol{an algebraic} \newWFHol{{a proper analytic}}
%proper
%submersion of smooth \oldWFHol{algebraic} \newWFHol{{analytic}} varieties.\oldWFHol{ %Suppose $\fpt{p}:\fpt{X} \to \fpt{Y}$ is proper.} Let \oldWFHol{$\xi \in %\Sc^*(\fpt{X})$}
%\newWFHol{$\xi \in \Sc^*(X)$}
%  be a WF-holonomic distribution. Then
% \oldWFHol{$\fpt{(p})_*(\xi)$ }
% \newWFHol{$p _*(\xi)$ }
% is WF-holonomic.
%\end{proposition}
%Proposition \ref{prop:WF_Hol} (2) follows from Proposition \ref{prop:wf_push}
%and the following \Rami{lemma.
%one:

%\begin{proposition}
%Let $p:X \to Y$ be a proper submersion. Let $C \subset T^*(X)$ be a conic %Lagrangian algebraic subvariety then $p_*(C)$ is Lagrangian.
%\end{proposition}
%This proposition in turn follows from the lemma:

%}
%\begin{lemma} \label{lem:Lag_dir_im}
%Let $p:X \to Y$ be a
%%(??I think we do not need proper??)
%\RamiR{regular} {map}  of smooth \RamiJ{algebraic} varieties,
% and let $A \subset X$  be a smooth \RamiJ{algebraic} subvariety.
%Then $p_*(CN_A^X)$ is contained in a
%{union
%\begin{equation}\label{eq:un}
%\bigcup_{i=1}^k {CN_{A_i}^Y},
%\end{equation}
%where $A_i \subset X$ are some (locally closed) smooth subvarieties.
%}
%Lagrangian subvariety of $T^*Y$.
%\end{lemma}
%%it is remain to prove  lemma ??
%\begin{proof}%[proof of lemma ??]
%%\Rami{Let $p:=P_{T^*X} $}
%The proof is by induction on the dimension of $p(A)$. Let {$B=\overline{p(A)}$ %be the Zariski closure of p(A)
%}.
%\Rami{Consider $q:=p|_{A}:A \to B$ as a map from $A$ to $B$. Let $B' \subset B$ be the {set of the smooth points of $B$ which are regular values of $q$}. Let $A' :=q^{-1}(B')$.

% and $B'$ be the set of smooth points of $B$. let $A'=p^{-1}(B')$. Consider %the morphism $p|_{A'}:A' \to B'$  and let $B'' \subset B'$ be the set of %its regular values. Let $A''=p^{-1}(B'')$.
%Stratify $A$ to a union of smooth subvarieties $A=A' \cup \bigcup_i A_i$. Clearly $CN_{A} ^{X}\subset CN_{A'} ^{X} \cup \bigcup_i CN_{A_i} ^{X}$ and $\dim(p(A_i))< \dim(p(A))$. Thus by the induction hypothesis, it is enough to show that $p_*(CN_{A'} ^{X} )$ is {contained in a union as in (\ref{eq:un}).}
%Lagrangian.
%This immediately follows from the fact that  $p_*(CN_{A'} ^{X} )=CN_{B'} ^{Y}$.
%}
%\end{proof}



%\RamiI{
\subsection{\RamiAH{Equivalent definitions of isotropic  {subsets}}}
Let us now prove Lemma \ref{lem:isotr_discrip}.
%
%For this we will need the following proposition.
%\begin{proposition}[{Proposition 1.3.30 \cite{CG}}] \label{prop:sub_iso}
%A subvariety of an isotropic variety is isotropic.
%\end{proposition}
%\begin{remark}
% This proposition was proved in \cite{CG} for the algebraic case, but the
%  same proof works for analytic varieties.
%\end{remark}
%
%
%\begin{proof}[Proof of Lemma \ref{lem:isotr_discrip}]

Clearly (\ref{4}) $\Rightarrow$ (\ref{3}) $\Rightarrow$ (\ref{2}). By Proposition \ref{prop:sub_iso},
(\ref{2}) $\Rightarrow$ (\ref{1}).
%and by Proposition \ref{prop:sub_iso}, we have \ref{5} $\Leftrightarrow$  \ref{1}.


