\firstname{Christian
\firstname{Nguyen Viet
\firstname{Fr\'ed\'eric
The pull-back, push-forward and multiplication of smooth functions can be extended to distributions if their wave front set satisfies some conditions. Thus, it is natural to investigate the topological properties of these operations between spaces $\calD'_\Gamma$ of distributions having a wave front set included in a given closed cone $\Gamma$ of the cotangent space. As discovered by S. Alesker, the pull-back is not continuous for the usual topology on $\calD'_\Gamma$, and the tensor product is not separately continuous. In this paper, the pseudo-topology of $\calD'_\Gamma$ defined by H\"ormander is shown to be a bornology and a new topology is defined for which the pull-back and push-forward are continuous, the tensor product, the multiplication and the convolution product of distributions are hypocontinuous.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\title
[Functional properties
of the fundamental operations
on $\mathcal{D}^\prime_\Gamma$]
{Boundedness and continuity 
of the fundamental
operations on 
distributions having a specified wave front set.\\
\footnotesize{(with a counter example by Semyon Alesker)}}
% 
\author{\firstname{Christian} \lastname{Brouder}}
\address{ Institut de Min\'eralogie, de Physique des Mat\'eriaux et de
Cosmochimie, \\
Sorbonne Universit\'es,
UMR CNRS 7590, UPMC Univ. Paris 06,
\\ Mus\'eum National d'Histoire
Naturelle, IRD UMR 206,\\
4 place Jussieu, F-75005 Paris, France.\\}
\author{\firstname{Nguyen Viet} \lastname{Dang}}
\address{Laboratoire Paul Painlev\'e (U.M.R. CNRS 8524)\\
UFR de Math\'ematiques\\
Universit\'e de Lille 1\\
59 655 Villeneuve d'Ascq C\'edex France.}
\author{\firstname{Fr\'ed\'eric} \lastname{H\'elein}}
\address{Institut de Math\'ematiques de Jussieu Paris Rive Gauche, UMR CNR
S 7586\\ Universit\'e Denis Diderot Paris 7, UFR de Math\'ematiques,\\ Case 7012
, B\^atiment Sophie
Germain 75205 Paris Cedex 13, France.}


\keywords{microlocal analysis, functional analysis, mathematical physics, renormalization}  
  
\begin{document}  

\begin{abstract}
The pull-back, push-forward and multiplication
of smooth functions can be extended to distributions if 
their wave front set satisfies some conditions.
Thus, it is natural to investigate the topological
properties of these operations between 
spaces $\calD'_\Gamma$ of distributions having a wave front set
included in a given closed cone $\Gamma$ of the cotangent space.
As discovered by S. Alesker, the pull-back is not continuous
for the usual topology on $\calD'_\Gamma$,
and the tensor product is not separately continuous.
In this paper, the pseudo-topology of $\calD'_\Gamma$ 
defined by H\"ormander
is shown to be a bornology and a new topology is
defined for which the pull-back and push-forward
are continuous, the tensor product, the multiplication
and the convolution product
of distributions are hypocontinuous.
\end{abstract}
\maketitle 


\section{Introduction}
 The motivation
of our work comes
from the renormalization
of QFT in
curved space times,
indeed the
question adressed
in this
paper cannot
be avoided in this context 
and
also the technical
results of this paper
form the core 
of the proof
that perturbative
quantum field theories
are renormalizable
on curved space times \cite{Dangthese,DabrowskiDang}.

 Since L.~Schwartz, we know that the tensor product of distributions
is continuous~\cite[p.~110]{Schwartz-66}
and the product of a distribution by a smooth function
is hypocontinuous~\cite[p.~119]{Schwartz-66} (see definition \ref{hypodef}), although 
it is not jointly continuous~\cite{Kucera-81,Larcher-13}.

However, in many applications (for instance the multiplication of distributions
involved in quantum field theory),
we cannot work with all distributions and we must consider the 
subsets $\calD'_\Gamma$ of distributions whose wave front set is included
in some closed subset $\Gamma$ of 
$\dotT^*\bbR^n=\{(x;\xi)\in T^*\bbR^n\telque \xi\not=0\}$.
In the following, $\Gamma$ will always
denote a \textbf{cone}
i.e. $(x;\xi)\in\Gamma
\implies (x;\lambda\xi)\in\Gamma$
for every 
$\lambda\in\mathbb{R}_{>0}$.
Indeed the spaces $\calD'_\Gamma$
are widely used in 
microlocal analysis because wave front
set conditions
rule the fundamental 
operations on distributions 
(multiplication,
pull-back, push-forward and restriction) 
which are known to be sequentially 
continuous~\cite{Chazarain}.

H\"ormander himself, who introduced the concept of a wave front 
set~\cite{Hormander-71}, equipped $\calD'_\Gamma$ with a 
\emph{pseudo-topology}~\cite[p.~125]{Hormander-71},
which is just a rule describing the convergence of sequences and not a topology.
In particular, when H\"ormander writes that the
pullback is continuous~\cite[p.~263]{HormanderI}, 
he means ``sequentially continuous'', precisely because he does not
define any topology on $\calD'_\Gamma$
(in \cite{Hormander-71} H\"ormander states explicitly that
the fundamental operations are sequentially continuous).

Duistermaat's famous lecture notes~\cite{Duistermaat} are more
ambiguous because they define a locally convex topology on 
$\calD'_\Gamma$ and they state that the pull-back~\cite[p.~19]{Duistermaat},
the push-forward~\cite[p.~20]{Duistermaat} and 
the product of
distributions~\cite[p.~21]{Duistermaat} are continuous maps.
A similar confusion exists in the modern literature,
where Heifetz states that the 
pull-back by a smooth map is continuous~\cite{Heifetz-85}.

The purpose of the present paper is to clarify the situation, to
determine the precise regularity of the operations with distributions
and to define a new topology for which the fundamental operations
have optimal continuity properties.
Our paper is divided in two parts.

 In the first part, we review H\"ormander's
pseudo-topology and we
show that the right
framework to understand
it is the natural bornology
of $\mathcal{D}^\prime_\Gamma$ \cite{Dabrouder-13}.
Indeed this concept,
introduced by 
Bourbaki~\cite{Bourbaki-50},
is enough to control
the sequential continuity
in $\mathcal{D}^\prime_\Gamma$.
However, sequential continuity and
bornology
are not well 
adapted to 
applications in 
renormalization of QFT
on curved space times and
valuation theory 
\cite{Alesker-10}.
For such applications,
it is desirable
to have available
a topology on
$\mathcal{D}^\prime_\Gamma$ for
which the usual operations
of pull--back
and push--forward are
continuous 
and the tensor product
is hypocontinuous.
And most importantly, we
need to study situations
where the maps depend
smoothly on
extra parameters
so that
pull--back
and push--forward
should be \emph{uniformly continuous}
w.r.t. the 
extra parameters. 
Duistermaat, in his lecture notes
\cite{Duistermaat},
proposed a locally convex
topology on $\mathcal{D}^\prime_\Gamma$.
However, we shall present
in subsection \ref{aleskercounterex}
a counterexample due to S. Alesker
showing that
the tensor product
is not separately continuous
and the pull--back is not continuous
for this topology.

This prompts us to introduce
in the second part another, 
finer topology on $\mathcal{D}^\prime_\Gamma$.
This topology, called
the \emph{normal}
topology of $\mathcal{D}^\prime_\Gamma$
by Dabrowski and Brouder \cite{Dabrouder-13},
is defined in section \ref{normaltopsect}. 
In this new topology, we prove in Theorem \ref{hypotens} 
that the tensor product
is hypocontinuous. 
In sections \ref{pullbacksect} and \ref{prodpushsect}, 
we use 
the functional properties of 
$\calD'_\Gamma$ \cite{Dabrouder-13}, 
to give conceptually transparent
proofs, without hard analytic 
estimates and 
using
only the geometry
of wave front sets,
that:
\begin{itemize}
\item the pull-back 
by a smooth map is continuous
(Proposition \ref{pullbackprop})
\item the pull--back
by a family of smooth maps depending 
smoothly on parameters is uniformly
continuous (Theorem
\ref{pbenfamille})  
\item the push-forward  by a smooth map is
also continuous (Theorem \ref{pushforwardthm})
\item the push--forward by a family
of smooth maps
(Theorem \ref{pshfwdenfamille}) depending
smoothly on
parameters is uniformly
continuous 
\item the multiplication of distributions
(Theorem \ref{hypocontprod}) and the
convolution product (Theorem \ref{convolprod}) 
are hypocontinuous.
\end{itemize}

Finally in section \ref{Gaborsect}, we discuss how 
the wave front set of distributions
on manifolds can be defined in an intrinsic way.

In appendices, we prove important technical results
concerning the covering of the complement of $\Gamma$,
the topology of $\calD'_\emptyset$ and
the fact that the additional seminorms used to define
the topology of $\calD'_\Gamma$ can be taken to be 
countable.
\\
\begin{center}
 \textsc{First part}
\end{center}




\section{Sequential continuity in $\calD'_\Gamma$}
In his fundamental paper~\cite{Hormander-71}, H\"ormander
defines, for every open subset $\Omega$ of $\bbR^n$
and every closed cone $\Gamma\subset\dotT^*\Omega$,
 the space
$\calD'_\Gamma=\{u\in \calD'(\Omega)\telque \WF(u)
\subset\Gamma\}$. 
He endowes $\calD'_\Gamma$ with
a \emph{pseudo-topology}, a concept due to
Choquet~\cite{Choquet-48,Marinescu-63} where conditions
of convergence of sequences are given but no topology
is defined (for example by a family  of neighborhoods or of
open sets).
More precisely~\cite[p.~262]{HormanderI}, 
\begin{dfn}
\label{convdef}
Let
$\Omega$ be some open set in $\mathbb{R}^n$.
We say that a sequence $(u_j)\in \calD'_\Gamma(\Omega)$ 
converges to $u$ in $\calD'_\Gamma$ in
the sense of H\"ormander iff
\begin{itemize}
\item[(i)] $(u_j)$ converges to $u$ for the weak topology
   of $\calD'(\Omega)$,
\item[(ii)] for every $\chi\in \calD(\Omega)$,
every $N\in\mathbb{N}$ and every closed cone
  $V$ in $\bbR^n$ such that 
   $\supp\chi\times V \cap \Gamma=\emptyset$,
\begin{eqnarray*}
\sup_{k\in V} (1+|k|)^N |\widehat{u_j\chi}(k)-\widehat{u\chi}(k)|
  \to 0\text{ for }j\to\infty.
\end{eqnarray*}
\end{itemize}
\end{dfn}
A map $f:\calD_{\Gamma_1}(\Omega_1)\to \calD_{\Gamma_2}(\Omega_2)$
is said to be \emph{sequentially continuous} iff
the image of a sequence $(u_j)$ that converges to $u$
is a sequence $\big(f(u_j)\big)$ that converges to $f(u)$.


It was noticed by Hogbe-Nlend~\cite[p.~10]{Hogbe-71}
that a pseudo-topology can often be interpreted
in terms of a bornology. 
In this section, we prove that this is the case for
$\calD'_\Gamma$ and that a map is sequentially
continuous iff it is bounded.

We give some basic definitions of bornological concepts.


\subsection{Bornology}
\label{bornosect}
A bornology on $X$ is a family of bounded subsets
of $X$. The abstract definition is
\begin{dfn}
A \emph{bornology} on a set $X$ is a family
$\calB$ of subsets of $X$ (called the
\emph{bounded (sub)sets} of $X$) such that:
(i) every one-element subset of $X$ belongs to $\calB$;
(ii) if $A\in\calB$ and $B\subset A$ then $B\in\calB$
and (iii) if $A$ and $B$ are in $\calB$ then
$A\cup B\in \calB$.
\end{dfn}
These axioms are very natural properties of
bounded sets: a point is bounded, a subset of
a bounded set is bounded and the union of two
bounded sets is bounded.


 Among 
topological
vector spaces,
it is well known
that the locally
convex ones have nice 
properties. 
A similar 
concept holds 
for bornologies
but we need first to define a disked hull:
\begin{dfn}
A subset $B$ of a vector space $E$ over $\bbK$ ($\bbK=\bbR,\bbC$) is a 
\emph{disk} iff $\lambda x + \mu y\in B$ whenever
$x\in B$,  $y\in B$,  $\lambda\in \bbK$, $\mu\in \bbK$  and
$|\lambda|+|\mu|\le 1$.
The \emph{disked hull} of a subset $B$ of a 
vector space $E$ is the smallest disk of $E$ containing 
$B$.
\end{dfn}
Now,
a class
of bornological
vector space
which has good
properties
is the class
of convex
bornological spaces.
\begin{dfn}
Let $E$ be a vector space over $\bbK$. A bornology
$\calB$ on $E$ is said to be
a \emph{convex bornology}
if, for every $A$ and $B$ in $\calB$ and every
$\lambda\in \bbK$, the
sets $A+B$, $\lambda A$ and the disked hull of $A$
belong to $\calB$.
Then $E$ or $(E,\calB)$ is 
called a convex bornological space.
\end{dfn}

 Let $E$
be a topological vector space,
then by definition
the bounded sets are the subsets
$B$
of $E$ such that
for any neighborhood
$V$ of $0\in E$,
there exists $t\geqslant 0$ such that
for all $\lambda$ with $\vert \lambda\vert\geqslant t$,
$B\subset \lambda V$.
In particular, if $E$
is a \emph{locally convex}
space whose topology
is defined by a family of seminorms $(p_\alpha)_{\alpha\in I}$,
the bounded sets of $E$ are the 
sets that are bounded for every seminorm $p_\alpha$
and they form
a natural bornology, called the \emph{von Neumann} bornology of $E$.
%{\blue This bornology is called the \emph{von Neumann} bornology of $E$.}


It was proved
in~\cite[Thm 33]{Dabrouder-13} that 
all families of seminorms on
$\calD'_\Gamma$ which define 
topologies which are finer than the weak topology
and coarser than the Mackey topology
(see \cite[p.~IV.4]{Bourbaki-TVS} 
for the definition of
the Mackey topology
)
give the
same von Neumann bornology
(i.e. define the same bounded sets).
Hence this canonical von Neumann bornology 
will be called the
bornology of $\calD'_\Gamma$.

The interesting maps between two topological spaces 
are the continuous maps. The analogous objects for 
bornological spaces are the bounded maps:
\begin{dfn}
A map $f: E \to F$ between two bornological spaces is said
to be \emph{bounded} if the image of every bounded set
of $E$ is a bounded set of $F$.
\end{dfn}
We will show that sequential continuity and boundedness are
equivalent for $\mathcal{D}^\prime_\Gamma$.

\subsection{Bornological convergence}
The concept of convergence relevant for bornological
spaces was proposed
in 1938 by Fichtenholz~\cite{Fichtenholz-38}
and was elaborated by Mackey in his PhD thesis.
This definition can be motivated as
follows.
In a normed space a sequence $(x_n)$ converges
to zero iff the sequence $(||x_n||)$ converges to
zero~\cite[p.~9]{Horvath}. This happens iff there is a
sequence $(\beta_n)$ of positive real numbers tending to zero such
that $||x_n||\le \beta_n$ for every integer $n$.
In other words, the sequence $(x_n)$ converges to zero
iff  there is a
sequence $(\beta_n)$ of positive real numbers tending to zero such
that $x_n\in \beta_n B(1)$, where
$B(1)=\{y\telque ||y||\le 1\}$ is the unit ball.
In a bornological space there is generally no unit ball
and we must extend this definition by noticing that
a sequence $(x_n)$ converges to zero iff  there is a
sequence $(\alpha_n)$ of positive real numbers tending to zero
and a bounded set $B$ such that
$x_n\in \alpha_n B$. Indeed, the boundedness of $B$ implies
that there is a number $M$ such that
$||b||\le M$ for all $b\in B$. Thus,
$||x_n||\le  \beta_n$, where $\beta_n=M\alpha_n$
is a sequence of positive real numbers tending to zero.

We have now reached a definition that is valid in any
convex bornological space~\cite[p.~12]{KrieglMichor}:
\begin{dfn}
Let $E$ be a convex bornological space. A sequence $(x_n)$ in
$E$ is said to \emph{converge bornologically} (or
to \emph{Mackey-converge}) to zero if there
exists a bounded disk $B\subset E$ and a sequence
$(\alpha_n)$ of positive real numbers tending to zero, such that
$x_n\in \alpha_n B$ for every integer $n$.
\end{dfn}
To ensure that the limit of a 
convergent sequence is unique we say that a
convex bornological space is \emph{separated}
if the only vector space of $\calB$ is $\{0\}$.
In other words, it is separated if no
straight line is bounded.
As a consequence~\cite[p.~28]{Hogbe-77},
a convex bornological space is separated iff every Mackey-convergent
sequence has a unique limit. The von Neumann bornology
of $\calD'_\Gamma$ is separated.

\paragraph{H\"ormander convergence is the bornological (Mackey) convergence.}
The first link between pseudo-topology and bornology
comes from the following
\begin{prop}
\label{seq-Mackey}
A sequence $u_j$ in $\calD'_\Gamma$ 
converges to $u$ in the sense
of H\"ormander iff it Mackey-converges to $u$
for the bornology 
of $\mathcal{D}^\prime_\Gamma$.
\end{prop}
The
above result 
relates
a topological notion
of convergence 
with 
a bornological 
notion of convergence.
We postpone
the proof
of Proposition \ref{seq-Mackey}
which is given
in section \ref{Hormander-sect}.
This brings us to the main 
result of this subsection, 
namely the 
identification between
bounded maps and sequentially continuous maps:
\begin{prop}
\label{seq-bound}
A linear map $f:\calD'_{\Gamma_1}(\Omega_1)\to
\calD'_{\Gamma_2}(\Omega_2)$ is sequentially continuous
iff it is bounded.
\end{prop}
\begin{proof}
We must show that the map $f$ is bounded iff it transforms
any convergent sequence of $\calD'_{\Gamma_1}(\Omega_1)$
in the sense of H\"ormander into 
a convergent sequence of $\calD'_{\Gamma_2}(\Omega_2)$
in the sense of H\"ormander.
Because of Prop.~\ref{seq-Mackey}, this is equivalent to
saying that $f$ transforms a Mackey-convergent sequence
into a Mackey-convergent sequence.
If $E$ is a convex bornological space and
$F$ a locally 
convex 
space endowed with 
the corresponding
von Neumann bornology, 
then a linear map from $E$ to $F$ is bounded
iff the image of every Mackey-convergent sequence
is a Mackey-convergent sequence.
The direct sense, \emph{bounded $\rightarrow$ Mackey sequentially
continuous}, is easy to
prove since
a linear
bounded map
$f$
sends bounded
disked hulls
to bounded
disked hulls.
We prove 
the converse 
by a 
contradiction
argument.
Assume $f$
maps Mackey convergent
sequences to Mackey 
convergent sequences
but $f$ is not
bounded.
Then 
\begin{eqnarray*}
\exists B\subset E 
\text{ bounded s.t.
}
\exists p_\alpha\text{ seminorm of }F \text{ s.t. } 
\forall n, \exists x\in B,  p_\alpha(f(x))\geqslant n .
\end{eqnarray*}
Therefore we can find
some sequence 
$(x_n)_n$
in $B$
such that 
$p_\alpha(f(x_n))\geqslant n$.
Define the sequence $(\frac{x_n}{n})_{n\in \mathbb{N}}$
which is Mackey
convergent in $E$ 
since
$\frac{x_n}{n}\in \frac{1}{n}D$
where $D$
is the disked hull
of $B$.
By assumption,
$f(\frac{x_n}{n})\in F$
is Mackey convergent,
therefore we should
have 
$p_\alpha(f(\frac{x_n}{n}))\rightarrow 0$
but this is in
contradiction
with the 
estimate
$p_\alpha(f(x_n))\geqslant n\Leftrightarrow p_\alpha(f(\frac{x_n}{n}))\geqslant 1$.

Hence
the proposition
is proved.
\end{proof}




We saw in section~\ref{bornosect} that the bornology of 
$\calD'_\Gamma$ is compatible
with many topologies (from the weak to the 
Mackey topology) which are all quasi-complete.
Thus,  every Cauchy sequence converges
for these topologies and the
space $\calD'_\Gamma$ is complete as a bornological
space~\cite[p.~46]{Hogbe-77}.


%As a convex bornological space, $\calD'_\Gamma$
%is complete and nuclear~\cite{Dabrouder-13}.



\subsection{Boundedness of the fundamental operations}
It is very convenient to be able to manipulate
bounded sets of distributions in $\calD'_\Gamma$.
For example, renormalization of quantum field theory
in curved spacetimes requires to control the
scaling of distributions, which is defined
in terms of bounded sets of distributions~\cite{Meyer-98}.

By using Prop.~\ref{seq-bound}, we can translate
the results of sequential continuity of operations
with distributions into statements about their boundedness.
This is the object of the present section.
In the sequel, for any open set
$\Omega\subset \mathbb{R}^d$, we will
denote by $\dot{T}^*\Omega$
the cotangent space $T^*\Omega$
minus the zero section.

