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\begin{document}

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\section*{tensor product of distributions}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{tensor product of distributions} is the generalization to [[distributions]] of the [[tensor product]] of [[smooth functions]], hence it defines for two distributions $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ a new distribution $u \otimes v \in \mathcal{D}'(X \times Y)$ on the [[Cartesian product]] space which, as a [[generalized function]] behaves like $(u \otimes v)(x,y) = u(x) \cdot v(y)$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{TensorProductOfSmoothFunctions}\hypertarget{TensorProductOfSmoothFunctions}{}
\textbf{(tensor product of smooth functions)}

For $X_1, X_2$ two [[open subsets]] of some [[Cartesian space]], there is an [[injective function|injection]] from the [[tensor product of modules|tensor product]] of the [[real vector spaces]] of [[smooth functions]] on the separate spaces to that on the [[Cartesian product]] space:

\begin{displaymath}
\itexarray{
    C^\infty(X_1) \otimes_{\mathbb{R}} C^\infty(X_2)
      &\overset{}{\hookrightarrow}&
    C^\infty(X_1 \times X_2)
    \\
    (f_1, f_2)
    &\mapsto&
    f_1 \otimes f_2
  }
\end{displaymath}
with

\begin{displaymath}
(f_1 \otimes f_2)(x_1, x_2) \coloneqq f_1(x_1) \cdot f_2(x_2)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{TensorProductOfDistributions}\hypertarget{TensorProductOfDistributions}{}
\textbf{(tensor product of distributions)}

Let $u_1 \in \mathcal{D}'(X_1)$ and $u_2 \in \mathcal{D}'(X_2)$ be [[distributions]]. Then there is a unique [[distribution of two variables]] $u_1 \otimes u_2 \in \mathcal{D}'(X_1 \times X_2)$ such that for all pairs of [[bump functions]] $b_1 \in C_c^\infty(x)$ and $b_2 \in C^\infty(X_2)$ its value on their tensor product according to def. \ref{TensorProductOfSmoothFunctions} is

\begin{displaymath}
u(b_1 \otimes b_2)
  =
  u_1(b_1) \cdot u_2(b_2)
  \,.
\end{displaymath}
This $u_1 \otimes u_2$ is called the \emph{tensor product} of $u_1$ with $u_2$

\end{prop}
(\hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 5.1.1})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{example}
\label{WaveFrontOfTensorProductDistribution}\hypertarget{WaveFrontOfTensorProductDistribution}{}
\textbf{([[wave front set]] of tensor product distribution)}

Let $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ be two distributions. then the [[wave front set]] of their tensor product distribution $u \otimes v \in \mathcal{D}'(X \times Y)$ (def. \ref{TensorProductOfDistributions}) satisfies

\begin{displaymath}
WF(u \otimes v)
  \;\subset\;
  \left(
     WF(u) \times WF(v)
  \right)
    \cup
  \left(
    \left( supp(u) \times \{0\} \right)
    \times WF(v)
  \right)
    \cup
   \left(
     WF(u)
     \times
     \left(
       supp(v) \times \{0\}
     \right)
   \right)
  \,,
\end{displaymath}
where $supp(-)$ denotes the [[support of a distribution]].

\end{example}
(\hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 8.2.9})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Schwartz kernel theorem]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Lars Hörmander]], section 5.1 of \emph{The analysis of linear partial differential operators}, vol. I, Springer 1983, 1990 (\href{http://www.ctr.maths.lu.se/media/MATP11/2014vt2014/distho.pdf}{pdf})

\end{itemize}
[[!redirects tensor products of distributions]]

[[!redirects tensor product distribution]] [[!redirects tensor product distributions]]



\end{document}