% It is easy to see that $\ref{4} $\Rightarrow$  \ref{5}.  Indeed, Let $$C %\subset \bigcup_i {CN_{A_i}^X}$$ as in \ref{4}. We may assume that $C$ is %irreducible. Then $C \cap CN_{A_i}^X$ is dense in $C$ for certain  $i$. %So it is enough to show that $C \cap CN_{A_i}^X$ is isotropic. This follows %from the fact that for any smooth point $x \in  C \cap CN_{A_i}^X$ the tangent %space $T_x C$ is contained in $T_x (CN_{A_i}^X)$ which is Lagrangian.

Now it is remains to show that \RamiK{(\ref{1})} $\Rightarrow$  (\ref{4}). We have to show that any isotropic $C \subset T^*X$
is contained in a union as in (\ref{4}). We will prove it by induction on $\dim (C)$ and $\dim P_{T^*X}(C)$\RamiAM{, where $P_{T^*(X)}:T^*(X)\to X$ is the projection}.
Let $C'$ be the set of smooth points in $C.$ Note that $\dim(C-C')< \dim (C)$ and by Proposition \ref{prop:sub_iso},
 $C-C'$ is isotropic. Thus, by induction, we may assume that  $C-C'$ satisfies (\ref{4}). Therefore it is enough
 to prove that $C'$ \RamiA{satisfies} (\ref{4}).


Consider the map $q:=P_{T^*X}|_{C'} $ as a map from $C'$ to
$\overline{P_{T^*X}(C')}$.   Let $U \subset \overline{P_{T^*X}(C')}$
be the  \RamiM{set of} those smooth points of $\overline{P_{T^*X}(C')}$ which are regular values of $q$,
 and let $C''=q^{-1}(U)$. Note that by the algebraic
 Sard lemma,
 $\dim(P_{T^*X}(\overline{C'-C''})){=\dim(\overline{P_{T^*X}({C'})}-U)}< \dim (P_{T^*X}(C')) =\dim (P_{T^*X}(C))$
 and by Proposition \ref{prop:sub_iso}, $\overline{C'-C''}$ is isotropic.
 Thus, by induction, we may assume that  $\overline{C'-C''}$ (and thus also ${C'-C''}$)
satisfies (\ref{4}). Therefore it is enough to prove that $C''$  satisfies (\ref{4}).
 We will prove that $C'' \subset {CN_{U}^X}$. For this let $x \in U $  and let $Y:=q^{-1}(x) \subset T^*_x X$.
 Fix any $y \in Y$. We know that $T_yC''$ is isotropic, i.e. $T_yC'' \bot (T_yC'')$.
Thus  we have $T_y Y = \ker d_y q \subset  (\Im d_y q)^\bot= CN_{U,x}^X.$ This implies that any connected
 component of $Y$ is a subset of a \RamiAH{shift} of $CN_{U,x}^X$. Since $\overline{Y}$  is conical, this shows that
 $Y \subset CN_{U,x}^X$.
%\end{proof}
%}

\subsection{\RamiAH{Co-normal bundle to a subbundle.}}
\RamiAH{Finally, let us prove Lemma \ref{l:symplectic_lem}. First recall its formulation:
}
\RamiCPE{
\begin{lemma}    \label{l:symplectic_lem_pf}
Let $V$ be a finite dimensional vector space over $F$ and $X$ be a manifold.
Let $E$ be a vector bundle over $X$ which is a subbundle of the trivial vector bundle $X \times V$.
Then $CN_{E^\bot}^{X \times V^*}=CN_E^{X \times V}$.

Here
$$CN_E^{X \times V} \subset T^*(X \times V)=T^*(X) \times V \times V^*,$$
$$CN_{E^\bot}^{X \times V^*} \subset T^*(X \times V^*)=T^*(X) \times V^* \times V,$$
and the symplectic manifolds $T^*(X) \times V \times V^*$ and $T^*(X) \times V \times V^*$ are identified via the map
$$V \times V^*{\buildrel{\sim}\over{\longrightarrow}} V^* \times V, \quad (v,w)\mapsto (w,-v).$$
\end{lemma}


\begin{proof}
Set $L:=CN_E^{X \times V}$. Without loss of generality we may assume that $X$ is irreducible. Then so is $L$.
Clearly $L$ is a closed Lagrangian submanifold of
$T^*(X \times V)=T^*(X) \times V \times V^*$. %\DrinK
{It is easy to check that the image of $L$ in $X \times V^*$ equals $E^\bot$. So it remains to show that
$L$ is conic as a submanifold of $T^*(X \times V^*)$. In terms of the action of
${\mathbb{G}}_m^3$ on $T^*(X) \times V \times V^*$, we have to show that $L$ is stable with respect to
the subgroup
\begin{equation}   \label{e:the_subgroup}
\{(\lambda, \lambda ,1)\,|\,\lambda\in {\mathbb{G}}_m\}\subset{\mathbb{G}}_m^3 \, .
\end{equation}