\subsubsection{Pull-back}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ and $\Omega_2\subset
\mathbb{R}^{d_2}$
be two open sets. For any smooth map $f:\Omega_1
\to \Omega_2$ we define the \emph{pull-back operation} 
$f^*: C^\infty(\Omega_2)\to  C^\infty(\Omega_1)$ by
$u\mapsto f^*u:= u\circ f$ and we define the
set of normals of $f$ by
$N_f =  \{(f(x); \eta)\in \Omega_2\times \bbR^{d_2}\telque
  \eta\circ df_x =0\}$,
where we abused notation by writing
$\bbR^{d_2}$ for its dual $(\bbR^{d_2})^*$
and where
\begin{eqnarray*}
 \eta\circ df_x &:=&  
\sum_{j=1}^{d_2} \eta_jd(y^j \circ f)_x  = 
\sum_{j=1}^{d_2} \eta_jdy^j \circ df_x\\ 
&=& 
 \sum_{j=1}^{d_2}  \eta_jdf^j_x 
 =\sum_{j=1}^{d_2} \sum_{i=1}^{d_1}  
   \eta_j \frac{\partial f^j}{\partial x^i} dx^i.
\end{eqnarray*}
Therefore, $\eta\circ df_x=0$ iff
$ \sum_{j=1}^{d_2} \eta_j \partial f^j /  \partial x^i =0$
for every $i=1,\dots, d_1$.

\begin{example}\label{embedex}
Let $M,N$ be smooth manifolds
where $\dim M\leqslant \dim N$, 
$f\in C^\infty(M,N)$ 
is a
smooth embedding and 
let $S=f(M)$.
Let us note 
that 
$N_f=\left(f_*(TM)\right)^\perp$ 
is 
nothing but the conormal
bundle of $S$. 
Thus, $N_f$ can be seen as a generalization of
the conormal bundle.
\end{example}


H\"ormander proved the following~\cite[Thm 8.2.4]{HormanderI}
\begin{prop}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ and $\Omega_2\subset \mathbb{R}^{d_2}$
be two open sets. For any smooth map
$f:\Omega_1 \to \Omega_2$ it 
is possible to extend the pull-back operation to
the distributions 
$u\in \mathcal{D}'(\Omega_2)$
which satisfy $N_f\cap WF(u) = \emptyset$ in a unique way.
Moreover the wave front set of $f^*u$ is contained in the set
$f^*WF(u) = \{(x; \eta\circ df_x)|(f(x);\eta)\in WF(u)\}$.
\end{prop}

We can give an equivalent criterion
for which the pull--back theorem
holds true:
\begin{lem}\label{lempullbackconicset}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ and $\Omega_2\subset
\mathbb{R}^{d_2}$
be two open sets and 
$\Gamma$ a closed cone in $\dotT^*\Omega_2$. For any smooth map
$f:\Omega_1 \to \Omega_2$,
$N_f\cap \Gamma = \emptyset$ is equivalent
to $f^*\Gamma\cap \underline{0}=\emptyset$ where $\underline{0}$
denotes the zero section $\underline{0}\in T^*\Omega_1$.
\end{lem}
\begin{proof}
We have the equality
$ N_f\cap \Gamma=\{(f(x);\eta) | (f(x);\eta)\in\Gamma \text{ and } \eta\circ df(x)=0\}$ by definition of $N_f$.
But we also have $f^*\left(\Gamma\cap N_f \right)= \{(x;\eta\circ df(x)) | (f(x);\eta)\in \Gamma\cap N_f \}=\{(x;0) | (f(x);\eta)\in\Gamma \text{ and } \eta\circ df(x)=0 \}=f^*\Gamma\cap\underline{0}$.
\end{proof}
\begin{example}
If we go back to example \ref{embedex}, we can pull--back
a distribution $u\in \mathcal{D}^\prime(N)$
by an embedding
$f$ if $WF(u)$ does not
meet the conormal 
of the embedded submanifold $S=f(M)$.
\end{example}

The sequential continuity of 
the pull-back~\cite[Thm 8.2.4]{HormanderI}
and Prop.~\ref{seq-bound} imply
\begin{prop}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ and $\Omega_2\subset
\mathbb{R}^{d_2}$
be two open sets and 
$\Gamma$ a closed cone in $\dotT^*\Omega_2$. For any smooth map
$f:\Omega_1 \to \Omega_2$ such that
$N_f\cap \Gamma = \emptyset$, the pull-back operation
$f^*: \calD'_\Gamma(\Omega_2)\to \calD'_{f^*\Gamma}(\Omega_1)$,
where $f^*\Gamma = \{(x; \eta\circ df_x)|(f(x);\eta)\in \Gamma\}$,
is sequentially continuous and bounded.
\end{prop}
The proof of this theorem is
sketched by H\"ormander and 
Duistermaat~\cite[p.~19]{Duistermaat} and given in detail
in~\cite[p.~155]{Friedlander}.

Note that, in the case of distributions on manifolds,
the influence of orientation on the pull-back is often not correctly
taken into account~\cite{Stampfer-11}.

\subsubsection{Push-forward}

Let $\Omega_1\subset \mathbb{R}^{d_1}$ and $\Omega_2\subset \mathbb{R}^{d_2}$
be two open sets. For any proper smooth map $f:\Omega_1
\to \Omega_2$ we define the \emph{push-forward operation} 
$f_*: \calD(\Omega_1)\to  C^\infty(\Omega_2)\subset\calD'(\Omega_2)$ by
$\langle f_*g,h\rangle=\langle g,h\circ 
f\rangle=\langle g,f^*h\rangle$~\cite[p.~527]{Chazarain}.
We can already note
that the above definition
means that the push--forward
is the adjoint
of the pull--back.
\begin{prop}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ and $\Omega_2\subset \mathbb{R}^{d_2}$
be two open sets. For any smooth map $f:\Omega_1
\to \Omega_2$, the push-forward operation can be
extended to any distribution $u\in \calD'$ such that
$f$ is a proper map from $\supp u$ to 
$\Omega_2$ and
$\WF(f_*u)\subset\{(y;\eta)\in \dotT^*\Omega_2\telque
\exists x\in \Omega_1\text{ with } y=f(x)\text{ and }
(x;\eta\circ df_x)\in WF(u)\cup \supp u\times \{0\}\}$~\cite[p.~528]{Chazarain}.
\end{prop}
We define the projections
$\overline{\pi}:(x;\xi)\in T^*\mathbb{R}^{d_1} 
\mapsto \xi\in\mathbb{R}^{d_1}$ and
$\underline{\pi}:(x;\xi)\in T^*\mathbb{R}^{d_1} 
\mapsto x\in\Omega$.
Moreover, 
\begin{prop}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ and $\Omega_2\subset \mathbb{R}^{d_2}$
be two open sets and $\Gamma$ a closed cone
in $\dotT^*\Omega_1$. For any smooth map $f:\Omega_1
\to \Omega_2$ and any closed subset 
$C$ of $\Omega_1$ such that 
$f|_C: C\to \Omega_2$ is proper and 
$\underline{\pi}(\Gamma)\subset C$,
then $f_*$ is sequentially continuous and bounded from
$\{u\in \calD'_\Gamma\telque \supp u \subset 
 C\}$ to $\calD'_{f_*{\Gamma}}$,
where 
$f_*\Gamma=\{(y;\eta)\in \dotT^*\Omega_2\telque
\exists x\in \Omega_1\text{ with } y=f(x)\text{ and }
(x;\eta\circ df_x)\in \Gamma \cup \supp u\times \{0\}\}$~\cite[p.~528]{Chazarain}.
\end{prop}


\subsubsection{Geometric interpretation.}
Our goal is to give
some geometric
formalism
which unifies
the various
transformations
in cotangent space.
To a map $f\in C^\infty(\Omega_1,\Omega_2)$,
we associate
a subset of the 
Cartesian product 
$T^*\Omega_1\times T^*\Omega_2$
called 
\textbf{relation}
which is
defined
as follows:
\begin{eqnarray*}
\text{Rel}(f)=\{ \left((x;\xi),(y;\eta )\right)\telque y=f(x),\xi=\eta\circ df,\eta\neq 0  \}\subset T^*\Omega_1\times T^*\Omega_2\setminus \left(\underline{0}\times\underline{0}\right).
\end{eqnarray*}

It is convenient to
introduce
the following
notation
$$\text{Rel}(f)^\prime= \{ (x,y;\xi,-\eta )  \telque (x,y;\xi,\eta )\in \text{Rel}(f)  \}\subset T^*\Omega_1\times T^*\Omega_2$$
we also denote by $\omega_i,(i=1,2)$
the symplectic form
of $T^*\Omega_i,(i=1,2)$.
Then we give a symplectic 
interpretation of $\text{Rel}(f)$,
$\text{Rel}(f)^\prime$
is nothing but 
the conormal
$\{(x,f(x);-\eta\circ df,\eta) \telque x\in\Omega_1,\eta\neq 0 \}$
of the graph of $f$ and
is therefore a 
\textbf{Lagrangian submanifold}
of the symplectic 
manifold 
$T^*\Omega_1\times T^*\Omega_2$
with
symplectic
form $\omega_1+\omega_2$
whereas $\text{Rel}(f)$ is Lagrangian
for $\omega_1-\omega_2$.
Let $\Gamma_2$ (resp $\Gamma_1$) 
be a closed conic set
in $\dot{T}^*\Omega_2$ 
(resp $\dot{T}^*\Omega_1$)
and 
$f\in C^\infty(\Omega_1,\Omega_2)$,
then we define
the three sets
$(N_f,f^*\Gamma_2,f_*\Gamma_1)$ 
in $(\dot{T}^*\Omega_2,T^*\Omega_1,\dot{T}^*\Omega_2)$ 
respectively
in terms of the \textbf{relation} $\text{Rel}(f)$:
\begin{eqnarray*}
N_f&=&\{  (y;\eta)\in \dot{T}^*\Omega_2 \telque ((x;0),(y;\eta))\in \text{Rel}(f) \}\\
f^*\Gamma_2&=&\{ (x;\xi)\in T^*\Omega_1 \telque ((x;\xi),(y;\eta))\in \text{Rel}(f), (y;\eta)\in\Gamma_2  \} \\
f_*\Gamma_1&=&\{ (y;\eta)\in \dot{T}^*\Omega_2 \telque ((x;\xi),(y;\eta))\in \text{Rel}(f), (x;\xi)\in\Gamma_1\cup\underline{0}  \}.
\end{eqnarray*} 
Note that the definition of 
$\text{Rel}(f)$ is coordinate
free since the conormal of the graph
of $f$ is an intrinsic object, therefore 
the three above sets are intrinsically defined.



\subsubsection{Tensor product}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ and 
$\Omega_2\subset \mathbb{R}^{d_2}$
be two open sets, 
$u\in \calD'(\Omega_1)$, 
$v\in \calD'(\Omega_2)$.
The tensor product of $u$ 
and $v$ is the unique distribution
$u\otimes v\in \calD'(\Omega_1\times\Omega_2)$ such that
for all
$f\in \calD(\Omega_1)$ and $g\in \calD(\Omega_2)$,
$\langle u\otimes v,f\otimes g\rangle=\langle u,f\rangle
\langle v,g\rangle$~\cite[p.~109]{Schwartz-66}.
The wave front set of the tensor product satisfies
\begin{eqnarray*}
\WF(u\otimes v) \subset    
\big(\WF(u)\times \WF(v)\big)
&\cup& 
\big((\supp u\times\{0\})\times \WF(v)\big)\\
 &\cup &\big(\WF(u)\times (\supp v\times\{0\})\big).
\end{eqnarray*}

The tensor product is then sequentially continuous in the
following sense:
\begin{prop}
Let $\Omega_1\subset \mathbb{R}^{ d_1}$,  
$\Omega_2\subset \mathbb{R}^{d_2}$
be two open sets and $\Gamma_1\subset \dotT^*\Omega_1$, 
$\Gamma_2\subset \dotT^*\Omega_2$ be two closed cones.
Then, the mapping
$(u,v)\mapsto u\otimes v$ is sequentially continuous and
bounded from $\calD'_{\Gamma_1}(\Omega_1)\times 
  \calD'_{\Gamma_2}(\Omega_2)$
to $\calD'_{\Gamma}(\Omega_1\times\Omega_2)$, where
\begin{eqnarray*}
\Gamma &= & 
\big(\Gamma_1 \times \Gamma_2\big) \cup 
\big((\Omega_1\times\{0\})\times \Gamma_2\big)
\cup \big(\Gamma_1\times (\Omega_2\times\{0\})\big).
\end{eqnarray*}
\end{prop}
\begin{proof}
The sequential continuity of the tensor product
is well known~\cite[p.~511]{Chazarain}.
The boundedness follows from the hypocontinuity of the
tensor product for the normal topology, which is proved
in section~\ref{tensprodsect}.
Indeed, if $f: E\times F\to G$ is a hypocontinuous
bilinear map, then $f(A\times B)$ is bounded in $G$
if $A$ is bounded in $E$ and $B$ in 
$F$~\cite[p.~III.31]{Bourbaki-TVS}.
\end{proof}

\subsubsection{Multiplication of distributions}
The H\"ormander famous product Theorem~\cite[p.~267]{HormanderI}
states that:
\begin{thm}
Let $\Omega\subset \bbR^{ d}$ be an open set, 
$u$ and $v$ two distributions in $\calD'(\Omega)$
such that $\WF(u)\cap \WF(v)'=\emptyset$,
then the product of distributions $uv$ is well
defined in $\calD'(\Omega)$ and
\begin{equation}\label{conclusion-mthm}
 \WF(uv)\subset S_+ \cup S_u \cup S_v,
\end{equation}
where $S_+=\{(x; \xi+\eta)| 
(x ;\xi)\in \WF(u)\text{ and }
(x ;\eta)\in \WF(v)\}$,
$S_u=\{(x ;\xi)| (x;\xi)\in \WF(u)
\text{ and } x\in \supp(v)\}$ and
$S_v=\{(x;\xi)| (x;\xi)\in \WF(v) \text{ and } x\in \supp(u)\}$.
\end{thm}

Moreover~\cite[p.~526]{Chazarain}
\begin{prop}
Let $\Omega\subset \bbR^n$ be an open set
and $\Gamma_1,\Gamma_2$ be two closed cones in
$\dotT^*\Omega$ such that $\Gamma_1\cap \Gamma_2'=\emptyset$.
Then the product of distributions 
is sequentially continuous and bounded from
$\calD'_{\Gamma_1}\times\calD'_{\Gamma_2}$
to $\calD'_\Gamma$, where
\begin{eqnarray*}
\Gamma &= & 
\big(\Gamma_1 +_\Omega \Gamma_2\big) \cup 
\big((\Omega\times\{0\})+_\Omega \Gamma_2\big)
\cup \big(\Gamma_1+_\Omega (\Omega\times\{0\})\big),
\end{eqnarray*}
where, for any pair of cones $\Gamma_1$ and $\Gamma_2$
in $\dotT^*\Omega$, $\Gamma_1+_\Omega\Gamma_2=
\{(x; \xi_1+\xi_2)\telque
(x;\xi_1)\in \Gamma_1\text{ and } (x;\xi_2)\in\Gamma_2\}$.
\end{prop}
\begin{proof}
This follows from the fact that the product of distributions
is the composition of the tensor product and of the
pull-back by the diagonal map 
$x\mapsto (x,x)$~\cite[p.~267]{HormanderI}.
\end{proof}

\section{Continuity for the H\"ormander topology}
\label{Hormander-sect}

A pseudo-topology is  not always compatible with a
topology~\cite{Dolecki} but 
Duistermaat~\cite[p.~18]{Duistermaat} noticed that
H\"ormander's  pseudo-topology is compatible with
a topology defined in terms of 
the following seminorms~\cite[p.~80]{Grigis}:
\begin{itemize}
\item[(i)] All the seminorms on $\calD'(\bbR^n)$ for the weak 
topology: $||u||_\phi=|\langle u,\phi\rangle|$ for
all $\phi\in\calD(\bbR^n)$.
\item[(ii)] The seminorms of the form
\begin{eqnarray*}
||u||_{N,V,\chi} &=& \sup_{k\in V} (1+|k|)^N |\widehat{u\chi}(k)|,
\end{eqnarray*}
where $N\ge 0$, $\chi\in \calD(\bbR^n)$, and $V\in \bbR^n$ is a closed
cone with $\supp\chi\times V \cap \Gamma=\emptyset$.
\end{itemize}
These seminorms equip $\calD'_\Gamma$ with the structure of a
locally convex space and the corresponding topology is usually
called \emph{H\"ormander's topology}.
It probably first appeared
 in the 1970-1971 lecture notes
by Duistermaat~\cite{Duistermaat}, although the seminorms
$||\cdot||_{N,V,\chi}$ are already mentioned by
H\"ormander~\cite[p.~128]{Hormander-71}.
The topological convergence 
of sequences in \emph{H\"ormander's topology} 
is indeed equivalent to
their convergence in the sense
of H\"ormander~\cite[p.~262]{Dabrouder-13}
as in definition
\ref{convdef}.
We can now 
recall the statement of Proposition 
\ref{seq-Mackey} and give the proof:
\begin{prop}
A sequence $u_j$ in $\calD'_\Gamma$ 
converges to $u$ in the sense
of H\"ormander iff it Mackey-converges to $u$.
\end{prop}
\begin{proof}
 It follows
from the
definition 
\ref{convdef}
and the definition
of the 
H\"ormander
topology that
a
sequence converges in the sense of H\"ormander
iff it converges topologically for the
H\"ormander topology~\cite[p.~65]{Treves}.

Moreover, Lemma 21 in
\cite{Dabrouder-13} states that a
sequence of elements of $\calD'_\Gamma$
converges topologically iff it Mackey-converges.
Thus, a sequence converges in the sense of H\"ormander
iff it Mackey-converges.
\end{proof}

The functional properties of $\calD'_\Gamma$ with the
H\"ormander topology were
studied in detail recently~\cite{Dabrouder-13}
and it was found that, although in $\calD'(\Omega)$
every bounded map is continuous, this is no longer
the case in $\calD'_\Gamma$.
Although we now know that the fundamental operations 
with distributions are bounded, we must investigate
whether or not they are topologically continuous.

Duistermaat defines the H\"ormander 
topology~\cite[p.~18]{Duistermaat}
and claims without proof 
that the pull-back by a smooth map is 
continuous~\cite[p.~19]{Duistermaat}.
However the following counterexample due to Alesker shows that 
this is not the case.

\subsection{Alesker's counterexample}
\label{aleskercounterex}
The following
is a transcription of a letter sent to us by
Semyon Alesker on 7 October 2013.
The notation was slightly changed to make it
consistent with the rest of the paper.

\emph{I will give an example of a map $f: X \to Y$
and of a closed conic subset 
$\Gamma\subset\dotT^*Y$ such that the pull-back
map $f^*:\calD_\Gamma(Y) \to \calD_{f^*\Gamma}(X)$
is \emph{not} topologically continuous for
the H\"ormander topology. We will show
as a corollary that the tensor product is not 
topologically continuous (even separately).}

\begin{prop}
Let $f:\bbR^2\to \bbR$ be the projection to the first 
coordinate. Let $\Gamma=\dotT^*\bbR$, so that
$\calD'_\Gamma(\bbR)=\calD'(\bbR)$, then
$f^*\Gamma=\{(x_1,x_2;\xi_1,0)\}$. We claim that
the map $f^*: \calD'_\Gamma(\bbR) \to \calD'_{f^*\Gamma}(\bbR^2)$
is not topologically continuous for the H\"ormander topology.
\end{prop}
\begin{proof}
Let $\varphi\in \calD(\bbR)$ such that 
$\varphi|_{[-1,1]}=1$. Take
$\chi=\varphi\otimes\varphi$, $V=\{(\xi_1,\xi_2)\in \bbR^2
   \telque |\xi_1|\le |\xi_2|\}$ and $N=0$. 
The intersection of $V$ with
$\{(\xi_1,0)\telque \xi_1\not=0\}$ is empty
because $|\xi_1|\le |\xi_2|=0$ implies $\xi_1=\xi_2=0$.
Therefore, 
$||\cdot||_{N,V,\chi}$ is a seminorm of 
$\calD'_{f^*\Gamma}$ and, if $f^*$ were continuous,
it would be possible to bound
$||f^*u||_{N,V,\chi}$ with 
$\sup_i |\langle u,f_i\rangle|$ for a finite set of $f_i\in
\calD(\bbR)$ and every $u\in \calD'(\bbR)$.

We are going to show that this is not the case.
We have
\begin{eqnarray*}
 ||f^* u||_{0,V,\chi} &=& \sup_{\xi\in V} 
  |\widehat{\varphi u}(\xi_1)|\,
|\widehat{\varphi}(\xi_2)|
= 
\sup_{\xi_1} 
  |\widehat{\varphi u}(\xi_1)|\,
  \omega(\xi_1),
\end{eqnarray*}
where  $\omega(\xi_1)=\sup_{|\xi_2|\ge|\xi_1|}
|\widehat{\varphi}(\xi_2)|$.
It is clear that $\omega(\xi_1)>0$ everywhere since
$\widehat{\varphi}$ is a real analytic function.
Thus we should show that the map 
$\calD'(\bbR)\to\bbR$ given by 
$u\mapsto \sup_{\xi\in \bbR} |\widehat{\varphi u}(\xi)|\omega(\xi)$
is not continuous (for a fixed $\omega>0$).

If the pull-back were continuous, there would
be a finite set $\chi_1,\dots,\chi_t$ of functions in $\calD(\bbR)$
such that
\begin{eqnarray*}
||f^* u||_{0,V,\chi} & \le & \sup_{i=1,\dots,t} 
   |\langle u,\chi_i\rangle|.
\end{eqnarray*}
We can find $\xi$ such that the functions 
$\chi_1,\dots,\chi_t$ and $\varphi(x) e^{-ix\xi}$
are linearly independent. Then there exists
$u\in \calD'(\bbR)$ such that
$\langle u,\chi_i\rangle = 0$ for $i=1,\dots,t$
and 
$\widehat{u\varphi}(\xi)=\langle u,\varphi e_\xi\rangle
= 1+ 1/\omega(\xi)$,
where $e_\xi(x)=e^{ -i\xi.x}$.
Then, 
$||f^*u||_{0,V,\chi}= 1 + \omega(\xi)$ and we reach
a contradiction.
\end{proof}
Thus, the pull-back is not continuous.
Moreover, the same example can be considered
as the map
$u\to u\otimes 1$. This shows that the tensor
product is not separately continuous
for the H\"ormander topology.
\\
\begin{center}
 \textsc{Second part}
\end{center}

\section{The normal topology}
\label{normaltopsect}
 The H\"ormander topology was chosen to be
compatible with the convergence of sequences in the
sense of H\"ormander.
But in fact, for any topology that is finer than the
weak topology and coarser than the Mackey topology
of $\calD'_\Gamma$,  a sequence is topologically
convergent iff it is convergent in the sense of 
H\"ormander~\cite{Dabrouder-13}.
Therefore, we can look for an alternative topology
for which the fundamental operations have better
continuity properties.

