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\bigskip

Prof.\ Dr.\ H.\ Abels, Prof.\ Dr.\ B.\ Ammann \hfill SoSe 2012 \\
Dr.\ C.\ Neira Jimenez  \hfill 19.06.2012 \\
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 7} 
\end{center}

\bigskip

\begin{enumerate}

\let\witi\widetilde
\item Let $M$ be compact manifold. 
A parametrization $\phi:U\to V\subset M$ 
is a good parametrization if it can be extended to a parametrization
$\witi\phi:\witi U\to \witi V$, such that $U$ is a subset of $\witi U$ 
with compact closure (and thus the same holds for $V$ and $\witi V$).

\begin{enumerate}
\item Show that a linear map $P:C^\infty(M)\to C^\infty(M)$ is a pseudo-differential operator if and only if for any good parametrization $\phi:U\to V$ the linear map $P^\phi$ given by composition
\begin{equation*}  
C_c^\infty(V) \stackrel{(\phi^{-1})^*}{\to}C_c^\infty(U)\hookrightarrow C^\infty(M) 
\stackrel{P}{\to} C^\infty(M)\stackrel{\phi^*}{\to}C^\infty(U)
\end{equation*}
extends to a pseudo-differential operator on $\mathbb{R}^n$.
\item Assume that $\{\phi_1,\ldots,\phi_k\}$ is an atlas. Is it sufficient
to check whether all $P^{\phi_i}$ extend to a pseudo-differential operator?
\end{enumerate}

\item Let $M$ be a compact manifold with a fixed volume form, 
and $E$ and $F$ two vector bundls over $M$.
Let $P:\Gamma(E)\to \Gamma(F)$ be a pseudo-differential operator of symbol 
class $S^m_{1,0}$. Then there exists an pseudo-differential operator
$P^*: \Gamma(F^*)\to \Gamma(E^*)$ which is (formally) adjoint to $P$, i.e.\
  $$(\psi,P\phi)_{L^2}= (P^*\psi,\phi)_{L^2}.$$

\item Let $u$ be a distribution on $M$, which is a
compact manifold or Euclidean space. Show that $u$ is smooth on an open subset 
$U$ of $M$ 
in the sense of the lecture if and only if for all $x\in U$ there is an open
neighborhood $V_x$ of $x$ in $U$ such that $u$ is smooth on $V_x$.


\item 
Let $P$ be an elliptic pseudodifferential operator on Euclidean space or on a compact manifold. Show for all $u\in {\cal D}'$ 
  $${\rm sing\ supp\ } (Pu)={\rm sing\ supp\ }u.$$ 
Note: This property is called \emph{hypo-ellipticity}.
\end{enumerate}

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