Since $E\subset X\times V$ is
${\mathbb{G}}_m$-stable the submanifold $L$ is stable with respect to the subgroup
\[
\{(1,\lambda,\lambda^{-1})\,|\,\lambda\in {\mathbb{G}}_m\}\subset{\mathbb{G}}_m^3\, .
\]
Clearly $L$ is conic as a submanifold of $T^*(X \times V)$, which means that $L$ is stable with respect to the subgroup
\[
\{(\lambda, 1, \lambda )\,|\,\lambda\in {\mathbb{G}}_m\}\subset{\mathbb{G}}_m^3\, .
\]
But $(\lambda, \lambda ,1)=(1,\lambda,\lambda^{-1})\cdot (\lambda, 1, \lambda )$,
so $L$ is stable with respect to the subgroup~\eqref{e:the_subgroup}.
}
\end{proof}

\begin{rem}
Here is a sketch of a slightly different proof of Lemma~\ref{l:symplectic_lem_pf}. The subbundle $E$ defines a map
\[
f:X\to \{\mbox{the Grassmannian of all subspaces of } V\}.
\]
Its differential at $x\in X$ is a linear map $T_xX\to\Hom (E_x,V/E_x)=\Hom (E_x,(E_x^\bot)^*)$; it defines a
bilinear map $B_x:E_x\times E_x^\bot\to T_x^*X$.
%\RamiQW{ Should I add ``given by $B_x(v,w)(t)=\langle(d_xf(t))(v),w\rangle$''?}.
One checks that $CN_E^{X \times V}$ has the following description in terms of $B_x$:
let $(x,\xi )\in T^*X$, $v\in V$, $w\in V^*$, then
\begin{equation}
(x,\xi ,v,w)\in CN_E^{X \times V} \Leftrightarrow v\in E_x, \, w\in E_x^\bot , \, \xi =-B_x (v,w).
\end{equation}
%\RamiQW{ Should I add the following explanation: ``Indeed,
%\begin{equation}
%(t,v') \in T_{x,v}E \Leftrightarrow v' +E_x = d_xf(t)(v)
%\end{equation}
%Thus
%\begin{multline}
%(x,\xi ,v,w)\in CN_E^{X \times V} \Leftrightarrow v\in E_x, \, \forall t,v' \text{ s.t. }v' +E_x = d_xf(t)(v) \text{ we
%have }, \phi(t)+\langle w,v'\rangle=0\\
%\Leftrightarrow v\in E_x,w\in E_x^\bot ,\forall t \text{ we have } \phi(t)+\langle(d_xf(t))(v),w\rangle=0
%\Leftrightarrow v\in E_x, w\in E_x^\bot , \, \xi =-B_x (v,w).
%\end{multline}
%'?
%}
Lemma~\ref{l:symplectic_lem_pf} follows from this description.
\end{rem}


\begin{rem}
Lemma \ref{l:symplectic_lem_pf} is closely related to the following fact: if $\xi\to X$ is any vector bundle and $\xi^*\to X$ is the dual bundle then
there is a canonical symplectomorphism between the cotangent bundles of $\xi$ and $\xi^*$ (see
\cite[Theorem 5.5]{MX} and also  \cite[Section 3.4]{Ro}).
\end{rem}
}


\begin{thebibliography}{Kly84}
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%Archimedean analog of Jacquet-Rallis' Theorem,} Duke Mathematical Journal,  149/3, 509-567 (2009). See also arXiv: 0812.5063[math.RT].
\RamiE{

\bibitem[\href{http://arxiv.org/abs/0811.2768}{Aiz}]{Aiz} A. Aizenbud, {\it A partial analog of the integrability theorem for distributions on p-adic spaces and applications}, arXiv:0811.2768, 2008. {To appear in the Israeli Math. Journal.}

\RamiCPA{

\bibitem[\href{http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160230202/abstract;jsessionid=2EDA339B5C15DA082B30A9B1A4066301.d03t01}{Ati}]{Ati}
M. F. Atiyah,
\emph{Resolution of singularities and division of distributions.}
Comm. Pure Appl. Math. {\bf 23}, 145-150, 1970.