However, if we choose a topology that is
too fine, then many maps $f$ from a 
locally convex space $E$ to $\calD'_\Gamma$ 
will not be continuous (because there are 
too many $f^{-1}(U)$ where $U$ runs
over the open sets of $\calD'_\Gamma$) and, conversely,
if the topology is too coarse, then many maps $f$ 
from $\calD'_\Gamma$ to $E$ will not be continuous.
The choice of the proper topology is a subtle matter
and we follow the advice given by Laurent
Schwartz~\cite[p.~10]{Schwartz-57} and define a
\emph{normal topology} on $\calD'_\Gamma$.
\begin{dfn}\label{normaltopdef}
A Hausdorff locally convex space $E$ is said to be
a \emph{normal space of distributions} if there
are continuous injective linear maps $i:\calD(\Omega)
\hookrightarrow E$
and $j:E\hookrightarrow \calD'(\Omega)$,
where $\calD'(\Omega)$ is equipped with its strong topology,
such that: (i) The image of $i$ is dense in $E$,
(ii) for any $f$ and $g$ in $\calD(\Omega)$,
$\langle j\circ i (f),g\rangle=\int_\Omega 
f(x) g(x)dx$~\cite[p.~319]{Horvath}.
\end{dfn}
The point of having
$\calD(\Omega)\hookrightarrow E \hookrightarrow \calD'(\Omega)$
is that the dual $E'$ can also be equipped with a
normal topology and many interesting relations arise
from that duality.
In this paper, we choose to
use \emph{the coarsest topology} 
among all topologies defined in (\ref{normaltopdef}).
In~\cite{Dabrouder-13}, it was
shown that this topology is defined
by the seminorms $||\cdot||_{N,V,\chi}$
of the H\"ormander topology
and the seminorms $p_B$ of the strong topology of
$\calD'(\Omega)$: $p_B(u)=\sup_{f\in B} |\langle u,f\rangle|$,
where $B$ runs over the bounded sets of 
$\calD(\Omega)$~\cite{Dabrouder-13}.
This topology will be simply called 
\textbf{the normal topology} on $\calD^\prime_\Gamma$
in the following.
From now on, the space of distributions $\calD'(\Omega)$
will always be equipped with the strong topology.

With the normal topology, it will be shown that
the pull-back and push-forward are continuous
and that the tensor product and distribution product
are hypocontinuous. 
We cannot expect the distribution product
to be (jointly) continuous because, in the
case of $\Gamma_1=\emptyset$ and 
$\Gamma_2=\dotT^*(\Omega_2)$, appendix~\ref{topequiCinfty}
states that $D'_{\Gamma_1}=C^\infty(\Omega)$ and
$D'_{\Gamma_2}=\calD'(\Omega)$ and
it is known that the product 
$C^\infty(\Omega)\times \calD'(\Omega)
\to \calD'(\Omega)$ is hypocontinuous~\cite[p.~119]{Schwartz-66}
but not (jointly) continuous~\cite{Kucera-81,Larcher-13}.

Let us recall the definition of hypocontinuity
which, for many purposes, is as good as 
continuity~\cite[p.~104]{Schwartz-66}:
\begin{dfn}
\label{hypodef}
\cite[p.~423]{Treves}
Let $E$, $F$ and $G$ be topological vector spaces. 
A bilinear map
$f:E\times F\to G$ is said to be \emph{hypocontinuous}
if: (i) for
every neighborhood $W$ of zero in $G$
and every bounded set $A\subset E$ there is
a neighborhood $V$ of zero in $F$
such that
$f(A\times V)\subset W$ and
(ii) for
every neighborhood $W$ of zero in $G$ and
every bounded set $B\subset F$ there is
a neighborhood $U$ of zero in $E$
such that
$f(U\times B)\subset W$.
\end{dfn}
If $E$, $F$ and $G$ are locally convex spaces
with topologies defined by the families
of seminorms $(p_i)_{i\in I}$, $(q_j)_{j\in J}$ and
$(r_k)_{k\in K}$, respectively, the definition of hypocontinuity 
can be translated into
the following two conditions: (i) For every bounded
set $A$ of $E$ and every seminorm $r_k$, there
is a constant $M$ and a finite set of seminorms
$q_{j_1},\dots,q_{j_n}$ 
(both depending only on $k$ and $A$) 
such that 
\begin{eqnarray}
\forall x\in A, r_k\big(f(x,y)\big) & \le & M 
  \sup\{ q_{j_1}(y),\dots , q_{j_n}(y)\};
\label{hypogenA}
\end{eqnarray}
and (ii) For every bounded
set $B$ of $F$ and every seminorm $r_k$, there
is a constant $M$ and a finite set of seminorms
$p_{i_1},\dots, p_{i_n}$ 
(both depending only on $k$ and $B$) 
such that
\begin{eqnarray}
 \forall y\in B, r_k\big(f(x,y)\big) & \le & M 
  \sup\{ p_{i_1}(x),\dots , p_{i_n}(x)\}.
\label{hypogenB}
\end{eqnarray}
Equivalently~\cite[p.~155]{Kothe-II}, 
for every bounded set $A$ of $E$ and every
bounded set $B$ of $F$ the sets of maps
$\{f_x\telque x\in A\}$ and 
$\{f_y\telque y\in B\}$ are equicontinuous,
where $f_x: E\to G$ and $f_y: F\to G$ are defined
by $f_x(y)=f_y(x)=f(x,y)$.

%This means that, for every bounded set $A$ of
%$E$, the set of maps $\{f_x\telque x\in A\}$
%is equicontinuous, where $f_x: F\to G$ is defined
%by $f_x(y)=f(x,y)$, and for every bounded set $B$
%of $F$, the set {\pink of maps $\{f_y\telque y\in A\}$
%is equicontinuous.}

Important properties of hypocontinuous maps are 
given in
\begin{prop}
\cite[p.~359]{Horvath}
Let $E$, $F$ and $G$ be topological vector spaces and
$f:E\times F\to G$ a hypocontinuous map. Then,
for every bounded set $A$ of $E$ and $B$ of $F$:
(i) $f$ is continuous on $A\times F$ and $E\times B$;
(ii) $f$ is uniformly continuous on $A\times B$;
(iii) $f(A\times B)$ is bounded in $G$.
\end{prop}

\begin{lem}
\label{Linflem}
Let $\Omega$ be an open set of $\bbR^d$
and $B$ a bounded set in $\mathcal{D}'(\Omega)$, then for every
$\chi\in \mathcal{D}(\Omega)$ there exist an integer $M$ 
and a constant $C$ (both depending only on $B$ and on
an arbitrary relatively compact open neighborhood of $\supp\chi$)
such that
\begin{eqnarray*}
\sup_{u\in B}\sup_{\xi\in
\bbR^n}(1+|\xi|)^{-M}|\widehat{u\chi}(\xi)| < 
 2^M C\, \mathrm{Vol}(K)\, \pi_{M,K}(\chi),
\end{eqnarray*}
where $K=\supp\chi$.
\end{lem}
\begin{proof}
Let $\Omega_0$ be a relatively compact open
neighborhood of $K=\supp \chi$.
According to Schwartz~\cite[p.~86]{Schwartz-66},
for any bounded set $B$ in $\calD'(\Omega)$,
there is an integer $M$ (depending only on $B$
and $\Omega_0$) such that every $u\in B$
can be expressed in $\Omega_0$ as $u=\partial^\alpha f_u$
for $|\alpha|\le M$, where $f_u$ is a continuous function.
Moreover, there is a constant $C$ (depending only on
$B$ and $\Omega_0$) such that $|f_u(x)|\le C$ for all $x\in \Omega_0$
and $u\in B$.
Thus,
\begin{eqnarray*}
\widehat{u\chi}(\xi) &=&
\int_{\Omega_0} e^{ -i\xi\cdot x} \chi(x) \partial^\alpha f_u(x) dx
= (-1)^{|\alpha|}
\int_{\Omega_0} f_u(x) \partial^{\alpha}
    \big(e^{ -i\xi\cdot x} \chi(x)\big) dx
\\&=&
 (-1)^{|\alpha|}
\sum_{\beta\le\alpha} \binom{\alpha}{\beta}
  (i\xi)^\beta
  \int_{\Omega_0} f_u(x) e^{ -i\xi\cdot x}
   \partial^{\beta-\alpha}\chi(x) dx.
\end{eqnarray*}
By using $|(i\xi)^\beta|\le (1+|k|)^M$ if $|\beta|\le M$ we obtain
\begin{eqnarray*}
(1+|\xi|)^{-M}|\widehat{u\chi}(\xi) |&\le &
\sup_{|\alpha|\le M}
\sum_{\beta\le\alpha} \binom{\alpha}{\beta}
  \Big|\int_{\Omega_0} f_u(x) e^{ -i\xi\cdot x}
\partial^{\beta-\alpha}\chi(x)
dx\Big|
\\ &\le &
 2^M C\, \mathrm{Vol}(K)\, \pi_{M,K}(\chi).
\end{eqnarray*}
\end{proof}

\section{Tensor product of distributions.}
\label{tensprodsect}



Let $\Omega_1$ and $\Omega_2$ be open sets in 
$\bbR^{d_1}$ and $\bbR^{d_2}$, respectively, and
$(u,v)\in \calD'_{\Gamma_1}\times
\calD'_{\Gamma_2}$,
where $\calD'_{\Gamma_1}\subset\calD'(\Omega_1)$ and
$\calD'_{\Gamma_2}\subset\calD'(\Omega_2)$. Then the
tensor product $u\otimes v$ belongs to 
$\calD'_\Gamma\subset \calD'(\Omega_1\times\Omega_2)$ 
where 
\begin{eqnarray*}
\Gamma &= & 
\big(\Gamma_1 \times \Gamma_2\big) \cup 
\big((\Omega_1\times\{0\})\times \Gamma_2\big)
\cup \big(\Gamma_1\times (\Omega_2\times\{0\})\big)
\\&=&
\left(\Gamma_1\cup\{\underline{0}\}_1\right)\times 
  \left(\Gamma_2\cup\{\underline{0}\}_2\right)\setminus 
  \{(\underline{0},\underline{0})\},
\end{eqnarray*}
where $\{\underline{0}\}_1$ means $\Omega_1\times\{0\}$,
$\{\underline{0}\}_2$ means $\Omega_2\times\{0\}$
and $\{\underline{0},\underline{0}\}$ 
means $(\Omega_1\times\Omega_2)\times\{0,0\}$.
Our goal in this section is to show that the 
tensor product is hypocontinuous for the
normal topology.
We denote by $(z;\zeta)$ the coordinates
in $T^*(\Omega_1\times\Omega_2)$,
where $z=(x,y)$ with $x\in \Omega_1$ and $y\in \Omega_2$,
$\zeta=(\xi,\eta)$ with 
$\xi\in \bbR^{d_1}$ and $\eta\in \bbR^{d_2}$.
We also denote $d=d_1+d_2$, so that $\zeta\in \bbR^d$.

\begin{lem}
The seminorms of the strong topology of $\calD'_\Gamma$
and the family of seminorms:
\begin{equation}
\Vert t_1\otimes t_2\Vert_{N,V,\varphi_1\otimes
\varphi_2}=\sup_{\zeta\in V}
(1+|\zeta|)^N\vert \widehat{t_1\varphi_1}(\xi)\vert\,\vert
   \widehat{t_2\varphi_2}(\eta)\vert,
\end{equation}
where $\zeta=(\xi,\eta)$,
$(\varphi_1,\varphi_2)\in\calD(\Omega_1)\times \calD(\Omega_2)$ 
and $V\subset\mathbb{R}^{d}$, are such that
$\left(\supp\left(\varphi_1\otimes\varphi_2\right)\times V\right)
\cap \Gamma=\emptyset$,
are a fundamental system of seminorms for the normal
topology of $\calD'_\Gamma$.
\end{lem}
\begin{proof}

We use the following lemma~\cite[p.~80]{Grigis}
\begin{lem}
Let $\Omega$ be an open set in $\bbR^n$.
If we have a family, indexed by $\alpha\in A$,
of $\chi_\alpha\in \calD(\Omega)$
and of closed cones $V_\alpha\subset (\bbR^n\backslash\{0\})$
such that 
$(\supp\chi_\alpha\times V_\alpha)\cap \Gamma=\emptyset$ and
\begin{eqnarray*}
\Gamma^c &=& \bigcup_{\alpha\in A} \{(x,\xi)\in \dotT^*\Omega
  \telque \chi_\alpha(x)\not=0, \xi\in \mathring{V_\alpha}\},
\end{eqnarray*}
then the topology of $\calD'_\Gamma$ is already defined by
the topology of $\calD'(\Omega)$ and 
the seminorms $||\cdot||_{N,V_\alpha,\chi_\alpha}$.
\end{lem}
It is clear that the family indexed by $\varphi_1\otimes\varphi_2$
and $V$ such that $\supp(\varphi_1\otimes\varphi_2)\times
V\cap\Gamma=\emptyset$ satisfies the hypothesis of the
lemma.
\end{proof}

To establish the hypocontinuity 
of the tensor product,
we consider an arbitrary bounded set 
$B\subset\calD'_{\Gamma_1}(\Omega_1)$ and,
according to eq.~\eqref{hypogenA}, we must
show that, for every seminorm $r_k$ of 
$ \calD'_\Gamma(\Omega_1\times\Omega_2)$, there is a constant $M$ and a 
finite number of seminorms $q_j$ such that
$r_k(u\otimes v)\leqslant M\sup_j q_j(v)$ for 
every $u\in  B$ and every $v\in \calD'_{\Gamma_2}(\Omega_2)$.
By Schwartz' 
theorem ~\cite[p.~110]{Schwartz-66} we already know that this
is true for every seminorm $r_k$ of the 
strong topology of $\calD'_\Gamma(\Omega_1\times\Omega_2)$. It remains
to show it for every
$||\cdot||_{N,V,\varphi_1\otimes\varphi_2}$.
This will be done by first defining 
a suitable partition of unity on $\Omega_1\times \Omega_2$ 
and its corresponding cones. Then, this partition of unity
will be used to bound the seminorms by standard microlocal
techniques.





\begin{lem}\label{lemm0}
Let $\Gamma_1,\Gamma_2$ be closed cones in 
$\dotT^*\Omega_1$ and $\dotT^*\Omega_2$, respectively.
Set $\Gamma=\left(\Gamma_1\cup\{\underline{0}\}\right)\times 
\left(\Gamma_2\cup\{\underline{0}\}\right)\setminus 
\{(\underline{0},\underline{0})\}\subset \dotT^*\bbR^d$
then for all closed cones $V\subset\bbR^{d}$ and 
$\chi\in\calD(\Omega_1\times\Omega_2)$ 
such that
$\left(\supp\chi\times V\right)\cap\Gamma=\emptyset$,
there exist a partition of unity 
$(\psi_{j1}\otimes\psi_{j2})_{j\in J}$ 
of $\Omega_1\times\Omega_2$,
which is finite on $\supp\chi$,
and a family of closed cones $(W_{j1}\times W_{j2})_{j\in J}$ in 
$(\bbR^{d_1}\setminus\{0\})
\times (\bbR^{d_2}\setminus\{0\})$ such that
\begin{eqnarray}
(\supp\psi_{j1}\times W_{j1}^c)\cap \Gamma_1 &=& 
  (\supp\psi_{j2}\times W_{j2}^c)\cap \Gamma_2
= \emptyset,\\
V\cap\left((W_{j1}\cup\{0\})\times(W_{j2}\cup\{0\})\right) &=&\emptyset,\\
\nonumber
\quad\text{ if }\,
\supp\chi\cap\supp(\psi_{j1}\otimes\psi_{j2})\not=\emptyset.
\end{eqnarray}
\end{lem}
\begin{proof} --- We first set some useful notations
and observations.
For any $D\in \mathbb{N}$, on the base of the identification
$T^*\mathbb{R}^D\simeq \mathbb{R}^D\oplus (\mathbb{R}^D)^*$, we denote by
$\underline{\pi}:T^*\mathbb{R}^D\longrightarrow \mathbb{R}^D$
the projection on the first factor and by
$\overline{\pi}:T^*\mathbb{R}^D\longrightarrow (\mathbb{R}^D)^*$
the projection on the second factor.
We use the distance
$d_\infty$ on $\mathbb{R}^D$ (or $(\mathbb{R}^D)^*$) defined by 
$d_\infty(u,v):= \sup_{1\leq i\leq D}|u^i-v^i|$. For $u\in \mathbb{R}^D$
and $r\geq 0$ we then set $\overline{B}(u,r) = \{v\in \mathbb{R}^D;
d_\infty(u,v)\leq r\}$ and, for any subset $Q\subset \mathbb{R}^D$, 
$Q_{,r}:= \{v\in \mathbb{R}^D;d_\infty(v,Q)\leq r\}$.
We note the useful property that, for any
pair of sets $Q_1\subset \mathbb{R}^{d_1}$
and $Q_2\subset \mathbb{R}^{d_2}$, 
$(Q_1\times Q_2)_{,r} = Q_{1,r}\times Q_{2,r}$ (in particular,
if $(x,y)\in \Omega_1\times \Omega_2$,
$\overline{B}((x,y),r) = \overline{B}(x,r) \times \overline{B}(y,r)$).
Lastly for any closed conic subset
$W\subset (\mathbb{R}^D)^*\setminus \{0\}$, 
we set $\overline{W}:= W\cup \{0\}$
for short and $UW:= S^{D-1}\cap W$. Similarly if $\Gamma$ is a conic
subset of $T^*\mathbb{R}^D$, we set $ U\Gamma = (\mathbb{R}^D\times S^{D-1})\cap \Gamma$
and $\overline{\Gamma}=\Gamma\cup \underline{0}\subset T^*\mathbb{R}^D$
where $\underline{0}$ is the zero
section of $T^*\mathbb{R}^D$.

We will prove that there exists a family of open ball
$(B_{j1}\times B_{j2})_{j\in J}$ which covers
$\Omega_1\cap\Omega_2$, which is finite over any compact subset
of $\Omega_1\times \Omega_2$ and in particular on $\supp\chi$
and such that
$(\overline{B_{j1}}\times W_{j1}^c)\cap \Gamma_1 =
(\overline{B_{j2}}\times W_{j2}^c)\cap \Gamma_2 = \emptyset$ and that
$V\cap(\overline{W}_{j1}\times \overline{W}_{j2}) = \emptyset$,
if $\supp\chi\cap(\overline{B}_{j1}\times \overline{B}_{j2}) \neq \emptyset$.
The conclusion of the lemma will then follow 
by constructing a partition of unity 
$(\psi_{j1}\otimes\psi_{j2})_{j\in J}$ 
such that $\supp\psi_{j1} = \overline{B}_{j1}$ and
$\supp\psi_{j2} = \overline{B}_{j2}$, $\forall j\in J$,
by using standard arguments.

\emph{Step 1} --- Hypothesis $(\supp\chi\times V)\cap \Gamma = \emptyset$
implies that there exists some $\delta>0$ such that
$d_\infty(\supp\chi\times UV,U\Gamma) \geq 4\delta$. Consider
$K:= (\supp\chi)_{,\delta}$ we note then that
$d_\infty(K\times UV,U\Gamma) \geq 3\delta$. W.l.g. we can assume
that $\delta$ has been chosen so that $K\subset \Omega_1\times \Omega_2$.
Obviously $\Omega_1\times \Omega_2$ is covered by
$(B((x,y),\delta))_{(x,y)\in \Omega_1\times \Omega_2}$.
Moreover all balls $B((x,y),\delta)$ are contained in $K$ if
$(x,y)\in \supp\chi$ and $\supp\chi$ is covered by the subfamily
$(B((x,y),\delta))_{(x,y)\in \supp\chi}$. Since $\supp\chi$ is
compact we can thus extract a countable family of balls
$(B_i)_{i\in I} = (B_{i1}\times B_{i2})_{i\in I}$ which covers
$\Omega_1\times \Omega_2$ and which is
finite over $\supp\chi$.

