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\bigskip

Prof.\ Dr.\ H.\ Abels, Prof.\ Dr.\ B.\ Ammann \hfill SoSe 2012 \\
Dr.\ C.\ Neira Jimenez  \hfill 03.07.2012 \\
Fakult\"at f\"ur Mathematik\\
Universit{\"a}t Regensburg \\ 

\vskip 15pt
\begin{center}
 \Large{\bf Pseudo-differential operators and applications}\\  \vskip 5pt
 \large{\bf Exercises 8} 
\end{center}

\bigskip

\begin{enumerate}
\item{\it Examples of wave front sets} 
\begin{enumerate}
\item Determine the wave front set of the delta distribution on $\mathbb{R}^n$.
\item Let $V$ be a linear subspace of $\mathbb{R}^n$ with the induced volume element
$dv$. Let $\delta_V$ be the distribution defined by $\<\delta_V,f\>=\int_V f\,dv$.
Determine the wave front set of $\delta_V$.
\item As above, but let $V$ be a submanifold of $\mathbb{R}^n$ or of a manifold
with an arbitrary volume form.
\end{enumerate}
\item Verify that the definition of a wave front set on a manifold does not depend on the choice
of parametrization, see Corollary 4.15-1)-a).
\item Let $U_1\subset\mathbb{R}^n$ and $U_2\subset\mathbb{R}^m$ be open subsets.
Show for $u_1\in \mathcal{D}(U_1)$ and $u_2\in \mathcal{D}(U_2)$ and some obvious rearrangment 
in the coordinates we have
  $$WF(u_1\otimes u_2)\subset 
    (WF(u_1)\times WF(u_2)\;\cup\; ((\mr{supp}\, u_1\times \{0\}) \times WF(u_2))$$
   $$ \kern2cm\;\cup\; (WF(u_1)\times (\mr{supp}\, u_2\times \{0\}))$$  
\item Show that $\delta(|x|^2-1)$ is well-defined as the pullback of the delta distribution
under the map $\mathbb{R}^n\to \mathbb{R}$, $x\mapsto |x|^2-1$. 
Investigate whether this distribution is 
related to a distribution studied in Exercise 1.
\end{enumerate}
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