%\bibitem[\href{}{??}]{p_Ati} \RamiQ{I should  find the reference for the p-adic analog of  Atyah's  proof of the analytic continuation of $p^\lambda$.}

\bibitem[\href{http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/bernstein-mod-dif-FAN.pdf}{Ber1}]{Ber_FT}
J. Bernstein,
\emph{Modules over the ring of differential operators; the study of
fundamental solutions of equations with constant coefficients.}
Functional Analysis and its Applications {\bf 5/2}, 1-16, 1972.

\bibitem[\href{http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf}{Ber2}]{ber_pl} J. Bernstein,
\emph{Analytic continuation of distributions with respect to a parameter.}
Functional Analysis and its Applications {\bf 6/4}, 26-40, 1972.


\bibitem[\href{http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-Gel-P-lam-FAN.pdf}{BG}]{BG}
J. Bernstein, S.I. Gelfand,
\emph{Meromorphic continuation of the function $P^{\lambda}$.}
Functional Analysis and its Applications {\bf 3/1}, 84-85, 1969.
}

%\bibitem[BZ76]{BZ}
\bibitem[BZ]{BZ}
%I.~N. Bern{\v{s}}te{\u\i}n and A.~V. Zelevinski{\u\i},
J. Bernstein, A.V. Zelevinsky,
\emph{Representations of
  the group {$GL(n,F),$} where {$F$} is a local non-Archimedean field},
  Uspehi Mat. Nauk {\bf 10/3}, 5-70, 1976. %\MR{MR0425030 (54  \#12988)}


\bibitem[CL]{CL} R. Cluckers, F. Loeser,
\emph{Fonctions constructibles exponentielles, transformation de Fourier motivique et principe de transfert.}
C. R. Math. Acad. Sci. Paris {\bf 341/12}, 741-746, 2005.


{
\bibitem[Con]{Con_Nag} B. Conrad,
%, Brian(1-MI)
Deligne's notes on Nagata compactifications.
J. Ramanujan Math. Soc. {\bf 22/3}, 2007;
%, no. 3, 205–257.
see also
B. Conrad,
\emph{Erratum for ``Deligne's notes on Nagata compactifications'',}
J. Ramanujan Math. Soc. {\bf 24/4}, 2009.
}

\RamiI{
\bibitem[\RamiAF{\href{http://www.springer.com/birkhauser/mathematics/book/978-0-8176-4937-1}{CG}}]%
{CG} N. Chriss, V. Ginzburg, \emph{Representation Theory and Complex Geometry}.
Birkh\"auser, Boston, MA, 1997.
%\href{http://books.google.com/books?id=lwS59rR78eIC&lpg=PR9&ots=8FVtuyIIRM&dq=Representation\%20Theory\%20and\%20Complex\%20Geometry\%20victor\%20ginzburg&lr&pg=PP1#v=onepage&q&f=false}
}


\bibitem[\href{http://www.jstor.org/stable/1996307?seq=1}{Gab}]{Gab} A. Gabor, Remarks on the wave front set of a distribution, Trans. Amer. Math. Soc., {\bf 170}, 239-244, 1972.

\bibitem[G]{G}V.~Ginzburg,
\emph{The global nilpotent variety is Lagrangian.}
Duke Math. J.  {\bf 109} (2001), no. 3, 511-519.



{

\bibitem[{Har}]{Har}
R. Hartshorne,
%Robin
 \emph{Algebraic geometry.} Graduate Texts in Mathematics, No. {\bf 52}. Springer-Verlag, New York-Heidelberg, 1977.

\bibitem[\href{http://projecteuclid.org/euclid.pjm/1102707065}{Hef}]{Hef} D. B. Heifetz, {\it $p$-adic oscillatory integrals and wave front sets,} Pacific J. Math. {\bf 116/2}, 285-305, 1985.

\bibitem[\href{http://www.ams.org/mathscinet-getitem?mr=199184}{Hir}]{Hir}
 H. Hironaka, %Heisuke
 \emph{Resolution of singularities of an algebraic variety over a field of characteristic zero.}
  I, II. Ann. of Math. {\bf 79/2}, 109-326, 1964. %, 109203 ; ibid. (2) 79 1964 205326.