We now set
$\gamma:= \overline{\pi}(\underline{\pi}^{-1}(K)\cap \Gamma)$
and $U\gamma:= \overline{\pi}(\underline{\pi}^{-1}(K)\cap U\Gamma)$
and we estimate the distance of $U\gamma$ to $UV$:
\[
 \begin{array}{ccl}
  d_\infty[U\gamma,UV]
  & = & \displaystyle \inf_{\xi\in \overline{\pi}(\underline{\pi}^{-1}(K)\cap U\Gamma)}
  \quad \inf_{\eta\in UV}d_\infty(\xi,\eta)\\
  &  = & \displaystyle \inf_{(u,\xi)\in U\Gamma;u\in K}
  \quad \inf_{(v,\eta)\in K\times UV}d_\infty(\xi,\eta)\\
  & = & \displaystyle \inf_{(u,\xi)\in U\Gamma;u\in K}
  \quad \inf_{(v,\eta)\in K\times UV}d_\infty((u,\xi),(v,\eta)),
 \end{array}
\]
where the last equality is due to the fact that one can choose $v=u$ in the minimization.
We deduce that, by removing the constraint $u\in K$ in the minimization,
\[
 \begin{array}{ccl}
  d_\infty[U\gamma,UV]
  & \geq & \displaystyle \inf_{(u,\xi)\in U\Gamma}
  \quad \inf_{(v,\eta)\in K\times UV}d_\infty((u,\xi),(v,\eta))\\
  & = & d_\infty(K\times UV,U\Gamma) \geq 3\delta.
 \end{array}
\]
\emph{Step 2} --- Since $\gamma$
and $V$ are cones, we deduce from the previous inequality that
\[
 \forall \xi\in \gamma,\quad d_\infty(\xi,V)\geq 2\|\xi\|\delta.
\]
For any $i\in I$ such that the ball $B_i$ is centered at a point
in $\supp\chi$, the inclusion $\overline{B_i}\subset K$ implies
$\overline{\pi}(\underline{\pi}^{-1}(\overline{B_i})\cap \Gamma)\subset \gamma$.
We hence have also
\begin{equation}\label{distance-securite}
\forall \xi\in \overline{\pi}(\underline{\pi}^{-1}(\overline{B_i})\cap \Gamma)
\quad d_\infty(\xi,V)\geq 2\|\xi\|\delta.
\end{equation}
We now set $\overline{W}_{i1}:=
\{\xi_1\in (\mathbb{R}^{d_1})^*;
d_\infty(\xi_1,\overline{\pi}(\underline{\pi}^{-1}(\overline{B_{i1}})\cap \Gamma_1))
\leq \|\xi_1\|\delta\}$,
$\overline{W}_{i2}:=
\{\xi_2\in (\mathbb{R}^{d_2})^*;
d_\infty(\xi_2,\overline{\pi}(\underline{\pi}^{-1}(\overline{B_{i2}})\cap \Gamma_2))
\leq \|\xi_2\|\delta\}$
and $W_{i1}:= \overline{W}_{i1}\setminus\{0\}$, $W_{i2}:= \overline{W}_{i2}\setminus\{0\}$.
By the definition of $W_{i1}$, $W_{i1}^c\cap \overline{\pi}(\underline{\pi}^{-1}(\overline{B_{i1}})\cap \Gamma_1) = \emptyset$,
which is equivalent to $(\overline{B}_{i1}\times W_{i1}^c)\cap \Gamma_1 = \emptyset$. Similarly
$(\overline{B}_{i2}\times W_{i1}^c)\cap \Gamma_2 = \emptyset$.

On the other hand, since 
\begin{eqnarray*}
\overline{\pi}(\underline{\pi}^{-1}(\overline{B_{i1}})\cap\overline{\Gamma_1})
\times \overline{\pi}(\underline{\pi}^{-1}(\overline{B_{i2}})\cap\overline{\Gamma_2})
&=& \overline{\pi}[\underline{\pi}^{-1}(\overline{B_{i1}}\times \overline{B_{i2}})\cap(\overline{\Gamma_1}\times \overline{\Gamma_2})]\\
&=& \overline{\pi}[\underline{\pi}^{-1}(\overline{B_i})\cap\overline{\Gamma}],\,\ 
\overline{B_i}=\overline{B_{i1}}\times \overline{B_{i2}}
\end{eqnarray*}
because
\[
\{\xi_1;\exists (x_1;\xi_1)\in\overline{\Gamma_1} , x_1\in \overline{B_{i1}} \}
\times
\{\xi_2;\exists (x_2;\xi_2)\in\overline{\Gamma_2} , x_2\in \overline{B_{i2}} \}\]
\[=\{(\xi_1,\xi_2);
\exists (x_1,x_2;\xi_1,\xi_2)\in\overline{\Gamma_1}\times\overline{\Gamma_2},
(x_1,x_2)\in \overline{B_{i1}}\times \overline{B_{i2}}\}
\]
we also have
\begin{eqnarray*}
 \overline{W}_{i1}\times \overline{W}_{i2}& =& \{(\xi_1,\xi_2)\in (\mathbb{R}^d)^*;
 d_\infty[(\xi_1,\xi_2),\overline{\pi}(\underline{\pi}^{-1}(\overline{B_i})\cap\Gamma)]\\
 &\leq &\sup (\|\xi_1\|,\|\xi_2\|) \delta\}.
\end{eqnarray*}
Hence by (\ref{distance-securite}), we deduce that $\overline{W}_{i1}\times \overline{W}_{i2}$ does not meet $V$.
\end{proof}
In the
remaining part of the 
paper, we may identify 
abusively
$\bbR^d$ and $(\bbR^d)^*$.

 We also introduce
the notation
$e_\zeta(x,y)=e^{i(\xi.x+\eta.y)}$
where $\zeta=(\xi,\eta)$.

To estimate $||u\otimes v||_{N,V,\varphi_1\otimes\varphi_2}$,
we use Lemma~\ref{lemm0} to find a partition of unity
$(\psi_{j1}\otimes\psi_{j2})_{j\in J}$ which is finite
on $\supp(\varphi_1\otimes\varphi_2)$ to write
\begin{eqnarray*}
\widehat{u\varphi_1}(\xi) 
\widehat{v\varphi_2}(\eta) =
\calF(u\varphi_1\otimes v\varphi_2)(\zeta) &=&
\langle u\otimes v,(\varphi_1\otimes \varphi_2) e_\zeta\rangle
\\=\sum_j 
\langle u\otimes v,(\varphi_1\psi_{j1}\otimes \varphi_2\psi_{j2}) 
e_\zeta\rangle
&=& \sum_j \widehat{u\varphi_1\psi_{j1}}(\xi) \,
 \widehat{v\varphi_2\psi_{j2}}(\eta).
\end{eqnarray*}
Therefore
$||u\otimes v||_{N,V,\varphi_1\otimes\varphi_2}
\leqslant \sum_j ||u\otimes v||_{N,V,\varphi_1\psi_{j1}\otimes
  \varphi_2\psi_{j2}}$, where the sum over $j$ is finite.
Each seminorm on the right hand side is bounded by the
following lemma.
\begin{lem}
Let $\Gamma_1$, $\Gamma_2$ and $\Gamma$ be closed
cones as in the previous lemma, $\psi_1\in \calD(\Omega_1)$
$\psi_2\in \calD(\Omega_2)$ such that
$ (\supp(\psi_1\otimes\psi_2) \times V ) \cap\Gamma=\emptyset$
and closed cones
$W_1$ and $W_2$ in $\bbR^{d_1}\backslash\{0\}$ and
$\bbR^{d_2}\backslash\{0\}$ such that
\begin{eqnarray}
(W_1\cup\{0\})\times (W_2\cup\{0\})\cap V &=& \emptyset,\\
(\supp\psi_k\times W_k^c)\cap\Gamma_k &=& \emptyset,
\text{ for }\, k=1,2.
\end{eqnarray}
Then, for every bounded set $A\subset\calD'_{\Gamma_1}$
and every integer $N$, there are constants
$m$, $M_1$, $M_2$ and a bounded set
$B\subset \calD(K)$, where $K$ is an arbitrary
compact neighborhood of $\supp\psi_2$,  such that
\begin{eqnarray*}
||t_1\otimes t_2||_{N,V,\psi_1\otimes\psi_2} & \le &
  M_1 
   ||t_2||_{N,C_\beta,\psi_2}
+ 
 M_2
   ||t_2||_{N+m,C_\beta,\psi_2} 
+ p_{B}(t_2),
\end{eqnarray*}
for every $t_1\in A$ and $t_2\in \calD'_{\Gamma_2}$,
where $C_\beta$ is an arbitrary compact neighborhood
of $W_2$.
\end{lem}
\begin{proof}
We want to calculate
\begin{eqnarray*}
||t_1\otimes t_2||_{N,V,\psi_1\otimes\psi_2}
= \sup_{\zeta\in V}  
    (1+|\zeta|)^N
    |\calF(t_1\psi_1\otimes t_2\psi_2)(\zeta)|.
\end{eqnarray*}    
We denote $u=t_1\psi_1$, $v=t_2\psi_2$
and $I=\widehat{u\otimes v}$. From $e_{(\xi,\eta)}=e_\xi\otimes e_\eta$
we find that
$I(\xi,\eta)=\langle t,e_{(\xi,\eta)}\rangle
=\langle u\otimes v, e_\xi\otimes e_\eta\rangle
=\langle u, e_\xi\rangle\langle v, e_\eta\rangle
=\widehat{u}(\xi)\widehat{v}(\eta)$.
By the shrinking lemma we can slightly enlarge
$W_1$ and $W_2$ to closed cones having the same properties.
Thus, there are two homogeneous functions of degree zero
$\alpha$ and $\beta$ on $\bbR^{d_1}$ and $\bbR^{d_2}$,
respectively, which are smooth except at the origin, 
non-negative and bounded by 1, such that:
(i) $\alpha|_{W_1\cup\{0\}}=1$ and $\beta|_{W_2\cup\{0\}}=1$;
(ii) $(\supp\alpha\times\supp\beta)\cap V=\emptyset$;
(iii) $(\supp\psi_1\times \supp(1-\alpha))\cap \Gamma_1=\emptyset$;
(iv) $(\supp\psi_2\times \supp(1-\beta))\cap \Gamma_2=\emptyset$.
We can write $I=I_1+I_2+I_3+I_4$ where (recalling that 
$\zeta=(\xi,\eta)$)
\begin{eqnarray*}
I_1(\zeta) &=&
\alpha(\xi)\widehat{u}(\xi)\beta(\eta)\widehat{v}(\eta),\\
I_2(\zeta) &=&
\alpha(\xi)\widehat{u}(\xi)(1-\beta)(\eta)\widehat{v}(\eta),\\
I_3(\zeta) &=&
(1-\alpha)(\xi)\widehat{u}(\xi)\beta(\eta)\widehat{v}(\eta),\\
I_4(\zeta) &=&
(1-\alpha)(\xi)\widehat{u}(\xi)(1-\beta)(\eta)\widehat{v}(\eta).
\end{eqnarray*}
The term $I_1(\zeta)=0$ because, by condition (ii)
$\alpha(\xi)\beta(\eta)=0$ for $(\xi,\eta)\in V$.
Condition (iii)  implies that
\begin{eqnarray*}
|(1-\alpha)(\xi)\widehat{u}(\xi)| 
  &\le & \sup_{\xi\in C_\alpha} 
   |\widehat{t_1\psi_1}(\xi)| 
  \le (1+|\xi|)^{-N} ||t_1||_{N,C_\alpha,\psi_1},
\end{eqnarray*}
where $\xi\in C_\alpha=\supp(1-\alpha)$.
This gives us immediately, with $C_\beta=\supp(1-\beta)$
\begin{eqnarray*}
|I_4(\zeta)| & \le & 
(1+|\xi|)^{-N}(1+|\eta|)^{-N} 
   ||t_1||_{N,C_\alpha,\psi_1}
   ||t_2||_{N,C_\beta,\psi_2}
\\&\le &
(1+|\zeta|)^{-N}
   ||t_1||_{N,C_\alpha,\psi_1}
   ||t_2||_{N,C_\beta,\psi_2},
\end{eqnarray*}
because 
$1+|(\xi,\eta)|\le 1 + |\xi|+ |\eta|\le 
(1+|\xi|)(1+|\eta|)$.
Since the set $A$ is bounded in $\calD'_{\Gamma_1}$
there is a constant $M_1=\underset{t_1\in A}{\sup}\Vert t_1\Vert_{N,C_\alpha,\psi_1}$ such that
$|I_4(\zeta)| \le (1+|\zeta|)^{-N} M_1 ||t_2||_{N,C_\beta,\psi_2}$.

To estimate $I_2$, we use the fact that, $u=t_1\psi_1$ being a compactly
supported distribution there is an integer $m$ such that,
for all $t_1\in A$,
\begin{eqnarray*}
|\alpha(\xi) \widehat{u}(\xi)| \le
|\widehat{u}(\xi)| \le
(1+|\xi|)^m ||\theta^{-m} \widehat{u}||_{L^\infty}.
\end{eqnarray*}
Moreover, as for the estimate of $I_4$, we have
$|(1-\beta)(\eta)\widehat{v}(\eta)| 
  \le (1+|\eta|)^{-N-m} ||t_2||_{N+m,C_\beta,\psi_2}$.
The set 
$\overline{\{\zeta\in \supp\alpha\times
C_\beta\telque |\zeta|=1\}\cap V}$
is \textbf{compact} and
avoids the set of all
elements of the
form $\zeta=(\xi,0),
\xi\in\text{supp }\alpha\setminus\{0\}$. 
Otherwise, we would find some
sequence $(\xi_n,\eta_n)\rightarrow (\xi,0)
\in  ((\text{supp }\alpha\times \{0\})\cap V)\subset
((\text{supp }\alpha\times \text{supp }\beta)\cap V)$
which contradicts
the condition (ii).
Let $\epsilon>0$ be 
the smallest value 
of $|\eta|$ in this set.
Then, the functions 
$\alpha$ and $\beta$ being
homogeneous of degree zero, 
$\supp\alpha\times C_\beta\cap V$ is a cone
in $\bbR^d$ and 
$|\eta|/|\zeta|  \ge \epsilon$ for all
$\zeta=(\xi,\eta)$ in the set
$\supp\alpha\times C_\beta\cap V$.
Thus, $(1+|\eta|)^{-N-m} \le \epsilon^{-N-m}
 (1+|(\xi,\eta)|)^{-N-m}$ and 
\begin{eqnarray*}
|I_2(\zeta)| &\le & ||\theta^{-m} \widehat{t_1\psi_1}||_{L^\infty}
   ||t_2||_{N+m,C_\beta,\psi_2} \epsilon^{-N-m}
(1+|\zeta|)^{-N},
\end{eqnarray*}
for every $\zeta\in V$,
because $|\xi|\le|(\eta,\xi)|$.
By lemma~\ref{Linflem}, 
$||\theta^{-m} \widehat{t_1\psi_1}||_{L^\infty}$
is uniformly bounded for 
$t_1\in A$ by a constant 
$M'_2=\underset{t_1\in A}{\sup}\Vert\theta^{-m}\widehat{t_1\psi_1}\Vert_{L^\infty},
\theta=1+\vert\xi\vert$.
Therefore, there is a constant 
$M_2=M'_2 \epsilon^{-N-m}$ such that,
for every $t_1\in A$ and 
every $t_2\in \calD'_{\Gamma_2}$,
$|I_2(\zeta)| \le M_2 
||t_2||_{N+m,C_\beta,\psi_2}$.

The term $I_3$ is treated differently because we want to
get the following result:
for every bounded set $A$ in $\calD'_{\Gamma_1}$ 
and every seminorm $||\cdot||_{N,V,\chi}$,
there is a bounded set $B\in \calD(\Omega_2)$
such that
$\forall \zeta\in V, I_3(\zeta) 
  \le p_B(t_2) (1+\vert\zeta\vert)^{-N}$ for every 
$t_2\in \calD'_{\Gamma_2}$.
This special form of eq.~\eqref{hypogenA}
is possible because the
union of bounded sets is a bounded set and
the multiplication of a bounded set by
a positive constant $M$ is a bounded set.

Therefore, we write
$I_3(\zeta)=\langle t_2, f_{\zeta}\rangle$,
where 
$$f_{(\xi,\eta)}(y)=(1-\alpha)(\xi)\widehat{u}(\xi)
\beta(\eta) \psi_2(y) e_\eta(y)$$ and
we must show that the set
$B=\{(1+|\zeta|)^N f_{\zeta}\telque \zeta\in V\}$ is
a bounded set of $\calD(\Omega_2)$.
A subset $B$ of $\calD(\Omega_2)$ is bounded iff
there is a compact set $K$ and
a constant $M_n$ for every integer $n$ such that
$\supp f\subset K$ and
$\pi_{n,K}(f)\le M_n$ for every $f\in B$.
All $f_\zeta$ are supported on 
$\supp\psi_2$ and are smooth functions
because $\psi_2$ and $e_\eta$ are smooth.
We have to prove that, if $t_1$ runs over a bounded
set of $\calD'_{\Gamma_1}$, then there are
constants $M_n$ such that
$\pi_{n,K}(f_\zeta)\le M_n$ for all
$\zeta\in V$, where $K$ is a compact
neighborhood of $\supp\psi_2$.
We start from
\begin{eqnarray*}
\pi_{n,K}(f_{(\xi,\eta)})  &=& 
\big|(1-\alpha)(\xi)\widehat{u}(\xi)
\beta(\eta)\big| \pi_{n,K}(\psi_2 e_\eta).
\end{eqnarray*}
We notice that
$\pi_{n,K}(\psi_2 e_\eta) \le 2^n\pi_{n,K}(\psi_2)
\pi_{n,K}(e_\eta)$ and that
$\pi_{n,K}(e_\eta) \le |\eta|^n$.
As for the estimate of $I_2$, we have
$|(1-\alpha)(\xi)\widehat{u}(\xi)|\le (1+|\xi|)^{-N-n}
||t_1||_{N+n,C_\alpha,\psi_1}$ because
$(\supp\varphi_1\times \supp(1-\alpha)) \cap \Gamma_1=\emptyset$
and $(1+|\xi|)^{-N-n}\le \epsilon^{-N-n}
(1+|(\xi,\eta)|)^{-N-n}$
for
some $\varepsilon$
because $(\xi,\eta)\in (V\cap \supp(1-\alpha)\times\supp\beta)$.
Therefore
\begin{eqnarray*}
\pi_{n,K}(f_\zeta) & \le &
  ||t_1||_{N+n,C_\alpha,\psi_1} 2^n \pi_{n,K}(\psi_2)
 \epsilon^{-N-n} (1+|\zeta|)^{-N},
\end{eqnarray*}
because $|\eta|^n(1+|(\xi,\eta)|)^{-n}\le 1$.
If $t_1$ belongs to a bounded set $A$ of
$\calD'_{\Gamma_1}$, then for each $N$
$||t_1||_{N,C_\alpha,\psi_1}$ is uniformly
bounded.
The estimate of $I_3$ is finally
\begin{eqnarray*}
|I_3(\zeta)| & \le & p_{B}(t_2) (1+|\zeta|)^{ -N}.
\end{eqnarray*}
\end{proof}
For each $j$, the conditions of the lemma hold if we
put $\psi_1=\varphi_1\psi_{j1}$,
$\psi_2=\varphi_2\psi_{j2}$, 
$W_1=W_{j1}$ and $W_2=W_{j2}$. 
Thus,  for every bounded set $A$ in $\calD'_{\Gamma_1}$,
every $u\in A$ and every $v\in \calD'_{\Gamma_2}$ we have
\begin{eqnarray*}
||u\otimes v||_{N,V,\varphi_1\otimes\varphi_2} & \le &
\sum_j
||u\otimes v||_{N,V,\varphi_1\psi_{j1}\otimes
  \varphi_2\psi_{j2}} \\& \le & \sum_j
  M_{1j} 
   ||v||_{N,C_{\beta_j},\varphi_2\psi_{j2}}
+ 
 M_2
   ||v||_{N+m,C_{\beta_j},\varphi_2\psi_{j2}} 
+ p_{B_j}(v).
\end{eqnarray*}
Since the sum over $j$ is finite, this means that
the family of maps $u\times v \mapsto u\otimes v$,
where $u\in A$, is equicontinuous for any
bounded set $A\subset\calD'_{\Gamma_1}$.
Because of the symmetry of the problem, we can prove 
similarly that
the family of maps $u\times v \mapsto u\otimes v$,
where $v\in B$, is equicontinuous for any
bounded set $B\subset\calD'_{\Gamma_2}$.
Finally, we have proved


\begin{thm}\label{hypotens}
Let $\Omega_1\subset \bbR^{d_1}$,
$\Omega_2\subset \bbR^{d_2}$ be open sets,
$\Gamma_1\in \dotT^*\Omega_1$,
$\Gamma_2\in \dotT^*\Omega_2$ be closed cones and 
\begin{eqnarray*}
\Gamma &= & 
\big(\Gamma_1 \times \Gamma_2\big) \cup 
\big((\Omega_1\times\{0\})\times \Gamma_2\big)
\cup \big(\Gamma_1\times (\Omega_2\times\{0\})\big).
\end{eqnarray*}
Then, the tensor product 
$(u,v)\mapsto u\otimes v$
is hypocontinuous from
$\calD'_{\Gamma_1}\times\calD'_{\Gamma_2}$ to $\calD'_\Gamma$,
in the normal topology.
\end{thm}

\section{The pull-back}
\label{pullbacksect}
The purpose of this section is to prove
\begin{prop}\label{pullbackprop}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ and $\Omega_2\subset
\mathbb{R}^{d_2}$
be two open sets and 
$\Gamma$ a closed cone in $\dotT^*\Omega_2$. 
Let $f:\Omega_1 \to \Omega_2$ be a smooth map such that
$N_f\cap \Gamma = \emptyset$, where
$N_f =  \{(f(x); \eta)\in \Omega_2\times \mathbb{R}^n\telque
  \eta\circ df_x =0\}$ and 
$f^*\Gamma = \{(x; \eta\circ df_x)|(f(x);\eta)\in \Gamma\}$.
Then, the pull-back operation
$f^*: \calD'_\Gamma(\Omega_2)\to \calD'_{f^*\Gamma}(\Omega_1)$
is continuous for the normal topology.
\end{prop}

We could prove that the pull-back is continuous by 
proving that $p_B(f^* u)$ is continuous for all
the seminorms of the strong topology of $\calD'$ and 
then that 
$||f^*u ||_{N,V,\chi}$ is continuous 
for every $N$ and every $V$ and $\chi$ such that
$ (\supp\chi\times V)\cap f^*\Gamma=\emptyset$.
However, it will be much simpler to 
use more topological methods
and to just prove that $\langle f^*u,v\rangle$ is continuous for 
every $v$ in an equicontinuous set.
This is the purpose of the following section.


\subsection{Equicontinuous bornology}
Let $\Omega$ be open in $\bbR^d$ and
$\Gamma$ be a closed cone in $\dotT^*\Omega$.
We define the open cone
$\Lambda=\{(x,\xi)\in \dotT^*\Omega\telque
(x,-\xi)\notin \Gamma\}$ and the
space
$\calE'_\Lambda(\Omega)$ of compactly supported distributions
$v\in \calE'(\Omega)$ such that $\WF(v)\subset \Lambda$.

