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\bigskip

Prof.\ Dr.\ H.\ Abels, Prof.\ Dr.\ B.\ Ammann \hfill SoSe 2012 \\
Dr.\ C.\ Neira Jimenez  \hfill 22.05.2012 \\
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 4} 
\end{center}

%\vskip 15pt


\bigskip

\begin{enumerate}
\item {\bf (Ellipticity for homogeneous symbols)}\\
    Let $p\in S^m_{1,0}(\Rn\times\Rn)$, $m\in\R$, be a symbol that is 
    \emph{homogeneous of degree $m$ for $|\xi|\geq 1$}, i.e.,
    \begin{equation*}
      p(x,r\xi)= r^mp(x,\xi) \qquad \text{for all}\ r,|\xi|\geq 1.
    \end{equation*}
    Show that $p$ is elliptic if and only if $p(x,\xi)\neq 0$ for 
    all $|\xi|=1$, $x\in\Rn$ and 
    \begin{equation*}
      \inf_{x\in\Rn,|\xi|=1}|p(x,\xi)|>0.
    \end{equation*}
\item {\bf (Commutator of Pseudodifferential Operators)}\\
    Let $p_j \in S^{m_j}_{1,0}(\Rn\times \Rn)$, $j=1,2$. 
    Show that there is some $r \in S^{m_1+m_2-1}_{1,0}(\Rn\times \Rn)$ 
    such that
    \begin{equation*}
      p_1(x,D_x)p_2(x,D_x)- p_2(x,D_x)p_1(x,D_x) = r(x,D_x).
    \end{equation*}
\item {\bf (Ellipticity stable under lower order symbols)}\\
    Let $p\in S^m_{1,0}(\Rn\times\Rn)$, $m\in\R$, be an elliptic symbol, 
    and let $q\in S^{m-\eps}_{1,0}(\Rn\times\Rn)$ for some $\eps>0$. 
    Prove that $p(x,\xi)+q(x,\xi)$ is an elliptic symbol of order $m$.
\newpage
\item {\bf (Products of polyhomogeneous symbols)}\\
    A symbol $p\in S^m_{1,0}(\Rn\times\Rn)$ is called \emph{polyhomogeneous}
    if $p$ admits a symbolic expansion
    \begin{equation*}
      p(x,\xi)\sim \sum_{k=0}^\infty p_k(x,\xi)\ :\Leftrightarrow \
      p(x,\xi)- \sum_{k=0}^N p_k(x,\xi)\in S^{m-N-1}_{1,0}(\Rn\times\Rn),
      \forall N\in\N, 
    \end{equation*}
    where $p_k\in S^{m-k}_{1,0}(\Rn\times\Rn)$ are homogeneous of degree $m-k$
    for $|\xi|\geq 1$. 
    Moreover, $p_0(x,\xi)$ is called \emph{principal symbol}
    of $p(x,\xi)$.
    \begin{enumerate}
       \item Show that $p(x,\xi)q(x,\xi)$ is a polyhomogeneous symbol of order
         $m_1+m_2$ if $p$ and $q$ are polyhomogeneous symbols of order $m_1$, $m_2$, 
         respectively and that
         \begin{equation*}
           p(x,\xi)q(x,\xi)\sim \sum_{k=0}^\infty \sum_{j+l=k} p_j(x,\xi) q_l(x,\xi),
         \end{equation*}
         where $p(x,\xi)\sim \sum_{j=0}^\infty p_j(x,\xi)$ and 
         $q(x,\xi)\sim \sum_{l=0}^\infty q_l(x,\xi)$ and $p_j$ and $q_l$ are 
         homogeneous of degree $m_1-j$, $m_2-l$, respectively, in $|\xi|\geq 1$.
       \item Show that $p\#q(x,\xi)$ is a polyhomogeneous symbol of order
         $m_1+m_2$ if $p$ and $q$ are polyhomogeneous symbols of order $m_1$, $m_2$, 
         respectively, and that
         \begin{equation*}
           (p\#q)(x,\xi)\sim \sum_{k=0}^\infty \sum_{|\alpha|+j+l=k} \frac1{\alpha!} 
           \partial_{\xi}^\alpha p_j(x,\xi) D_x^\alpha q_l(x,\xi).
         \end{equation*}
    \end{enumerate}
\end{enumerate}
\end{document}