\bibitem[\href{http://arxiv.org/abs/math/0510133}{HK}]{HK} E. Hrushovski, D. Kazhdan
\emph{Integration in valued fields.}
In: Algebraic geometry and number theory, Progr. Math. {\bf 253}, 261-405.
Birkh\"auser, Boston, MA, 2006.
(see also arXiv:math/0510133 [math.AG])
}

\bibitem[H\"or]{Hor} L. \Rami{H\"ormander}, {\it The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis.}
Grundlehren der Mathematischen Wissenschaften {\bf 256}. Springer-Verlag, Berlin, 1990.
{
\bibitem[\href{http://www.math.columbia.edu/~dejong/papers/ALTERATIONS.dvi}{Jon}]{Jon} J. de Jong,
\emph {Smoothness, semi-stability and alterations.}
 Inst. Hautes Etudes Sci. Publ. Math, {\bf 83}, 1996.
}

\RamiD{
\bibitem[{Kol}]{Hir_mod}
J. Kollar,
\emph{Lectures on resolution of singularities.}
Annals of Math. Studies. Princeton University Press, 2007.
}


\bibitem[{Mum}]{Mum}
D. Mumford,
% David
\emph{The red book of varieties and schemes.}
Lecture Notes in Mathematics, {\bf 1358}. Springer-Verlag, Berlin, 1988.

\RamiCPE{
\bibitem[MX]{MX} K. C. H. Mackenzie and Ping Xu, {\it Lie bialgebroids and Poisson groupoids,} Duke
Math. J., {\bf 73}, No.~2), 41--452, 1994.
}

\RamiCPE{
\bibitem[Ro\RamiAH{y}]{Ro} D.~Roytenberg, {\it Courant algebroids, derived brackets and even symplectic supermanifolds}, arXiv:math/9910078, 1999.
}

%\bibitem[Ser64]{ser}
\bibitem[Ser]{ser}
J.P. Serre, {\it Lie Algebras and Lie Groups.} Lecture Notes in Mathematics, {\bf 1500}. Springer-Verlag, New York, 1964.

}

\RamiAM{
\bibitem[Tre]{tre}
F. Treves, {\it Introduction to pseudodiferential Fourier integral operators.} Plenum Press, New York and London, 1980
}
\end{thebibliography}

\end{document}








\oldWFHol{
\begin{definition}
Lagrangian, \Rami{isotropic} subvariety??
\end{definition}
% \begin{definition}
% \Rami{Let $X$  be a variety. We say that $Y \subset T^* X$ is conic if it is invariant with respect to the homothety action of $F^\times$ on $T^* X$ given by $\rho_\lambda(x,v)=(x,\lambda v)$}.
% \end{definition}

\Rami{
We will need the following proposition.
\begin{proposition} [{Proposition 1.3.30 \cite{CG}}]
A subvariety of an isotropic variety is isotropic.
\end{proposition}


\begin{lemma} \label{lem:lag_discrip}
Let $X$ be an algebraic manifold.
Let $C \subset T^*(X)$ be a closed conic algebraic subvariety.
Then the following are equivalent:
\begin{enumerate}
\item $C$ is isotropic. \label{5}
\item $C$ \RamiA{is} contained in an isotropic \RamiA{sub}variety
 \RamiA{of $T^*(X)$}. \label{1}
\item $C$ \RamiA{is} contained in a Lagrangian \RamiA{sub}variety
\RamiA{of $T^*(X)$}. \label{2}
\item  There is a finite collection of smooth (locally closed) subvarieties $A_i \subset X$ such that $$C \subset \bigcup_i \overline{CN_{A_i}^X}$$\label{3}
\item There is a finite collection of smooth (locally closed) subvarieties $A_i \subset X$ such that $$C \subset \bigcup_i {CN_{A_i}^X}$$\label{4}

\end{enumerate}
\end{lemma}
\begin{proof}
Clearly \ref{4} $\Rightarrow$ \ref{3} $\Rightarrow$ \ref{2} $\Rightarrow$ \ref{1}, and by Proposition \ref{prop:sub_iso}, we have \ref{5} $\Leftrightarrow$  \ref{1}.


% It is esay to see that $\ref{4} $\Rightarrow$  \ref{5}.  Indid, Let $$C %\subset \bigcup_i {CN_{A_i}^X}$$ as in \ref{4}. We may assume that $C$ is %irredusebale. Then $C \cap CN_{A_i}^X$ is danse in $C$ for serten  $i$. %So it is enaf to show that $C \cap CN_{A_i}^X$ is isotropic. This follows %from the fact that for any smooth point $x \in  C \cap CN_{A_i}^X$ the tagent %space $T_x C$ is contained in $T_x (CN_{A_i}^X)$ which is Lagrangian.