The following theorem will be useful to
prove the continuity of linear maps~\cite[p.~200]{Horvath}:
\begin{thm}
If $E$ is a locally convex space and
$f: E \to \calD'_\Gamma(\Omega)$ is a linear map, then
$f$ is continuous iff, for every
equicontinuous set $H$ of $\calE'_\Lambda(\Omega)$
the seminorm $p_H:E\to\bbR$ defined by
\begin{eqnarray*}
p_H(x) &=& \sup_{v\in H} |\langle f(x),v\rangle|,
\end{eqnarray*}
is continuous.
\end{thm}
Similar results exist for multilinear
mappings~\cite{Concordido-04}.
Of course, this theorem can only be useful if the
equicontinuous sets are known.
 Let us
explain the 
concept of
equicontinuity~\cite[p.~200]{Horvath}
in the general context
of a 
locally 
convex topological vector space
$E$
with seminorms
$(p_\alpha)_{\alpha\in A}$. 
Let $E^*$
be its topological
dual, a set
$H$ in
$E^*$ is called
\emph{equicontinuous}
iff the family
of maps
$\ell_v:=u\longmapsto 
\left\langle
u,v
\right\rangle$
is \textbf{uniformly}
continuous
when $v$ runs
over the set $H$.
In other words,
for every 
\emph{equicontinuous}
set $H\subset E^*$,
there is a constant
$M$ and a finite family
of seminorms $(p_\alpha)_{\alpha\in A}$
of $E$
such that
\begin{eqnarray}
\forall v\in H,\,\ 
\vert \left\langle
u,v
\right\rangle\vert
\leq
M\left(\underset{\alpha\in A}{\sum} p_\alpha(u)\right). 
\end{eqnarray}

 Therefore a set $H$ is equicontinuous
in $\calE'_\Lambda(\Omega)$ (which is the topological dual
of $\mathcal{D}^\prime_\Gamma(\Omega)$) 
iff there is a finite number of 
seminorms 
$||\cdot||_{N_1,V_1,\chi_1},\dots,||\cdot||_{N_k,V_k,\chi_k}$
of $\mathcal{D}^\prime_\Gamma(\Omega)$,
a bounded subset $B_0$ of $\calD(\Omega)$
and a constant $M$  such that
\begin{eqnarray}
|\langle u,v\rangle| & \le & M \sup\{||u||_{N_1,V_1,\chi_1},
\dots, ||u||_{N_k,V_k,\chi_k},p_{B_0}(u)\}
\label{defequicont}
\end{eqnarray}
for every $u\in \calD'_\Gamma(\Omega)$ and every $v\in H$.
There is only one seminorm $p_{B_0}$ because these
seminorms are saturated~\cite[p.~107]{Horvath}
in $\calD'(\Omega)$ with the strong topology.

 We recall some useful terminology.
Let $X$ be a locally convex space 
and $ X^\prime$ its topological dual.
Then the strong topology $\beta( X^\prime,X)$
on $X^\prime$ is defined by the seminorms
$P_B(y)=\sup_{x\in B}\vert \left\langle x,y \right\rangle\vert $
where
$B$ runs over all bounded sets in $X$.
We also denote by
$\underline{\pi}$
the projection: 
$(x;\xi)\in T^*\Omega\longmapsto x\in\Omega$.
The following lemma characterizes the equicontinuous sets
of $\calE'_\Lambda(\Omega)$:
\begin{lem}
\label{charequilem}
A subset $B$ of $\calE'_\Lambda(\Omega)$ 
is equicontinuous
iff there is: (i) a compact set $K\subset\Omega$ containing
the support of all elements of $B$; (ii) a closed
cone $\Xi\subset\Lambda$ such that 
$B\subset \calD'_\Xi(\Omega)$, 
$B$ is bounded in $\calD'_\Xi(\Omega)$ 
and
$\underline{\pi}(\Xi)\subset K$.
\end{lem}
\begin{proof}
We first prove that every such $B$ is equicontinuous.
We showed in~\cite{Dabrouder-13} that
the space $\calE'_\Lambda(\Omega)$ is the inductive limit
of spaces $E_\ell=\{v\in \calE'_\Lambda(\Omega)\telque
\supp v\in L_\ell, \WF(v)\in \Lambda_\ell\}$,
where the compact sets $L_\ell$ exhaust $\Omega$
and the closed cones $\Lambda_\ell$ exhaust $\Lambda$.
Thus, there is an integer $\ell$ such that
$\Xi \subset \Lambda_\ell$ and 
$B\subset E_\ell$. The inclusion of $\Xi$ in $\Lambda_\ell$
implies that every seminorm $||\cdot||_{N,V,\chi}$ of
$E_\ell$ is also a seminorm of $\calD'_\Xi(\Omega)$ 
because
$\supp\chi\times V$ does not meet
$\Xi$ if it does not meet $\Lambda_\ell$. 
Thus, $B$ is bounded in $E_\ell$ and 
Eq.~(8) of~\cite{Dabrouder-13} gives us
\begin{eqnarray*}
\sup_{v\in B}|\langle u,v\rangle| & \le & \sum_j
\Big(
p_{B_j}(u) + ||u||_{m+n+1,V_j,\chi_j} C
I_n^{n+1}
+ ||u||_{n,V_j,\chi_j} M_{n,W_j,\chi_j}
  I_n^{2n} \Big),
\end{eqnarray*}
which can be converted to the equicontinuity
condition \eqref{defequicont}.

To show the converse, we denote by $B$ the
set of all $v\in \calE'_\Lambda(\Omega)$ that satisfy
Eq.~\eqref{defequicont}.
Then, by following exactly the proof of
Prop.~7 of \cite{Dabrouder-13},
we obtain that the support of all
elements of $B$ is included in a compact set
$K=\cup_j \supp\chi_j \cup K'$, where 
$K'$ is a compact set containing the support
of all $f\in B_0$.  Moreover, 
the wave front set of all elements of
$B$ is contained in
$\Xi=\cup_j \supp\chi_j\times (-V_j)$.
It remains to show that $B$ is bounded in
$\calD'_\Xi(\Omega)$ for the normal
topology.
We first notice that, if 
$\supp\chi\times (-V)\subset \Xi$, then
$||\cdot||_{N,V,\chi}$ is a continuous
seminorm of the strong dual $\calE'_{(\Xi')^c}(\Omega)$
of $\calD'_\Xi(\Omega)$.
Indeed, it was shown in the proof
of Prop.~7 of \cite{Dabrouder-13} that
$||u||_{N,V,\chi}=\sup_{\xi\in V}|\langle u,f_\xi\rangle|$,
where $f_\xi(x)=(1+|\xi|)^N \chi(x) e^{ -i\xi\cdot x}$
and the set $\{f_\xi,\xi\in V\}$ is bounded in 
$\calD'_\Xi(\Omega)$.
If $B'$ is a bounded set in $\calE'_{(\Xi')^c}(\Omega)$,
the continuous seminorms $||u||_{N,V,\chi}$
and $p_{B_0}(u)$ of $\calE'_{(\Xi')^c}(\Omega)$
appearing on the
right hand side of (\eqref{defequicont})
are bounded over $B'$.
Thus, for any bounded set $B^\prime$ in
$\calE'_{(\Xi')^c}(\Omega)$, taking $u\in B^\prime$
and taking the sup in (\eqref{defequicont})
over $u\in B^\prime$ yields that  
$\sup_{u\in B',v\in B} |\langle u,v\rangle|$ is bounded
and $B$ is a bounded subset of
$\calD'_\Xi(\Omega)$ when $\calD'_\Xi(\Omega)$ is equipped
with the strong $\beta(\calD'_\Xi,\calE'_{(\Xi')^c})$ topology.
It is shown in~\cite[Theorem 33]{Dabrouder-13} that the bounded
sets of $\calD'_\Gamma(\Omega)$ coincide for the strong and
the normal topologies. Thus, $B$ is bounded for
the normal topology.
\end{proof}

We obtain the following characterization of continuous 
linear maps:
\begin{thm}
\label{charcont}
Let $E$ be a locally convex space, $\Omega$ an open subset
of $\bbR^d$ and $\Gamma$ a closed cone in $\dotT^*\Omega$.
A linear map $f:E\to \calD'_\Gamma(\Omega)$ is continuous iff every map
$f_B: E\to \bbR$ defined by 
$f_B(x)=\sup_{v\in B}|\langle f(x),v\rangle|$
is continuous, 
where $B$ is equicontinuous in 
$\mathcal{E}^\prime_\Lambda
(\Omega),$
$\Lambda=\left(\Gamma^\prime\right)^c$.
\end{thm}
Recall $\Lambda=\left(\Gamma^\prime\right)^c$.
The equicontinuous sets of $\calE'_\Lambda(\Omega)$ intervene
also because of the following theorem~\cite[p.~157]{Kothe-II}:
\begin{thm}\label{thmdualityequi}
Let the duality pairing  $\calD'_\Gamma(\Omega)\times\calE'_\Lambda(\Omega)\to\bbK$
be defined by $u\times v\to f(u,v)=\langle u,v\rangle$.
Then, for every bounded set $A$ of $\calD'_\Gamma(\Omega)$ and every
equicontinuous set $B$ of $\calE'_\Lambda(\Omega)$ the sets of maps
$\{f_u\telque u\in A\}$ and 
$\{f_v\telque v\in B\}$ are equicontinuous.
\end{thm}

We cannot expect the
duality pairing to be (jointly) continuous because
this happens only for normable spaces~\cite[p.~359]{Horvath}.
The spaces $\calD'_\Gamma(\Omega)$ and $\calE'_\Lambda(\Omega)$ are not normable.
In fact, a nuclear space is normable if and only if it is finite
dimensional~\cite[p.~520]{Treves}.



\paragraph{Strategy of proof 
that the pull--back is continuous.}
Let $\Omega_1$ and $\Omega_2$ be open sets in 
$\bbR^{d_1}$ and $\bbR^{d_2}$, respectively.
Let $f:\Omega_1\to\Omega_2$ be a smooth map and
$\Gamma$ be a closed cone in $\dotT^*\Omega_2$.
We want to show that the pull-back 
$f^*:\calD'_\Gamma(\Omega_2)
\to \calD'_{f^*\Gamma}(\Omega_1)$ is continuous
for the normal topology.
According to Theorem~\ref{charcont},
the pull-back is continuous iff,
for every equicontinuous set 
$B\subset \calE'_\Lambda(\Omega_1)$
(where $\Lambda=(f^*\Gamma)^{',c}$) the family of maps
$(\rho_v)_{v\in B}$, defined by
$\rho_v:u\mapsto \langle f^*u,v\rangle$,
is equicontinuous 
which implies that
$\sup_{v\in B}\vert 
\left\langle f^*u,v \right\rangle
\vert$
is continuous in
$u$. 
By Lemma~\ref{charequilem}, we know that there is a compact set
$K\subset \Omega_1$ and a closed cone $\Xi\subset(f^*\Gamma')^c$
such that $\supp v\subset K$ and $\WF(v)\subset \Xi$ for all
$v\in B$. Choose a function  $\chi\in \calD(\Omega_1)$ 
such that $\chi|_K=1$. If $(\varphi_i)_{i\in I}$ is a partition of unity
of $\Omega_2$, we can write
$\langle f^*u,v\rangle=\sum_i\langle f^*(u\varphi_i),v\chi\rangle$.
The image of $\supp\chi$ by 
$f$ being compact~\cite[p.~19]{Bredon}, only a finite
number of terms of this sum are nonzero and 
the family $\rho_v$ is equicontinuous iff,
for every $\varphi\in \calD(\Omega_2)$,
the family of maps $u\mapsto \langle f^*(u\varphi),v{\chi}\rangle$
is equicontinuous.
\paragraph{Stationary phase and Schwartz kernels.}



In order to calculate the pairing between $f^*(u\varphi)$ and $v$,
we first
notice that, 
when $u$ is a locally integrable function, then
$u\varphi(y)=\calF^{-1}(\widehat{u\varphi})(y)=
(2\pi)^{-d_2}\int_{\bbR^{d_2}} d\eta e^{i\eta\cdot y} 
\widehat{u\varphi}(\eta)$,
so that 
\begin{eqnarray*}
f^*\left(u\varphi\right)(x)&=&(2\pi)^{-d_2} \int_{\bbR^{d_2}} d\eta 
  e^{i\eta\cdot f(x)} \widehat{u\varphi}(\eta)
\end{eqnarray*}  
and
\begin{eqnarray*}
\langle f^*\left(u\varphi\right), \chi v\rangle &=& \frac{1}{(2\pi)^{d_2}}
\int_{\Omega_1} 
\int_{\bbR^{d_2}} 
 \chi(x) v(x)
e^{i\eta \cdot f(x)} \widehat{u\varphi}(\eta)
d x d \eta\\
&=&\frac{1}{(2\pi)^{d_2}}
\int_{\bbR^{d_2}}
\int_{\Omega_1} 
\int_{\bbR^{d_2}} 
 \chi(x) v(x)
e^{i\eta \cdot f(x)} e^{-iy\cdot\eta}u(y)\varphi(y)
dyd x d \eta .
\end{eqnarray*}
This definition can be extended to any distribution
$u\in \calD'_\Gamma$ as
\begin{eqnarray}
\label{fsuphiv}
\langle f^*(u\varphi), \chi v\rangle &=& \frac{1}{(2\pi)^{d}}
  \int_{\bbR^{d}} 
   u(y)v(x)I(x,y)dxdy,
\end{eqnarray}
where $d=d_1+d_2$, 
$I(x,y)=\int_{\bbR^{d_2}}
e^{i\eta \cdot( f(x)-y)} \varphi(y)\chi(x) d \eta$.
The duality pairing can also be written
\begin{eqnarray}
\label{fsuphiv2}
\langle f^*(u\varphi),v\chi\rangle &=& 
  \langle v\otimes u, I\rangle.
\end{eqnarray}
Note that 
$I(x,y)=(2\pi)^{-d_2}
\chi(x)\varphi(y)
\int d\eta e^{i\eta\cdot( f(x)-y)}$
is an oscillatory integral 
in the sense of H\"ormander \cite{HormanderI}\cite{ReedSimonII} with
symbol $\chi(x)\varphi(y)$ and
phase $\eta\cdot( f(x)-y)$ where 
$\eta\cdot( f(x)-y)$ is
homogeneous of degree
$1$ w.r.t. $\eta$, $\forall \eta\neq 0,
d\left(\eta\cdot( f(x)-y)\right)\neq 0$. 
Therefore, 
$I\in\mathcal{D}^\prime(\Omega_1\times\Omega_2)$ 
is the 
\textbf{Schwartz
kernel of the bilinear
continuous map:}
\begin{eqnarray}
(u,v)\in \mathcal{D}(\Omega_1)\times\mathcal{D}(\Omega_2) 
\longmapsto \left\langle f^*(u\varphi), v\chi \right\rangle.
\end{eqnarray}
 

\paragraph{ Proof of Proposition \ref{pullbackprop}.}

By Theorem~\ref{hypotens},
the map $(v,u)\mapsto v\otimes u$ is hypocontinuous
from $\calD'_\Xi\times \calD'_\Gamma$ to
$\calD'_{\Gamma_\otimes}$ where 
$\Gamma_\otimes=\Xi\times \Gamma \cup (\Omega_1\times\{0\})\times \Gamma
\cup \Xi\times(\Omega_2\times\{0\})$.
Let $\Lambda_{\otimes}$
be the open cone $\Gamma_{\otimes}^{\prime,c}$.
Therefore by Theorem \ref{thmdualityequi}, the 
family of duality pairings

\begin{eqnarray*}
u\otimes v\in\calD'_{\Gamma_\otimes} \longmapsto \left\langle u\otimes v,r \right\rangle
\end{eqnarray*}

is equicontinuous
from $\calD'_{\Gamma_\otimes}$ to $\bbK$
uniformly in $r\in B^\prime$
for every equicontinuous set $B'$ of $\calE'_{\Lambda_\otimes}$.
In particular, if $B'$ contains only the element
$I$, then 
the map $v\otimes u\mapsto \langle v\otimes u,I\rangle$
would be continuous  
if
$I$ were compactly supported, which is
the case because its support is included in 
$\supp\chi\times\supp\varphi$ and if its
wave front set  were contained in $\Lambda_\otimes$.
Thus, if $\WF(I)\subset \Lambda_\otimes$,
then the map $(v,u)\mapsto \langle v\otimes u,I\rangle$
is hypocontinuous because, by the next lemma,
the composition of a hypocontinuous
map by a continuous map is hypocontinuous.
In other words, the map
$(v,u)\mapsto \langle f^*(u\varphi),\chi v\rangle$ is hypocontinuous,
by item (i) of Definition~\ref{hypodef}, 
this implies that the family of maps
$\rho_v:u\mapsto \langle f^*(u\varphi),\chi v\rangle$
with $v\in B$ is equicontinuous.
\begin{lem}
\label{hypocontcont}
The composition of a hypocontinuous map by a continuous map is
hypocontinuous.
\end{lem}
This result is 
known~\cite{Glockner-09} 
but we could not
find a proof in
the literature.
\begin{proof}
Let $f:E\times F\to G$ be a hypocontinuous map
and $g:G\to H$ a continuous map.
The map $g\circ f$ is hypocontinuous iff, for every bounded set
$B\subset F$
and every neighborhood $W$ of zero in $H$, there is a neighborhood
$U$ of zero in $E$ such that $(g\circ f)(U\times B)\subset W$
(with the similar condition for $(g \circ f)(A\times
V)$)~\cite[p.~III.30]{Bourbaki-TVS}.
By the continuity of $g$, there is a neighborhood $Z$ of zero in
$G$ such that $g(Z)\subset W$. By the hypocontinuity of $f$,
there is a neighborhood $U$ of zero in $E$ such that
$f(U\times B)\subset Z$. Thus, $(g\circ f)(U\times B)
\subset g(Z)\subset W$.
\end{proof}

It just remains to check that
$\WF(I)\subset \Lambda_\otimes$, i.e. that
$\WF(I)'$ does not meet $\Gamma_\otimes$.
The wave front set of $I$ is 
$\WF(I)\subset \{(x,f(x);-\eta\circ d_x f,\eta)\telque 
x\in \supp\chi\}$~\cite[p.~260]{HormanderI}.
Recall that $\Xi\subset(f^*\Gamma')^c=\{(x,-\eta\circ df_x)\telque
  (f(x),\eta)\notin\Gamma\}$. By definition of
$\Gamma_\otimes$ we must satisfy the following three conditions:
\begin{itemize}
\item $\Xi\times \Gamma \cap \WF(I)'=\emptyset$ because
it is the set of points 
$(x,f(x);-\eta\circ d_x f,\eta)$ such that
$(f(x),\eta)\notin\Gamma$ by definition of $\Xi$
and $(f(x),\eta)\in \Gamma$ by definition of $\Gamma$;
\item $\Xi\times (\Omega_2\times\{0\}) \cap \WF(I)'=\emptyset$ because
we would need $\eta=0$ whereas $(y,\eta)\in \Gamma$ implies
$\eta\not=0$; 
\item
$(\supp\chi\times\{0\})\times \Gamma\cap \WF(I)'\subset
\{(x,f(x);0,\eta)\telque x\in\supp\chi, \eta\circ df_x=0,
(f(x),\eta)\in \Gamma\}$.
\end{itemize}
Thus, if $f^*\Gamma\cap N_f=\emptyset$, 
then $\WF(I)'\cap\Gamma_\otimes=\emptyset$ and
the pull-back is continuous.

\paragraph{How to write the pull--back operator in terms of the Schwartz kernel $I$ ? Relationship with the product of distributions.} \label{kernelfonctoriel}
We start from a linear operator 
$L:\mathcal{D}(\mathbb{R}^{d_2})\longmapsto \mathcal{D}^\prime(\mathbb{R}^{d_1}) $ 
with corresponding Schwartz kernel 
$K\in\mathcal{D}^\prime(\mathbb{R}^{d_1}\times\mathbb{R}^{d_2})$.
Using the standard
operations on distributions, we
can make sense of the well--known
representation formula
$Lu=\int_{\mathbb{R}^{d_2}}  K(x,y) u(y)dy $
for an operator $L:\mathcal{D}(\mathbb{R}^{d_2})\longmapsto \mathcal{D}^\prime(\mathbb{R}^{d_1})$
and its kernel $K\in\mathcal{D}^\prime(\mathbb{R}^{d_1}\times\mathbb{R}^{d_2})$.
Let us define the two projections
$\pi_2:=(x,y)\in\mathbb{R}^{d_1}\times \mathbb{R}^{d_2}\longmapsto y\in\mathbb{R}^{d_2}$
and
$\pi_1:=(x,y)\in\mathbb{R}^{d_1}\times \mathbb{R}^{d_2}\longmapsto x\in\mathbb{R}^{d_1}$,
then we define
$K(x,y)u(y)=K(x,y)\pi_2^*u(x,y)=K(x,y)\left(1(x)\otimes u(y)\right)$ 
where $\pi_2^*u=1\otimes u$ 
and $\int_{\mathbb{R}^{d_2}}dyf(x,y)=\pi_{1*}f(x)$.
Therefore 
\begin{equation}
Lu=\int_{\mathbb{R}^{d_2}}  K(x,y) u(y)dy=\pi_{1*}\left(K\left(\pi_2^* u\right) \right).
\end{equation}

The interest
of the formula $Lu=\pi_{1*}\left(K\left(\pi_2^* u\right) \right)$ 
is that everything 
generalizes
to oriented manifolds. Replace
$\mathbb{R}^{d_2}$ (resp $\mathbb{R}^{d_1}$) 
with a 
manifold $M_2$ (resp $M_1$) 
with smooth volume densities 
$\vert\omega_2\vert$ 
(resp $\vert\omega_1\vert$), 
the duality pairing is
defined as the extension
of the usual 
integration 
against the 
volume densities, for
instance:
$\forall (u,\varphi)\in C^\infty(M_1)\times \mathcal{D}(M_1),
\left\langle u,\varphi\right\rangle_{M_1}=\int_{M_1}(u\varphi)\omega_1 $.
Finally, for the 
linear continuous map $L:=u\in\mathcal{D}(\mathbb{R}^{d_2})\longmapsto \chi f^*(u\varphi)\in \mathcal{D}^\prime(\mathbb{R}^{d_1})$, we get the formula:
\begin{equation}
Lu= \pi_{1*}\left( I (\pi_2^* u)\right)
\end{equation}
where $I(x,y)=(2\pi)^{-d_2}
\chi(x)\varphi(y)
\int d\eta e^{i\eta\cdot(f(x)-y)}$ is the Schwartz kernel of $L$.

