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\begin{document}


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\bigskip

Prof.\ Dr.\ H.\ Abels, Prof.\ Dr.\ B.\ Ammann \hfill SoSe 2012 \\
Dr.\ C.\ Neira Jimenez  \hfill 05.06.2012 \\
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 5} 
\end{center}

\bigskip

\begin{enumerate}

\item 
\begin{enumerate}
 \item Let $p\in S^m_{1,0}(\R^{2n}\times\R^n)$ be a symbol of an operator in $(x,y)$-form such that $p(x,x,\xi)=0$ for all $x,\xi\in\R^n$. Prove that $p(x,D_x,x)=p_L(x,D_x)$, where $p_L\in S^{m-1}_{1,0}(\R^n\times\R^n)$.
 \item Prove that the adjoint of an operator $a(x,D_x,x)$ is $a^*(x,D_x,x)$ with\\ $a^*(x,y,\xi)=\overline{a(y,x,\xi)}$ for all $x,y,\xi$.
\end{enumerate}

\item Let $p\in S^m_{1,0}(\R^n\times\R^n)$. Prove that there exists $q\in S^m_{1,0}(\R^n\times\R^n)$ such that 
\begin{equation*}
 p(x,D_x)=q(D_x,x).
\end{equation*}

\item Let $X^s$, $s\in\R$, be Banach spaces such that 
\begin{enumerate}
 \item[(i)] $\mathcal{S}(\R^n)\subseteq X^s\subseteq \mathcal{S}'(\R^n)$ with continuous embeddings, where functions and regular distributions are identified in the standard way.
 \item[(ii)] $\langle D_x\rangle^m:X^{s+m}\to X^s$ is a bounded linear operator for all $s,m\in\R$.
 \item[(iii)] $p(x,D_x):X^0\to X^0$ for all $p\in S^0_{1,0}(\R^n\times\R^n)$.
\end{enumerate}
Prove that 
\begin{enumerate}
 \item $X^s=\{u\in\mathcal{S}'(\R^n):\langle D_x\rangle^su\in X^0\}$ and $X^s\subseteq X^t$ if $t\leq s$ with continuous embedding.
 \item $p(x,D_x):X^{s+m}\to X^s$ for all $p\in S^m_{1,0}(\R^n\times\R^n)$, $s,m\in\R$.
 \item If $p\in S^m_{1,0}(\R^n\times\R^n)$ is an elliptic symbol and 
\[
 p(x,D_x)u=f
\]
with $f\in X^s$ and $u\in X^{-\infty}:=\bigcup\limits_{s\in\R}X^s$, then $u\in X^{s+m}$.
\end{enumerate}

\end{enumerate}

\end{document}