Now it is remains to show that \ref{5} $\Rightarrow$  \ref{4}. We have to show that any isotropic $C \subset T^*X$ is contained in a union as in \ref{4}. We will prove it by induction on $\dim (C)$ and   $\dim P_{T^*X}(C).$
Let $C'$ be the set of smooth points in $C.$ Note that $\dim(C-C')< \dim (C)$ and by Proposition \ref{??}, $C-C'$ is isotropic. Thus, by induction, we may assume that  $C-C'$ satisfies \ref{4}. Therefore it is enough to prove that $C'$  \RamiA{satisfies} \ref{4}.

Consider the map $q:=P_{T^*X}|_{C'} $ as a map from $C'$ to {$\overline{P_{T^*X}(C')}$.   Let $U \subset \overline{P_{T^*X}(C')}$ be the set of {set of those smooth points of $\overline{P_{T^*X}(C')}$ which are regular values of $q$},} and let $C''=q^{-1}(U)$. Note that by {the algebraic} Sard lemma, $\dim(P_{T^*X}(\overline{C'-C''})){=\dim(\overline{P_{T^*X}({C'})}-U)}< \dim (P_{T^*X}(C')) {=\dim (P_{T^*X}(C))}$ and by Proposition \ref{prop:sub_iso}, $\overline{C'-C''}$ is isotropic.   Thus, by induction, we may assume that  $\overline{C'-C''}$ (and thus also ${C'-C''}$) satisfies \ref{4}. \RamiA{Therefore} it is enough to prove that $C''$  satisfies \ref{4}. We will prove that $C'' \subset {CN_{U}^X}$. For this let $x \in U $  and let $Y:=q^{-1}(x) \subset T^*_x X$. Fix any $y \in Y$. We know that $T_yC''$ is isotropic, i.e. $T_yC'' \bot (T_yC'')$. Thus  we have $T_y Y = \ker d_y q \subset  (\Im d_y q)^\bot= CN_{U,x}^X.$ This implies that any connected component of $Y$ is a subset of a sheaft of $CN_{U,x}
^X$. Since $\overline{Y}$  is conical, this shows that $Y \subset CN_{U,x}^X$.
\end{proof}

%The following lemma is well-known, see ??.

%\begin{lemma} \label{lem:lag_discrip}
%Let $C \subset T^*(X)$ be a conic Lagrangian closed algebraic subvariety. %Then there is a finite collection of smooth subvarieties $A_i \subset X$ %such that $$C = \bigcup_i \overline{CN(A_i)}$$
%\end{lemma}
}

\begin{definition}
Let $X$ be an algebraic variety. We call a distribution $\xi \in \Sc^*(X(F))$ WF-holonomic if \Rami{
the Zariski closure of $WF(\xi)$ satisfies the conditions of Lemma \ref{lem:lag_discrip}.
%
%there exists a Lagrangian algebraic variety $C \subset T^*(X)$ such that %$WF(\xi) \subset C$. Note that in this case we may assume that $C$ is conic.
}
\end{definition}
}


















\RamiI{
The following lemma gives several equivalent definitions of this notion:
\begin{lemma} \label{lem:lag_discrip}
Let $X$ be an \RamiJ{algebraic manifold}.
Let $C \subset T^*(X)$ be a closed conic \RamiJ{algebraic} subvariety.
Then the following are equivalent:
\begin{enumerate}
\item $C$ is isotropic. \label{5}
\item $C$ \RamiA{is} contained in an isotropic \RamiA{sub}variety
 \RamiA{of $T^*(X)$}. \label{1}
\item $C$ \RamiA{is} contained in a Lagrangian \RamiA{sub}variety
\RamiA{of $T^*(X)$}. \label{2}
\item  There is a finite collection of smooth (locally closed) subvarieties $A_i \subset X$ such that $$C \subset \bigcup_i \overline{CN_{A_i}^X}$$\label{3}
\item There is a finite collection of smooth (locally closed) subvarieties $A_i \subset X$ such that $$C \subset \bigcup_i {CN_{A_i}^X}$$\label{4}
\end{enumerate}
\end{lemma}
This lemma is standard. For completeness we include its proof in Appendix \ref{subapp:WFHol}.
}


 intend to say it there too. Just in the same way that we refered the reader