\subsection{Pull-back by families of smooth maps}

\begin{lem}\label{Gammaclosed}
Let $\Omega_1,\Omega_2,U$
be open sets
in
$\mathbb{R}^{d_1},\mathbb{R}^{d_2},\mathbb{R}^{n}$
respectively.
For any compact sets
$(K_1\subset \Omega_1,K_2\subset\Omega_2,A\subset U)$,
the conic set
\begin{eqnarray*}
\Gamma
=\{(x,f(x,a);-\eta\circ d_xf(x,a),\eta)\telque (x,a,f(x,a))\in K_1\times A\times K_2,\eta\neq 0 \}
\end{eqnarray*}
is closed 
in $\dot{T}^*\left(\Omega_1\times\Omega_2\right)$.
\end{lem}
\begin{proof}
Let $(x,y;\xi,\eta)\in\overline{\Gamma}$
such that $(\xi,\eta)\neq (0,0)$. 
Then there is
a sequence 
\begin{eqnarray*}
(x_n,f(x_n,a_n);-\eta_n\circ d_xf(x_n,a_n),\eta_n)\in\Gamma, 
(x_n,a_n,f(x_n,a_n))\in K_1\times A\times K_2
\end{eqnarray*}
which converges to $(x,y;\xi,\eta)$.
By compactness of $A$, we extract
a convergent subsequence $a_n\rightarrow a$. By continuity
of $d_xf$,
we find that
$\xi=-\eta\circ d_xf(x,a)$,
we also find that
$\underset{n\rightarrow \infty}{\lim} 
f(x_n,a_n)=f(x,a)\in K_2$
since $K_2$ is closed
and we finally 
note that
we must have 
$\eta\neq 0$
otherwise $\xi=0,\eta=0$.
Therefore 
$(x,y;\xi,\eta)\in\Gamma$
by definition.
Finally, 
$\overline{\Gamma}
\subset\Gamma$
hence $\Gamma$ is closed. 
\end{proof}
\begin{prop}\label{Ifaequicontinuous}
Let $\Omega_1$ be an open set in $\bbR^{d_1}$,
$A\subset U\subset \bbR^n$ 
where $A$ is compact, $U$ 
and $\Omega_2$ are open sets 
in $\bbR^{d_2}$.
Let $\chi\in \calD(\Omega_1)$, 
$\varphi\in \calD(\Omega_2)$ and
$f$ a smooth map
$f:\Omega_1\times U \to\Omega_2$.
\begin{enumerate}
\item Then
the family of distributions $(I_{f(.,a)})_{a\in A}$ 
formally defined by
\begin{eqnarray*}
I_{f(.,a)}(x,y) &=& 
 \chi(x)\varphi(y)
  \int_{\bbR^{d_2}} \frac{d\theta}{(2\pi)^{d_2}}
   e^{i\theta\cdot( f(x,a)-y)}
\end{eqnarray*}
is a bounded set in $\calD'_{\Gamma}$, 
where
$\Gamma$ 
is the 
closed cone in $\dotT^*(\Omega_1\times\Omega_2)$
defined by:
\begin{eqnarray*}
\Gamma &=&\{(x,f(x,a);-\eta\circ d_xf(x,a),\eta)\telque
\\ & &  x\in \supp\chi,
f(x,a)\in \supp\varphi,a\in A,\eta\neq 0 \}.
\end{eqnarray*}
\item For any
open cone $\Lambda$ 
containing $\Gamma$,
$(I_{f(.,a)})_{a\in A}$
is
equicontinuous in
$\mathcal{E}^\prime_{\Lambda}(\Omega_1\times\Omega_2)$.
\end{enumerate}
\end{prop}

We will use the 
pushforward theorem
\ref{pushforwardthm}
in the following proof.

\begin{proof}
 From Lemma
\ref{charequilem} and from the fact that
$(I_{f(.,a)})_{a\in A}$
is supported
in a fixed compact set 
$\supp\chi\times\supp\varphi$, we deduce that
conclusion (2) follows from the first claim
thus it suffices
to prove the
claim (1).

 The conic set $\Gamma$ is closed
by Lemma \ref{Gammaclosed}. 
To prove
that the family
$\left(I_{f(.,a)}\right)_{a\in A}$
is bounded in $\mathcal{D}^\prime_\Gamma$,
it suffices
to check that
\begin{eqnarray*}
\forall v\in \mathcal{E}^\prime_{\Gamma^{\prime c}}(\Omega_1\times\Omega_2),
\sup_{a\in A}\vert\left\langle I_{f(.,a)},v \right\rangle\vert  < +\infty
\end{eqnarray*}
because of 
\cite[Proposition 1]{Dabrouder-13}.


\textbf{Step 1}
Our goal is to study
the map
$a\longmapsto \int_{\Omega_1\times\Omega_2} I_f(x,y,a)v(x,y)$
where 
$I_{f}(x,y,a) = 
 \chi(x)\varphi(y)
  \int_{\bbR^{d_2}} \frac{d\theta}{(2\pi)^{d_2}}
   e^{i\theta\cdot(f(x,a)-y)}$.
Let 
$\pi_{12},\pi_3$
be projections
defined by the formulas
\begin{eqnarray*}
\pi_{12}:(x,y,a)\in\Omega_1\times\Omega_2\times U
&\longmapsto & (x,y)\in \Omega_1\times\Omega_2\\
\text{ and }\pi_3:(x,y,a)\in\Omega_1\times\Omega_2\times 
U&\longmapsto & a\in U.
\end{eqnarray*}
Using the dictionary explained 
in paragraph
\ref{kernelfonctoriel},
if $v$ were a 
test function,
then we would
find
that
\begin{eqnarray}\label{pushforwardkeyformula}
\int_{\Omega_1\times\Omega_2} I_f(x,y,.)v(x,y)
dxdy &=& \pi_{3*}\left(I_f\pi_{12}^*v \right)\in \mathcal{D}^\prime(U).
\end{eqnarray}
We want to prove 
that $a\longmapsto  
\int_{\Omega_1\times\Omega_2} I_f(x,y,a)v(x,y)dxdy$
is smooth 
in some open
neighborhood
of $A$ since this
would imply
that 
\begin{eqnarray*}
\sup_{a\in A}\vert 
\int_{\Omega_1\times\Omega_2} I_f(x,y,a)v(x,y)dxdy\vert
=\sup_{a\in A}\vert\left\langle
I_{f(.,a)},v\right\rangle\vert<+\infty.
\end{eqnarray*}
In order to do so, it suffices
to prove that the condition
$v\in \mathcal{E}^\prime_{\Gamma^{\prime,c}}$
implies that
the distributional 
product $I_f(x,y,a)v(x,y)=I_f\left(\pi_{12}^*v \right)(x,y,a)$
makes sense in $\mathcal{D}^\prime(\Omega_1\times \Omega_2\times U)$
and the push--forward 
$\pi_{3*}\left(I_f\pi_{12}^*v \right)= \int_{\Omega_1\times\Omega_2} I_f(x,y,.)v(x,y)dxdy$
has empty wave front set over
some open neighborhood
of $A$.

\textbf{Step 2}
The wave front sets of
$I_f$ and $\pi_{12}^*v$ are:
\begin{eqnarray*}
WF(I_f)&=&\{\left(
\begin{array}{ccc}
x&;&-\theta\circ d_xf\\
f(x,a)&;&\theta\\
a&;&-\theta\circ d_af  
\end{array} \right)\telque \\
& &  x\in \text{supp }\varphi,f(x,a)\in\text{supp }\chi, a\in U , \theta\neq 0 \} \\
WF(\pi_{12}^*v)=WF(v\otimes 1)&=&\{
\left(\begin{array}{ccc}
x&;&\xi\\
y&;&\eta\\
a&;&0 \end{array} \right) 
\telque 
\left(\begin{array}{ccc}
x&;&\xi\\
y&;&\eta
\end{array}\right)\in WF(v) \}. 
\end{eqnarray*}

One also have 
\begin{eqnarray*}
v\in\mathcal{E}^\prime_{\Gamma^{\prime c}}& \implies & WF(v)\cap \Gamma^\prime=\emptyset\\
&\implies &  \xi\neq -\eta\circ d_xf\\
&\implies & \forall \theta , \left(\begin{array}{c}
\xi-\theta\circ d_xf \\
\theta+\eta
\end{array}\right)=
\left(\begin{array}{c}
0 \\
0
\end{array}\right)
\text{has no solution}
\end{eqnarray*}
Observe that
\begin{eqnarray*}
WF(I_f)+WF(\pi_{12}^*v)=\{ \left(\begin{array}{ccc}
x&;& \xi-\theta\circ d_xf \\
f(x)&;&\theta+\eta \\
a&;& -\theta\circ d_af
\end{array}\right),\forall\theta \in\mathbb{R}^d\setminus \{0\} \}
\end{eqnarray*}
\begin{eqnarray*}
&\implies & \left(WF(I_f)+WF(\pi_{12}^*v)\right)
\cap \underline{0}=\emptyset\\
&\text{ and } & \left(WF(I_f)\cup WF(\pi_{12}^*v)\right)
\cap \underline{0}=\emptyset.
\end{eqnarray*}



\textbf{Step 3} In the last step,
we shall prove that the condition
$WF(v)\cap \Gamma^\prime=\emptyset$
actually implies that
$WF\left(\pi_{3*}\left(I_f\pi_{12}^*v\right)\right)$
is empty over some
open neighborhood $U^\prime$
of $A$. 
The condition
$
WF(v)\cap\Gamma^\prime=\emptyset 
$
implies the existence
of
some open 
neighborhood $U^\prime$
of $A$ s.t.
\begin{eqnarray*}
\forall a\in U^\prime, \forall (x,f(x,a);\xi,\eta)\in WF(v),\xi\neq -\eta\circ d_xf(x,a).
\end{eqnarray*}
Since $A$ and $\text{supp }v$ are compact
and $A\times WF(v)$ is closed,
we can find $\delta>0$ s.t. $$\forall (a,(x,f(x,a);\xi,\eta))\in A\times WF(v),
\,\ \vert\xi+\eta\circ d_xf(x,a)\vert\geqslant \delta\vert\eta\vert.$$
Define $U^\prime=\{a\in U \telque  \forall (x,f(x,a);\xi,\eta)\in WF(v),\vert\xi+\eta\circ d_xf(x,a)\vert > \frac{\delta}{2}\vert\eta\vert\}$.
Therefore, the
condition 
$WF(v)\cap\Gamma^\prime=\emptyset$
on the wave front set of
$v$
ensures that
$\pi_{3*}\left(I_f\pi_{12}^*v\right)$
is well defined in
$\mathcal{D}^\prime_\emptyset(U^\prime)=C^\infty(U^\prime)$.
Hence $a\longmapsto \left\langle v,I_{f(.,a)}\right\rangle
= \int_{\Omega_1\times\Omega_2} I_f(x,y,a)v(x,y)dxdy$
is smooth on $U^\prime$, a fortiori 
continuous on the compact set $A$
which means that
\begin{eqnarray*}
\sup_{a\in A}\vert \left\langle v,I_{f(.,a)}\right\rangle\vert <+\infty.
\end{eqnarray*} 
\end{proof}

\begin{thm}\label{pbenfamille}
Let $\Omega_1\subset \mathbb{R}^{d_1},\Omega_2\subset
\mathbb{R}^{d_2}$ 
be two open sets, $A\subset U\subset \bbR^n$ where
$A$ compact, $U$
open 
and 
$\Gamma$ a closed cone in $\dotT^*\Omega_2$. 
Let $f:\Omega_1\times U \to \Omega_2$ be a smooth map such that
$\forall a\in A,  f(.,a)^*\Gamma$
does not meet 
the zero section
$\underline{0}$ 
and set 
$\Theta=\bigcup_{a\in A}f(.,a)^*\Gamma$.
Then for all seminorms 
$P_B$ of $\mathcal{D}^\prime_\Theta(\Omega_1)$,
$\forall u\in\mathcal{D}^\prime_\Gamma(\Omega_2)$, 
the family $P_B(f(.,a)^*u)_{a\in A}$ is bounded.
\end{thm}

\begin{proof}
Let $B$ be equicontinuous
in $\mathcal{E}^\prime_{\Theta^{\prime,c}}(\Omega_1)$.
We need to prove that
$\sup_{(v,a)\in B\times A} \vert  \langle f(.,a)^*u, v\rangle\vert<+\infty$.
It follows from
Lemma \ref{charequilem}
that there exists 
some closed cone $\Xi$
such that $\Xi\cap\Theta^\prime=\emptyset$
and $B\subset \mathcal{D}_{\Xi}^\prime(\Omega_1)$.
Set $\Gamma_\otimes=\left(\Xi\times \Gamma\right) \cup \left((\Omega_1\times\{0\})\times \Gamma\right)
\cup \left(\Xi\times(\Omega_2\times\{0\})\right)$. 
Let $\Lambda_\otimes$ be the open cone
defined as $\Lambda_\otimes=(\Gamma_\otimes)^{\prime,c}$.
By proposition \ref{Ifaequicontinuous},
we can easily verify
as in the proof
of Proposition
\ref{pullbackprop} that
the 
family 
$(I_{f(.,a)})_{a\in A}$
is equicontinuous
in $\calE'_{\Lambda_\otimes}$.
Then we prove similarly
as in the proof
of the pull-back Proposition  
\ref{pullbackprop}
that the family of maps
$\rho_{v,a}:u\mapsto \langle f(.,a)^*(u\varphi),\chi v\rangle$
with $(v,a)\in B\times A$ is equicontinuous where
$B$ is equicontinuous
in $\mathcal{E}^\prime_{\Theta^{\prime,c}}(\Omega_1)$ and
$\chi$ is chosen in such a way that 
$\chi|_{\text{supp }B}=1$
and $\varphi|_{f(\text{supp }B)}=1$, therefore:
\begin{eqnarray*}
\sup_{(v,a)\in B\times A} \vert  \langle f(.,a)^*u, v\rangle\vert=
\sup_{(v,a)\in B\times A} \vert \langle f(.,a)^*(u\varphi),\chi v\rangle\vert<+\infty.
\end{eqnarray*}
and the result follows
from the 
characterization
of continuous linear maps
by Theorem \ref{charcont}.
\end{proof}
%In fact, the 
%assumption that
%$\forall a\in A, f(.,a)^*\Gamma\cap \underline{0}=\emptyset$
%implies that
%$\Theta$ is a closed conic set
%and several other 
%equivalent statements: 
%\begin{lem}\label{eqconditionforpbenfamille}
%Let $\Omega_1\subset \mathbb{R}^{d_1},\Omega_2\subset
%\mathbb{R}^{d_2}$ 
%be two open sets, $A\subset \bbR^n$ a compact subset 
%and 
%$\Gamma$ a closed cone in $\dotT^*\Omega_2$. 
%Let $f:\Omega_1\times A \to \Omega_2$ be a smooth map,
%then
%the claims
%\begin{enumerate}
%\item $\forall a\in A$, $f(.,a)^*\Gamma$ does not meet $\underline{0}$,
%\item $\left(\cup_{a\in A} N_{f(.,a)}\right)\cap \Gamma=\emptyset$,
%\item $\bigcup_{a\in A}f(.,a)^*\Gamma$
%is a closed conic set and does not meet $\underline{0}$ in $T^*\Omega_1$
%\end{enumerate}
%are equivalent.
%\end{lem}
%\begin{proof}
%By Lemma
%(\ref{lempullbackconicset})
%we have 
%\begin{eqnarray*}
%\forall a\in A, f(.,a)^*\Gamma\cap \underline{0}=\emptyset &\Leftrightarrow &\left(\bigcup_{a\in A}f(.,a)^*\Gamma\right)\cap \underline{0}=\emptyset\\ 
%&\Leftrightarrow &\left(\bigcup_{a\in A}N_{f(.,a)}\right) \cap\Gamma=\emptyset
%\end{eqnarray*} 
%which is equivalent to
%\begin{eqnarray*}
%\forall K\subset\Omega_1\text{ compact}, \forall a\in A, 
%\forall (f(x,a);\eta)\in\Gamma\cap T_{f(K\times A)}\Omega_2,\\
%\vert \eta\circ d_xf(x,a)\vert>0.
%\end{eqnarray*}
%Actually, we need a more
%quantitative version 
%of the above
%condition:
%for all compact $K\subset \Omega_1$ there exists $\varepsilon>0$
%such that 
%\begin{eqnarray}\label{conditionavecestime}
%\forall a\in A,\forall (f(x,a);\eta)\in \Gamma\cap T^*_{f(K\times A)}\Omega_2, 
%\vert\eta\circ d_xf\vert(x,a)>\varepsilon\vert\eta\vert .
%\end{eqnarray}
%Let us prove that we can find a uniform bound by using 
%compactness of $K,A$ and continuity of $f$. Assume the above
%condition \ref{conditionavecestime}
%does not hold,  we denote by $U\Gamma$
%the intersection of $\Gamma$
%with the unit cosphere bundle 
%$UT^*\Omega_2$:
%\begin{eqnarray*}
%\exists K\subset\Omega_1, \exists (x_n,a_n)\in K\times A,\\
%\exists (f(x_n,a_n);\eta_n)\in U\Gamma\cap T_{f(K\times A)}\Omega_2,\lim_n\vert \eta_n\circ d_xf(x_n,a_n)\vert=0.
%\end{eqnarray*}
%We extract convergent subsequences
%$x_n\rightarrow x,a_n\rightarrow a, \eta_n\rightarrow \eta$
%then $(f(x,a);\eta)\in  U\Gamma$ since $f$ continuous and
%$ U\Gamma$ is closed. However $\vert \eta\circ d_xf(x,a)\vert=0$
%by continuity of $(\eta,x,a)\longmapsto \eta\circ d_xf(x,a)$, we
%reach a contradiction.
%
%Now let us prove that condition $\ref{conditionavecestime} \implies \bigcup_{a\in A}f(.,a)^*\Gamma$ is closed.
%Let $(x;\xi)\in \overline{\bigcup_{a\in A}f(.,a)^*\Gamma},\xi\neq 0$ 
%then there exists a sequence $(x_n;\xi_n)\in \bigcup_{a\in A}f(.,a)^*\Gamma$ s.t.
%$(x_n;\xi_n)\rightarrow (x;\xi)$
%where $\xi_n=\eta_n\circ d_xf(x_n,a_n)$
%for some sequence $(x_n,a_n,\eta_n)$ 
%such that $(f(x_n,a_n);\eta_n)\in\Gamma$
%by definition of $\bigcup_{a\in A}f(.,a)^*\Gamma$.
%We set $K=\overline{\{x_n | n\in\mathbb{N}\}}$ 
%which is a compact set and condition
%\ref{conditionavecestime} 
%implies that we have the inequality
%$\vert\eta_n\circ d_xf(x_n,a_n)\vert\geqslant\varepsilon \vert\eta_n\vert $,
%but we also
%have 
%$\vert\eta_n\circ d_xf(x_n,a_n)\vert=\vert\xi_n\vert\rightarrow \vert\xi\vert$
%which implies that
%the sequence $(\vert\eta_n\vert)_n$
%remains bounded. 
%By extracting convergent
%subsequences of
%$(x_n,a_n,\xi_n,\eta_n)$,
%we may assume that
%$\eta_n\rightarrow \eta,a_n\rightarrow a$.
%Finally,
%we have $(f(x,a);\eta)\in\Gamma$ since $f$ is continuous and $\Gamma$ 
%is closed thus
%$(x;\eta\circ d_xf(x,a))=(x;\xi)\in f(.,a)^*\Gamma $
%by definition
%and $(x;\xi)\in \cup_{a\in A}f(.,a)^*\Gamma $, we just
%proved that $\overline{\cup_{a\in A}f(.,a)^*\Gamma}\subset \cup_{a\in A}f(.,a)^*\Gamma$
%which yields the final result.
%\end{proof}


\section{Product, convolution and push--forward.}
\label{prodpushsect}
H\"ormander noticed that the product of distributions $u$ and $v$ can be
described
as the composition of the tensor product $(u,v)\to u\otimes v$
with the pull-back by the map $f: x\mapsto (x,x)$.
If the wave front sets of $u$ and $v$ are contained in $\Gamma_1$ and
$\Gamma_2$, then
the wave front set of $u\otimes v$ is contained in
$\Gamma_\otimes=\left(\Gamma_1\times \Gamma_2 \right)\cup\left( (\Omega_1\times\{0\})\times
\Gamma_2\right)
 \cup \left( \Gamma_1 \times (\Omega_2\times\{0\})\right)$
and the pull-back is well-defined if
the set $N_f=\{ (x,x;\eta_1,\eta_2)\telque 
(\eta_1+\eta_2)  \circ  dx)=0 \}$, 
which is the conormal bundle 
of the diagonal 
$\Delta\subset \mathbb{R}^n\times\mathbb{R}^n$
as in example \ref{embedex},
does not meet $\Gamma$, i.e. if there is no point $(x;\eta)$ in
$\Gamma_1$
such that $(x;-\eta)$ is in $\Gamma_2$.
Therefore, the multiplication of distributions is
hypocontinuous because it is the composition of
a hypocontinuous map by a continuous map
(see Lemma~\ref{hypocontcont}).

\begin{thm}
\label{hypocontprod}
Let $\Omega\subset \bbR^n$ be an open set
and $\Gamma_1,\Gamma_2$ be two closed cones in
$\dotT^*\Omega$ such that $\Gamma_1\cap \Gamma_2'=\emptyset$.
Then the product of distributions
is hypocontinuous for the normal topology from
$\calD'_{\Gamma_1}\times\calD'_{\Gamma_2}$
to $\calD'_\Gamma$, where
\begin{eqnarray}
\Gamma &= &
\big(\Gamma_1 +_\Omega \Gamma_2\big) \cup
\big((\Omega\times\{0\})+_\Omega \Gamma_2\big)
\cup \big(\Gamma_1+_\Omega (\Omega\times\{0\})\big).
\label{Gammaprod}
\end{eqnarray}
\end{thm}
\begin{proof}
The product of distribution is the composition of the hypocontinuous
tensor product
with the continuous pull-back.
\end{proof}


This gives us the useful corollary:
\begin{cor}\label{prodsmooth}
Let $\Omega\subset \bbR^n$ be an open set
and $\Gamma$ be a closed cone in
$\dotT^*\Omega$.
Then the product of a smooth map and a distribution
is hypocontinuous for the normal topology from
$C^\infty(\Omega)\times \calD'_\Gamma$ to $\calD'_\Gamma$.
\end{cor}
\begin{proof}
We prove in section~\ref{topequiCinfty} that 
$C^\infty(\Omega)$ and $\calD'_\emptyset$ are
topologically isomorphic. Therefore, the 
corollary follows by applying 
Theorem~\ref{hypocontprod} to $\Gamma_1=\emptyset$
and $\Gamma_2=\Gamma$. 
Equation \eqref{Gammaprod} shows that the wave front
set of the product is in $\Gamma$.
\end{proof}


\subsection{The push--forward as a consequence of the pull--back theorem.}
\begin{thm}\label{pushforwardthm}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ and $\Omega_2\subset \mathbb{R}^{d_2}$
be two open sets and $\Gamma$ a closed cone
in $\dotT^*\Omega_1$. For any smooth map $f:\Omega_1
\to \Omega_2$ and any closed subset 
$C$ of $\Omega_1$ such that 
$f|_C: C\to \Omega_2$ is proper and $ \underline{\pi}(\Gamma)\subset C$,
then $f_*$ is continuous in the normal topology 
from
$\{u\in \calD'_\Gamma\telque \supp u \subset 
C\}$ to $\calD'_{f_*{\Gamma}}$,
where 
$f_*\Gamma=\{(y;\eta)\in \dotT^*\Omega_2\telque
\exists x\in \Omega_1\text{ with } y=f(x)\text{ and }
(x;\eta\circ df_x)\in \Gamma \cup \supp u\times \{0\}\}$.
\end{thm}
\begin{proof}
The idea of the proof is to think of a push--forward as the adjoint of a pull--back
(see \cite{Duistermaat}).
For all $B$ equicontinuous
in $\mathcal{E}^\prime_{\Lambda}$
where $\Lambda=f_*\Gamma^{\prime,c}$,
$\forall v\in B, \text{supp }\left(f^* v\right)\cap C$
is contained in a fixed compact set $K$ 
since $f$ is proper on $C$. 
Hence, for any $\chi\in\mathcal{D}(\mathbb{R}^d)$ such that $\chi|_C=1$
and $f$ is proper on $\text{supp }\chi$, 
we should have at least formally
$\left\langle f_*u, v \right\rangle=\left\langle  u,\chi f^* v \right\rangle$
if the duality couplings make sense.
On the one hand, 
if $v\in\mathcal{E}^\prime_\Lambda(\Omega_2)$
where
$f^*\Lambda$ 
does not meet the zero section 
$\underline{0}\subset T^*\Omega_1$
then
the pull--back $f^*v$ 
would be well defined by the pull--back 
Proposition 
(\ref{pullbackprop})
(which is equivalent to the fact that 
$N_f\cap \Lambda=\emptyset$).
On the other hand, the duality pairing 
$\left\langle u , \chi(f^* v) \right\rangle$ 
is well defined if 
$f^*\Lambda\cap \Gamma^\prime=\emptyset$. 
Combining
both conditions leads to the
requirement that 
$f^*\Lambda\cap \left(\Gamma^\prime\cup \underline{0}\right)=\emptyset$.
But 
note that:
\begin{eqnarray*}
&&\left(f^*\Lambda\right)\cap 
\left(\Gamma^\prime\cup \underline{0}\right)=\emptyset \\
&\Leftrightarrow & 
\{(x;\eta\circ df) | (f(x);\eta)\in\Lambda,(x;\eta\circ df)\in \Gamma^\prime\cup\{0\}\}=\emptyset\\
&\Leftrightarrow &
 (f(x);\eta)\in\Lambda\implies (x;\eta\circ df)\notin \Gamma^\prime\cup\{0\}\\
 &\Leftrightarrow &
(f(x);\eta)\in\Lambda^\prime\implies (x;\eta\circ df)\notin \Gamma\cup\{0\}.
\end{eqnarray*}
which
is equivalent to the fact that $\Lambda^\prime$ 
does not meet
$f_*\Gamma=\{(f(x);\eta)\telque (x;\eta\circ df) 
\in \left(\Gamma\cup \underline{0}\right),\eta\neq 0 \}
\subset T^*\Omega_2$
which is exactly the
assumption of our theorem.
Therefore, the set of distributions
$\chi f^* B$ is supported in a fixed compact
set, 
bounded in 
$\mathcal{D}^\prime_{f^*\Lambda}(\Omega_1)$
by the pull--back Proposition \ref{pullbackprop} 
applied to $f^*$,
the formal duality couplings are well defined and:
$$\sup_{v\in B}\vert \left\langle f_* u, v \right\rangle\vert=\sup_{v\in B^\prime}\vert \left\langle u, v \right\rangle\vert $$
where $B^\prime=\chi f^* B$ is equicontinuous in $\mathcal{E}^\prime_{\Gamma^{\prime,c}}(\Omega_1)$ (the support of the distribution is compact because $f$
is proper) which means
that $\sup_{v\in B^\prime}\vert \left\langle u, v \right\rangle\vert $ is 
a continuous seminorm
for 
the normal topology of 
$\mathcal{D}_\Gamma^\prime(\Omega_1)$.
\end{proof}
We state and prove
a parameter version
of the push--forward theorem
\begin{thm}\label{pshfwdenfamille}
Let $\Omega_1\subset \mathbb{R}^{d_1}$ 
and $\Omega_2\subset \mathbb{R}^{d_2}$
be two open sets, $A\subset U\subset \mathbb{R}^n$ 
where $A$
is
compact, $U$ is open and $\Gamma$ a closed cone
in $\dotT^*\Omega_1$. For any smooth map $f:\Omega_1\times U
\to \Omega_2$ and any closed subset 
$C$ of $\Omega_1$ such that 
$f: C\times A\to \Omega_2$ is proper 
and $\underline{\pi}(\Gamma)\subset C$,
then $f(.,a)_{*}$ is uniformly continuous  in the normal topology 
from
$\{u\in \calD'_\Gamma\telque \supp u \subset 
C\}$ to $\calD'_{\Xi}$,
where $\Xi=\cup_{a\in A }f(.,a)_{*}\Gamma$.
\end{thm} 
\begin{proof}
We have to check that 
$\Xi$ is closed over 
$\dot{T}^*_{f(C\times A)}\Omega_2$.
Let $(y;\eta)\in \overline{\Xi}
\cap \dot{T}^*_{f(C\times A)}\Omega_2$
then there exists a 
sequence $(y_n;\eta_n)\rightarrow (y;\eta)$
such that 
$(y_n;\eta_n)\in\Xi\cap 
\dot{T}^*_{f(C\times A)}\Omega_2$.
By definition,
$y_n=f(x_n,a_n)$
where 
$(x_n,\eta_n\circ d_x f(x_n,a_n))
\in\Gamma\cup \underline{0},
(x_n,a_n)\in C\times A$.
The central observation is that
$\overline{\{y_n| n\in\mathbb{N} \}}
 \subset f(C\times A)  $
and 
$\overline{\{(x_n,a_n)| f(x_n,a_n)=y_n,n\in\mathbb{N} \}}
\subset C\times A$
are compact sets because
$f$ is proper on $C\times A$.
Then we can extract 
convergent subsequences
$(x_n,a_n)\rightarrow (x,a)$
and $(x,\eta\circ d_x f(x,a))
\in\Gamma\cup \underline{0}$
since $\Gamma\cup \underline{0}$
is closed in $\dot{T}^*\Omega_2$
and $d_xf$
is continuous. 
By
definition of $\Xi$, this proves
that $(y;\eta)\in \Xi$
and we can 
conclude that
$\Xi$ is closed.
Now we can repeat 
the proof of the 
push--forward theorem
except
that we use 
the pull--back 
theorem
with parameters.
Let $B$ be 
equicontinuous
in $\mathcal{E}_{\Xi^{\prime,c}}^\prime(\Omega_2)$
hence all elements of $B$ have
support contained in
some compact.
We have
$\forall v\in B, \text{supp }(f^*v)\cap C\times A$
is compact, therefore
for $\chi=1$ on $\cup_{a\in A} \text{supp }(f(.,a)^*v)\cap C$,
the family $B^\prime=\{(\chi f(.,a)^*v)| a\in A,v\in B \}$
is equicontinuous in 
$\mathcal{E}^\prime_{\Theta}$
where $\Theta=\Gamma^{\prime,c}$
by the
parameter version of
the pull--back theorem.
Therefore,
$u\longmapsto \sup_{a\in A} 
\sup_{v\in B}\vert \left\langle f(.,a)_{*} u, v \right\rangle\vert=\sup_{v\in B^\prime}\vert \left\langle u, v \right\rangle\vert$
is continuous in $u$
since the right hand term
is a 
continuous seminorm
for the normal
topology of
$\mathcal{D}^\prime_{\Gamma}(\Omega_1)$. 
\end{proof}
\paragraph{Convolution product.}
In the same spirit as for the multiplication
of distributions, Duistermaat--Kolk~\cite[p.~121]{Duistermaat-10}
describe the convolution product $u*v$ as the
composition of the
tensor product $(u,v)\longmapsto u\otimes v$
with the push--forward by the
map $\Sigma:=(x,y) \longmapsto (x+y)$. 
For a closed subset $X\subset \mathbb{R}^n$,
let $\mathcal{D}_\Gamma^\prime(X)$ be the set
of distributions supported in $X$ with wave front
in $\Gamma$.
Therefore, we have the 
\begin{thm}\label{convolprod}
Let  
$\Gamma_1,\Gamma_2$ be
two closed conic sets in $\dot{T}^*\mathbb{R}^n$ and 
$X_1,X_2$
two closed subsets of $\mathbb{R}^n$
such that $\Sigma:X_1\times X_2 \longmapsto \mathbb{R}^n$
is proper.
Then the convolution product
of distributions is hypocontinuous
from\\ $\mathcal{D}^\prime_{\Gamma_1}(X_1)\times\mathcal{D}^\prime_{\Gamma_2}(X_2)$
to $\mathcal{D}^\prime_\Gamma(X_1+X_2)$
where
\begin{eqnarray}
\Gamma=\{(x+y;\eta) \telque (x;\eta)\in\Gamma_1,(y;\eta)\in\Gamma_2 \}.
\end{eqnarray}
\end{thm}
\begin{proof}
The convolution product of distribution is the composition of the hypocontinuous
tensor product with the continuous push--forward.
\end{proof}

\section{Coordinate invariant definition of the 
wave front set.}
\label{Gaborsect}

Duistermaat~\cite[p.~13]{Duistermaat} proposed a coordinate
invariant definition of the wave front set that corrects
a first attempt by Gabor~\cite{Gabor-72}.

\begin{defi}\label{duistermaatWF}
Let $\Omega\subset\mathbb{R}^d$ be an open set, 
$u\in\mathcal{D}^\prime(\Omega)$.
An element $(x;\xi)\notin WF_{D}(u)$ iff
for all $f\in C^\infty(\Omega\times \mathbb{R}^n)$ 
such that
$d_xf(x,a_0)=\xi$ for some
$a_0\in \mathbb{R}^n$, there exists 
some neighborhoods
$A$ of $a_0$ and $U$ of $x_0$
such that for all
$\varphi\in\mathcal{D}(U)$:
$\vert\left\langle u,\varphi e^{i\tau f(.,a)} \right\rangle\vert
=O(\tau^{-\infty})$
uniformly in some 
neighborhood
of $a_0$ in $A$.
\end{defi}

The definition \ref{duistermaatWF} looks
a priori stronger than the property
of being in the complementary
of $WF(u)$. It is however
equivalent. We give here
an alternative proof.
\begin{thm}
$WF_{D}(u)=WF(u)$.
\end{thm}
\begin{proof}
The inclusion $WF(u)\subset WF_D(u)$
is easy. If $(x;\xi)\notin WF_D(u)$, by applying
Duistermaat's definition
to the Fourier phase
$f(x,a)=\left\langle x,a \right\rangle,$
$a\in\mathbb{R}^{d}$,
$a_0=\xi$, $A$ is a conic neighborhood
of $a_0=\xi$ and $U$ is some neighborhood
of $x$ such that $(U\times A)\cap WF_D(u)=\emptyset$,
we find that $(x;\xi)\notin WF(u)$.

The converse follows from
the Proposition \ref{propduist}
below.
\end{proof}
For any smooth map $f\in C(\mathbb{R}^d,\mathbb{R})$, 
we denote by $\mathbf{Gr}d_xf$ the graph of $d_xf$ in 
$T^*\mathbb{R}^d$.
\begin{prop}\label{propduist}
Let $\Omega$ be an open set in $\mathbb{R}^d$, 
$A\subset U \subset \mathbb{R}^n$ where $A$ is compact and $U$ is open, 
$\Gamma\subset T^\bullet\mathbb{R}^d$
a closed conic set. 
For all $f\in C^\infty(\Omega\times U,\mathbb{R})$
such that 
$\underset{(x,a)\in \Omega\times A}{\inf}\vert d_xf(\cdot,a)\vert> 0$
and  
$\left(\bigcup_{a\in A}\mathbf{Gr}d_xf(.,a)\right) \cap\Gamma=\emptyset$: 
$$ \exists\varphi\in\mathcal{D}(\Omega), \forall u\in\mathcal{D}^\prime_\Gamma(\Omega), \vert\left\langle u,e^{i\tau f(.,a)}\varphi \right\rangle\vert=O(\tau^{-\infty}) $$
uniformly in $a\in A$. 
\end{prop}

\begin{proof}
The idea of the proof comes 
from the following observation:
given a function $f\in C^\infty(\Omega)$ such that 
$df|_\Omega\neq 0$, i.e. $f\in C^\infty(\Omega,\mathbb{R})$ 
is a submersion on $\Omega$, 
then
the concept of 
wave front set 
can be interpreted
geometrically as 
the study of the 
regularity of the distribution 
$u\in\mathcal{D}^\prime(\Omega)$ 
by \textbf{averaging 
on the level surfaces} of $f$.
In mathematically precise terms,
this will be 
given by the push--forward operation
$f_*$.
For all
$\varphi\in \mathcal{D}(\Omega)$,
the push--forward 
$f_* \left(u\varphi\right)\in\mathcal{D}^\prime(\mathbb{R})$
is well defined
and is the \textbf{Radon transform}
of $u\varphi$ on the level surfaces 
of $f$ (see
\cite{WF1} for 
the relationship of 
Radon transform
with the wave front set). 
By definition of the Fourier transform 
$$\widehat{f_*(u\varphi)}(\tau)=\int_{\mathbb{R}} dt \left(f_*(u\varphi)\right)(t)e^{it\tau}$$
then
$$\widehat{f_*(u\varphi)}(\tau)=\left\langle u\varphi, f^*\left(e^{it\tau}\right) \right\rangle =\left\langle u\varphi, e^{i\tau f} \right\rangle $$ 
by definition of the pull--back as the adjoint 
of the push--forward. 

If $WF(u)$ does not meet the graph
of $df$ 
in
$T^*\Omega$ then by
the push--forward theorem
$WF(f_*(u\varphi))\subset \{ (f(x);\tau) \telque  (x;\tau df) \in WF(u)\cup \underline{0},\tau\neq 0   \} $
but since $df\neq 0$ on $\Omega$, 
we have the better
estimate
$WF(f_*(u\varphi))\subset \{ (f(x);\tau) | (x;\tau df) \in WF(u)\cap (\dot{T}^* \Omega)\}
=\emptyset$ since $WF(u)\cap \mathbf{Gr}df=\emptyset$.
Therefore
$WF(f_*(u\varphi))$ is empty and since
$f_*(u\varphi)$ is compactly supported it is a smooth compactly supported
function and its Fourier transform $\widehat{f_*(u\varphi)}(\tau)$
has fast decrease
when $\tau\rightarrow +\infty$.
To be completely 
faithful to the idea of
Duistermaat, $f$ should depend on some auxiliary
parameter $a$ which lives in
a \textbf{compact space} $A\subset U$, i.e. $f\in C^\infty(\Omega\times U,\mathbb{R})$.
Then our discussion is exactly the same except
we should apply the parameter version of the
push--forward theorem
\begin{eqnarray*}
\widehat{f(.,a)_*(u\varphi)}(\tau) = \left\langle u\varphi, e^{i\tau f(.,a)} \right\rangle,
\end{eqnarray*}
if $\forall a\in A, \mathbf{Gr}d_xf(.,a) \cap WF(u\varphi)=\emptyset$
then the parameter version of the
push--forward theorem implies
that the family
$f(.,a)_*(u\varphi)_{a\in A}$
is bounded in 
$\mathcal{D}^\prime_\emptyset(\mathbb{R})
=C^\infty(\mathbb{R})$
uniformly in $a\in A$.
\end{proof}



\section{Acknowledgements}
We are very grateful to Yoann Dabrowski for his generous help
in the functional analysis parts of the paper.
We thank Camille Laurent-Gengoux for discussions
about the geometrical aspects of the pull-back.

\section{Appendix: Technical results}
This appendix gather different useful results.
Several of them are folkore results for which we could
find no proof in the literature.

\subsection{Exhaustion of the complement of $\Gamma$}
\label{exhaustionsubsec}
To prove that $\mathcal{D}^\prime_\Gamma$
is nuclear, we need the fact that the additional seminorms
can be taken in a countable set.
Therefore, we need to prove 
that
the complement $\Gamma^c=\dotT^*M\backslash\Gamma$
of any closed cone $\Gamma\subset \dotT^*M$
can be exhausted by a countable set of products
$U\times V$, where $U\subset M$ and $V$ is a conic subset
of $\bbR^n$.

First we introduce the sphere bundle (or unit cotangent bundle)
$UT^*M=\{(x;k)\in T^*M \telque |k|=1\}$,
a vector bundle over $M$.
We then define the set 
$U\Gamma_K=\{(x;k)\in \Gamma\telque x\in K, |k|=1\} = UT^*M|_K\cap \Gamma$, for any compact
subset $K\subset M$. Since $M$ can be covered by a countable
union of compact sets, we assume without loss of generality
that $K$ is covered by a single
chart $(U,\psi)$ such that $\psi(K)\subset Q$, 
where $Q=[-1,1]^n$ is an $n$-dimensional
cube. We hence can assume w.l.g. that $M=\mathbb{R}^n$ and $UT^*M = \mathbb{R}^n\times S^{n-1}$.
It will be convenient to use the norm $d_\infty$ on $\mathbb{R}^n$, defined by: $\forall x,y\in \mathbb{R}^n$, $d_\infty(x,y):=
\sup_{1\leq i\leq n}|x_i-y_i|$. We denote
by $\overline{B}_\infty(x,r) =\{y\in \mathbb{R}^n\telque d_\infty(x,y)\leq r\}$ the closed
ball of radius $r$ for this norm.
We also denote the restriction of $d_\infty$ to $S^{n-1}\times S^{n-1}$ by the same letter and,
lastly, for $(x;\xi),(y;\eta)\in UT^*\mathbb{R}^n$ we set 
$d_\infty((x;\xi),(y;\eta)) = \sup (d_\infty(x,y),d_\infty(\xi,\eta))$. 


We define cubes centered at rational points in $Q$:
let $q_{j}=[-\frac{1}{2^j},\frac{1}{2^j}]^n= \overline{B}_\infty(0,2^{-j})$ and 
$q_{j,m}=\frac{m}{2^j}+q_j = \overline{B}_\infty(2^{-j}m,2^{-j})$, where $m\in \mathbb{Z}^n \cap 2^j Q$.
In other words, the center of $q_{j,m}$ runs over a hypercubic lattice with
coordinates $(2^{-j}m_1,\dots,2^{-j}m_n)$, where
$-2^j \le m_i \le 2^j$. Note that, for each non-negative integer $j$,
the hypercubes $q_{j,m}$ overlap and cover $Q$:
\begin{eqnarray}
Q &\subset & \bigcup_{m\in \bbZ^n\cap 2^jQ} q_{jm}.
\label{QUqjm}
\end{eqnarray}



Denote by $\underline{\pi}:UT^*\mathbb{R}^n\longrightarrow \mathbb{R}^n$
and $\overline{\pi}:UT^*\mathbb{R}^n\longrightarrow S^{n-1}$
the projection maps defined by $\underline{\pi}(x;k)=x$ and $\overline{\pi}(x;k) = k$.
We define $F_{j,m}=\underline{\pi}^{-1}(q_{j,m})\simeq q_{j,m}\times S^{n-1}$
(see fig.~\ref{figexhaustion}).
The set $\overline{\pi}(F_{j,m}\cap U\Gamma_K)$ is compact because the projection
$\overline{\pi}$ is continuous and $F_{j,m}\cap U\Gamma_K$ is compact.
For any positive integer $\ell$, define the compact set
\begin{eqnarray*}
C_{j,m,\ell} &=& \big\{\eta\in S^{n-1}\telque 
d_\infty\big(\overline{\pi}(F_{j,m}\cap U\Gamma_K),\eta\big) \ge 1/\ell\big\}.
\end{eqnarray*}
This is the set of points of the sphere which are at least
at a distance $1/\ell$ from the projection of
the slice of $U\Gamma$ inside 
$F_{j,m}$ (see fig.~\ref{figexhaustion}).
We have
\begin{eqnarray*}
\bigcup_{\ell>0} C_{j,m,\ell} &=& S^{n-1}\backslash \overline{\pi}(F_{j,m}\cap
U\Gamma_K).
\end{eqnarray*}
Indeed, by definition, any element of $C_{j,m,\ell}$ is
in $S^{n-1}$ and not in $\overline{\pi}(F_{j,m}\cap U\Gamma_K)$,
conversely, the compactness of $\overline{\pi}(F_{j,m}\cap U\Gamma_K)$ implies
that any point $(x;\xi)$ in $S^{n-1}\backslash \overline{\pi}(F_{j,m}\cap U\Gamma_K)$ 
is at a finite distance $\delta$ from 
$\overline{\pi}(F_{j,m}\cap U\Gamma_K)$. If we take $\ell > 1/\delta$, 
we have $(x;\xi)\in C_{j,m,\ell}$.
Note that all $C_{j,m,\ell}$ are empty if
$\overline{\pi}(F_{j,m}\cap U\Gamma_K) = S^{n-1}$.

With this notation we can now state
\begin{lem}\label{approxlemma1}
\begin{eqnarray}\label{identity-qK=UmoinsGamma}
\bigcup_{j,m,\ell} q_{j,m}\times C_{j,m,\ell} &=& 
UT^*M|_K\backslash U\Gamma_K,\
\end{eqnarray}
and, by denoting 
$V_{j,m,\ell} = \{(x;k)\in \dotT^*M\telque 
(x;k/|k|)\in C_{j,m,\ell}\}$,
\begin{eqnarray}\label{identity-qV=TmoinsGamma}
\bigcup_{j,m,\ell} q_{j,m}\times V_{j,m,\ell} &=& 
(T^*M\backslash\Gamma)|_K.
\end{eqnarray}
\end{lem}
\begin{figure}
\begin{center}
\includegraphics[width=10.0cm]{dessin-exhaustion.eps}
\caption{Picture of the case $n=2$, where only one dimension
of the cube $[-1,1]^2$ is shown and the circle
$S^1$ is represented by the vertical segment $[0,1]$.
The green surface is $U\Gamma_K$, the pink square is
the ball $\overline{B}_\infty(p,\delta)$. 
We see that $q_{j,m}$ contains $x$ and is contained
in $\underline{\pi}(\overline{B}_\infty(p,\delta))$.
  \label{figexhaustion}}
\end{center}
\end{figure}
\begin{proof}
We first prove the inclusion $\subset$ in (\ref{identity-qK=UmoinsGamma}). Let 
$(x;k)\in \bigcup_{j,m,\ell} q_{j,m}\times C_{j,m,\ell}$, this means that there exist
$j\in \mathbb{N}$, $m\in \mathbb{Z}^n\cap 2^jQ$ and $\ell\in \mathbb{N}^*$
such that $(x;k)\in q_{j,m}\times C_{j,m,\ell}$. Hence $(x;k)\in F_{j,m}$ and,
by definition of $C_{j,m,\ell}$, $d_\infty(\overline{\pi}(F_{j,m}\cap U\Gamma_K),k)\geq 1/\ell$, which
implies that $(x;k)\notin U\Gamma_K$.

Let us prove the reverse inclusion $\supset$. Let $(x;k)\in UT^*M|_K\setminus U\Gamma_K$. Since this set
is open, there exists some $\delta>0$ such that $\overline{B}_\infty((x;k),\delta)\subset UT^*M|_K\setminus U\Gamma_K$.
Let $j\in \mathbb{N}^*$ s.t. $2^{-j+1}<\delta$.
Because the sets $q_{jm}$ cover $Q$
(see eq.~\eqref{QUqjm}), there 
is an $m$ such that $x\in q_{jm}$.
Moreover, $\forall y\in q_{j,m}$, 
we have $d_\infty(x,y) \leq d_\infty(x,2^{-j}m) + d_\infty(2^{-j}m,y)\leq 2^{-j}+ 2^{-j}<\delta$,
i.e. $y\in \overline{B}_\infty(x,\delta)$. Hence
 $q_{j,m}\subset \overline{B}_\infty(x,\delta)$.
We deduce that 
\[
q_{j,m}\times \overline{B}_\infty(k,\delta)\subset \overline{B}_\infty(x,\delta)\times \overline{B}_\infty(k,\delta)
= \overline{B}_\infty((x;k),\delta)\subset UT^*M|_K\setminus U\Gamma_K.
\]
This means that $(q_{j,m}\times \overline{B}_\infty(k,\delta)) \cap U\Gamma_K = \emptyset$ or equivalently
$F_{j,m}\cap \overline{\pi}^{-1}(\overline{B}_\infty(k,\delta))\cap U\Gamma_K = \emptyset$. The latter
inclusion implies that $\overline{B}_\infty(k,\delta)\cap \overline{\pi}(F_{j,m}\cap U\Gamma_K) = \emptyset$.
In other words, $d_\infty(k,\overline{\pi}(F_{j,m}\cap U\Gamma_K)) >\delta$. Hence by choosing $\ell\in \mathbb{N}^*$
s.t. $1/\ell \leq \delta$, we deduce that $d_\infty(k,\overline{\pi}(F_{j,m}\cap U\Gamma_K) > 1/\ell$, i.e.
that $k\in C_{j,m,\ell}$. Thus we conclude that $(x;k)\in q_{j,m}\times
C_{j,m,\ell}$ and Eq.~\eqref{identity-qK=UmoinsGamma} is proved. 

To prove (\ref{identity-qV=TmoinsGamma}), we notice that,
because of the conic property of $\Gamma$, each $C_{j,m,\ell}$
corresponds to a unique $V_{j,m,\ell}$. 
\end{proof}

Any nonnegative smooth function
$\psi$ supported on $[-3/2,3/2]^n$ and such that
$\psi(x)=1$ for $x\in [-1,1]^n$ enables us to
define scaled and shifted functions
$\psi_{j-1,m}(x)=\psi(2^j(x-m))$ supported on $q_{j-1,m}$
and equal to 1 on $q_{j,2m}$.


If $C_{j,m,\ell}$ is not empty,
we denote by $\alpha_{j,m,\ell}:S^{n-1}\to\bbR$ a smooth
function supported on $C_{j,m,\ell}$ and equal to 1
on $C_{j,m,\ell+1}$.  Note that, if 
$C_{j,m,\ell}$ is a proper subset of $S^{n-1}$, then
it is strictly included in $C_{j,m,\ell+1}$.

\subsection{Equivalence of topologies}
Grigis and Sj\"ostrand stated~\cite[p.~80]{Grigis} that
if we have a family $\chi_\alpha$ of test functions and closed
cones $V_\alpha$ such that
$(\supp \chi_\alpha\times V_\alpha)\cap\Gamma=\emptyset$
and $\cup_\alpha \{(x,k)\telque \chi_\alpha(x)\not=0
\text{ and } k\in \mathring{V}_\alpha\} =\Gamma^c$, then the topology
of $\calD'_\Gamma$ is the topology given by the
seminorms of the weak topology and the seminorms
$||\cdot||_{N,V_\alpha,\chi_\alpha}$. 
By covering $M$ with a countable
family
of compact sets $K_i$ described 
in section 
\ref{exhaustionsubsec}, we see that
Lemma \ref{approxlemma1}
gives us a family
of indices $\alpha=(i,j,\ell)$, functions
$\chi_{j,m,\ell}=\psi_{j,m}$ and
cones $V_{j,m,\ell}$ adapted to $K_i$
such that the conditions
of the Grigis-Sj\"ostrand
lemma are satisfied. Therefore,
the normal topology
is described by the
seminorms 
of the strong topology
of $\mathcal{D}^\prime(\Omega)$
and by the countable family
$(i,j,m,\ell)$ of seminorms.

\subsection{Topological equivalence $C^\infty(X)$ 
and $\calD'_\emptyset$}
\label{topequiCinfty}

As a first application of the previous lemma, we show
\begin{lem}
The spaces $C^\infty(X)$ and
$\calD'_\emptyset$ are topologically isomorphic.
\end{lem}
\begin{proof}
This property was also stated (without proof) by
Alesker~\cite{Alesker-10}.
The two spaces are identical as vector spaces because
a distribution $u$ whose wave front set is empty
is smooth everywhere, since its singular
support $\mathrm{sing\,supp}(u)=\underline{\pi}(\WF(u))$~\cite[p.~254]{HormanderI}
is empty~\cite[p.~42]{HormanderI}, and 
a distribution is a smooth function if and only if
its singular support is empty.

To prove the topological equivalence, we must show that
the two inclusions 
$\calD'_\emptyset\hookrightarrow C^\infty(X)$
and
$C^\infty(X)\hookrightarrow \calD'_\emptyset$
are continuous.
Recall that a system of semi-norms defining the
topology of $C^\infty(X)$ is
$\pi_{m,K}$, where $m$ runs over the integers and
$K$ runs over the compact
subsets of $X$~\cite[p.~88]{Schwartz-66}.
By a straightforward estimate, we 
obtain:
\begin{eqnarray*}
\pi_{m,K}(\varphi) & \le & C_{n} (2\pi)^{-n} 
\sum_{|\alpha|\le m} 
\sup_{k\in \bbR^n}
(1+ \vert k\vert^2)^p |k^\alpha \widehat{\varphi\chi}(k)|\\
&\le & C_{n} (2\pi)^{-n} 
\binom{m+n}{n}
||\varphi||_{m+2p,\bbR^n,\chi},
\end{eqnarray*}
where $\chi\in \calD(X)$ is equal to one on a compact
set whose interior contains $K$ where 
we used $(1+\vert k\vert^2)\le (1+|k|)^2$,
$|k^\alpha|\le (1+|k|)^m$ and
$\sum_{|\alpha|\le m} = \binom{m+n}{n}$.
Thus, every seminorm of $C^\infty(X)$ is bounded
by a seminorm of $\calD'_\emptyset$ and the injection
$\calD'_\emptyset\hookrightarrow C^\infty(X)$ is continuous.

Conversely, for any closed conic set $V$ and any
$\chi\in \calD(X)$ we have 
$||\varphi||_{N,V,\chi}\le ||\varphi||_{N,\bbR^n,\chi}$.
Thus, it is enough to estimate
$||\varphi||_{N,\bbR^n,\chi}$.
We also find that, for any 
integer $N$ and $\alpha=0$,
\begin{eqnarray*}
\sup_{k\in\bbR^n}
(1+k^2)^N |\widehat{\varphi\chi}(k)|
\le |K|2^N \pi_{2N,K}(\varphi\chi),
\end{eqnarray*}
where $K$ is the support of $\varphi$.
Then, the relation 
$1+|k|\le 2(1+|k|^2)$ and 
application of the Leibniz rule 
give us
$||\varphi||_{N,\bbR^n,\chi}\le 
|K|8^N \pi_{2N,K}(\chi) \pi_{2N,K}(\varphi)$ and
the seminorms $||\cdot||_{N,V,\chi}$ are indeed
controlled by the seminorms of $C^\infty(X)$.

It remains to show that this also holds
for the seminorms of $\calD'(X)$.
It is well known that the inclusion 
$C^\infty(X)\hookrightarrow \calD'(X)$ is
continuous, but we can prove it for completeness.
The weak topology of $\calD'(X)$ is generated
by the seminorms 
$p_f: u\mapsto |\langle u,f\rangle|$ where
$f$ runs over the test function space $\calD(X)$.
Let $K$ be a compact neighborhood of $\supp f$.
We have, for every $\phi\in C^\infty(X)$,
\begin{eqnarray*}
p_f(\phi) &=& |\langle \phi,f\rangle| 
=\left| \int \phi(x) f(x) dx\right| 
\le \pi_{0,K}(\phi) \pi_{0,K}(f) |K|,
\end{eqnarray*}
where $|K|$ is the volume of $K$.
Thus, the seminorms $p_f$ are also controlled
by those of $C^\infty(X)$. 
We can also consider the strong topology of 
$\calD'(X)$, for which the seminorms are
$p_B(\phi)=\sup_{f\in B}|\langle \phi,f\rangle|$,
where $B$ is a bounded set of $\calD(X)$.
Then, we have again
$P_B(\phi)\le \pi_{0,K}(\phi) M_0 |K|$, where
$K$ is a compact set containing the support of all
functions of $B$ and $M_0$ is the upper
bound of  $\pi_{0,K}(f)$ for all $f\in B$.
Therefore, 
the inclusion 
$C^\infty(X) \hookrightarrow \calD'_\emptyset$
is also continuous. We proved that
$C^\infty(X)$ and $\calD'_\emptyset$ are
topologically isomorphic, where
$\calD'_\emptyset$ can be equipped with the H\"ormander or the
normal topology.
\end{proof}
\def\bysame{\leavevmode ---------\thinspace}
\makeatletter\if@francais\providecommand{\og}{<<~}\providecommand{\fg}{~>>}
\else\gdef\og{``}\gdef\fg{''}\fi\makeatother
\def\cdrandname{\&}
\providecommand\cdrnumero{no.~}
\providecommand{\cdredsname}{eds.}
\providecommand{\cdredname}{ed.}
\providecommand{\cdrchapname}{chap.}
\providecommand{\cdrmastersthesisname}{Memoir}
\providecommand{\cdrphdthesisname}{PhD Thesis}
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\end{document}

1409.7662.extracted_bib
{Alesker-10 {\scshape S.~Alesker, {\og Valuations on manifolds and integral geometry\fg, Geom. Funct. Anal. 20 (2010), p.~1073-143.
1409.7662.extracted_bib
{Bourbaki-50 {\scshape N.~Bourbaki, {\og Sur certains espaces vectoriels topologiques\fg, Ann. Inst. Fourier 2 (1950), p.~5-16.
1409.7662.extracted_bib
{Bourbaki-TVS \bysame , Elements of mathematics. {Topological vector spaces, Springer, Berlin, 2003.
1409.7662.extracted_bib
{Bredon {\scshape G.~Bredon, Topology and geometry, Springer, New York, 1993.
1409.7662.extracted_bib
{WF1 {\scshape C.~Brouder, N.~V. Dang {\normalfont \cdrandname~F.~H\'elein, {\og A smooth introduction to the wavefront set\fg, Journal of Physics A: Mathematical and Theoretical 47, To appear.
1409.7662.extracted_bib
{Chazarain {\scshape J.~Chazarain {\normalfont \cdrandname~A.~Piriou, Introduction {\`a la th{\'eorie des {\'equations aux d{\'eriv{\'ees partielles, Gauthier-Villars, Paris, 1981.
1409.7662.extracted_bib
{Choquet-48 {\scshape G.~Choquet, {\og Convergences\fg, Annales de l'universit\'e de Grenoble 23 (1947-1948), p.~57-112.
1409.7662.extracted_bib
{Concordido-04 {\scshape C.~F.~R. Concordido {\normalfont \cdrandname~D.~P.~Pombo Jr, {\og Hypocontinuous multilinear mappings in topological modules\fg, Bull. Soc. Roy. Sci. Li{\`ege 72 (2004), p.~383-90.
1409.7662.extracted_bib
{Dabrouder-13 {\scshape Y.~Dabrowski {\normalfont \cdrandname~C.~Brouder, {\og Functional properties of {H\"ormander's space of distributions having a specified wavefront set\fg, Commun. Math. Phys. (2014), http://link.springer.com/article/10.1007/s00220-014-2156-0.
1409.7662.extracted_bib
{DabrowskiDang {\scshape Y.~Dabrowski {\normalfont \cdrandname~N.~V. Dang, {\og The extension of distributions on manifolds\fg, (2014), in preparation.
1409.7662.extracted_bib
{Dangthese {\scshape N.~V. Dang, Renormalization of quantum field theory on curved space-times, a causal approach, Ph.D. thesis, Paris Diderot University, 2013, http://arxiv.org/abs/1312.5674.
1409.7662.extracted_bib
{Dolecki {\scshape S.~Dolecki, {\og An initiation into convergence theory\fg, in Beyond Topology (F.~Mynard {\normalfont \cdrandname~E.~Pearl, \cdredsname), Contemp. Math., vol. 486, Amer. Math. Soc., 2009, p.~115-61.
1409.7662.extracted_bib
{Duistermaat {\scshape J.~J. Duistermaat, Fourier integral operators, Birkh{\"{auser, Boston, 1996.
1409.7662.extracted_bib
{Duistermaat-10 {\scshape J.~J. Duistermaat {\normalfont \cdrandname~J.~A.~C. Kolk, Distributions: Theory and Applications, Birkh{\"{auser, New York, 2010.
1409.7662.extracted_bib
{Fichtenholz-38 {\scshape G.~Fichtenholz, {\og Sur les fonctionnelles lin{\'eaires, continues au sens g\'en\'eralis\'e\fg, Recueil Math. 4 (1938), p.~193-214.
1409.7662.extracted_bib
{Friedlander {\scshape F.~G. Friedlander {\normalfont \cdrandname~M.~Joshi, Introduction to the theory of distributions, second \cdredname, Cambridge University Press, Cambridge, 1998.
1409.7662.extracted_bib
{Gabor-72 {\scshape A.~Gabor, {\og Remarks on the wave front of a distribution\fg, Trans. Am. Math. Soc. 170 (1972), p.~239-44.
1409.7662.extracted_bib
{Glockner-09 {\scshape H.~Gl{\"ockner, {\og Applications of hypocontinuous bilinear maps in infinite-dimensional differential calculus\fg, in Generalized Lie Theory in Mathematics, Physics and Beyond (Berlin) (S.~Silvestrov, E.~Paal, V.~Abramov {\normalfont \cdrandname~A.~Stolin, \cdredsname), Springer, 2009, p.~171-86.
1409.7662.extracted_bib
{Grigis {\scshape A.~Grigis {\normalfont \cdrandname~J.~Sj{\"ostrand, Microlocal analysis for differential operators, Cambridge University Press, London, 1994.
1409.7662.extracted_bib
{Heifetz-85 {\scshape D.~B. Heifetz, {\og $p$-oscillatory integrals and wave front sets\fg, Pacific J. Math. 116 (1985), p.~285-305.
1409.7662.extracted_bib
{Hogbe-71 {\scshape H.~Hogbe-Nlend, Th\'eories des bornologies et applications, Lecture Notes in Mathematics, vol. 213, Springer, Berlin, 1971.
1409.7662.extracted_bib
{Hogbe-77 \bysame , Bornologies and functional analysis, Notas de Matem{\'atica, vol.~26, North Holland, Amsterdam, 1977.
1409.7662.extracted_bib
{Hormander-71 {\scshape L.~H{\"ormander, {\og Fourier integral operators. {I\fg, Acta Math. 127 (1971), p.~79-183.
1409.7662.extracted_bib
{HormanderI \bysame , The {Analysis of {Linear {Partial {Differential {Operators {I. {Distribution {Theory and {Fourier {Analysis, second \cdredname, Springer Verlag, Berlin, 1990.
1409.7662.extracted_bib
{Horvath {\scshape J.~Horv{\'ath, Topological vector spaces and distributions, Addison-Wesley, Reading, 1966.
1409.7662.extracted_bib
{Kothe-II {\scshape G.~K{\"othe, Topological {Vector {Spaces {I{I, Springer Verlag, New York, 1979.
1409.7662.extracted_bib
{KrieglMichor {\scshape A.~Kriegl {\normalfont \cdrandname~P.~W. Michor, The convenient setting of global analysis, American Mathematical Society, Providence, 1997.
1409.7662.extracted_bib
{Kucera-81 {\scshape J.~Kucera {\normalfont \cdrandname~K.~McKennon, {\og Continuity of multiplication of distributors\fg, Intern. J. Math. Math. Sci. 4 (1981), p.~819-22.
1409.7662.extracted_bib
{Larcher-13 {\scshape J.~Larcher, {\og Multiplications and {Convolutions in {L. Schwartz' {Spaces of {Test {Functions and {Distributions and their {Continuity\fg, arXiv:1209.4174.
1409.7662.extracted_bib
{Marinescu-63 {\scshape G.~Marinescu, Espaces vectoriels pseudo-topologiques et th\'eorie des distributions, Deutsche Verlag der Wissenschaft, Berlin, 1963.
1409.7662.extracted_bib
{Meyer-98 {\scshape Y.~Meyer, Wavelets, vibrations and scalings, CRM Monograph Series, vol.~9, Amer. Math. Soc., Providence, 1998.
1409.7662.extracted_bib
{ReedSimonII {\scshape M.~Reed {\normalfont \cdrandname~B.~Simon, Methods of modern mathematical physics. {II {Fourier analysis, self-adjointness, Academic Press, New York, 1975.
1409.7662.extracted_bib
{Schwartz-57 {\scshape L.~Schwartz, {\og Th{\'eorie des distributions {\`a valeurs vectorielles. {I\fg, Ann. Inst. Fourier 7 (1957), p.~1-141.
1409.7662.extracted_bib
{Schwartz-66 \bysame , Th{\'eorie des distributions, Hermann, Paris, 1966.
1409.7662.extracted_bib
{Stampfer-11 {\scshape F.~Stampfer, {\og Pullback of measures on {Riemannian manifolds\fg, J. Math. Anal. Appl. 381 (2011), p.~812-20.
1409.7662.extracted_bib
{Treves {\scshape F.~Tr{\`eves, Topological vector spaces, distributions and kernels, Dover, New York, 2007.