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\bigskip

Prof.\ Dr.\ H.\ Abels, Prof.\ Dr.\ B.\ Ammann \hfill SoSe 2012 \\
Dr.\ C.\ Neira Jimenez  \hfill 24.04.2012 \\
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 2} 
\end{center}

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\bigskip

\begin{enumerate}
 \item \textbf{(Product of Symbols)}\\
  Let $p_j\in S^{m_j}_{1,0}(\Rn\times\Rn)$, $m_j\in\R$, $j=1,2$ and let  $p(x,\xi):=p_1(x,\xi)p_2(x,\xi)$ for all $x,\xi\in\Rn$. Prove that $p\in S^{m_1+m_2}_{1,0}(\Rn\times\Rn)$ and that for every $k\in\N$ one has $|p|_{k}^{(m_1+m_2)}\leq C_k |p_1|_k^{(m_1)}|p_2|_k^{(m_2)}$, where $C_k$ depends only on $k$ and $n$.

 \item \textbf{(Special Symbol)}\\
  Let $\weight{\xi}=\sqrt{1+|\xi|^2}$ for $\xi\in\Rn$. Prove that for any $m\in\R$, $\alpha\in\N_0^n$ there is some $C_{m,\alpha}>0$ such that
  \begin{equation*}
    |\partial_\xi^\alpha \weight{\xi}^m|\leq C_{m,\alpha} (1+|\xi|)^{m-|\alpha|}\qquad \text{for all}\ \xi\in\Rn.
  \end{equation*}
  {\bf Hint:} Consider the function $f(a,x):= (a^2+|x|^2)^{\frac{m}2}$, where
  $a\in \R$, $x\in\Rn$. Use that $f$ is {homogeneous of degree $m$}, i.e.,
  $
  f(ra,rx)= r^m f(a,x)
  $
  for all $r>0$, $a\in\R\setminus\{0\}$, $x\in\Rn\setminus\{0\}$.

 \item \textbf{(A Simple Functional Calculus)}\\
  Let $\mathsf{L}^ {\infty} (\Rn):=\{G:\Rn\to\C\,\vert\, G\text{ is bounded and measurable}\}$. For all functions $G \in \mathsf{L}^ {\infty} (\Rn)$ set 
  \begin{equation}\label{DefG(Dx)}
   G(D_x)  f :=  \mathcal{F}^ {-1} [G(\xi) \hat{f}(\xi) ]\qquad \text{for all}\ f \in L^2 (\R^n).
  \end{equation}

  \begin{enumerate}
  \item  Prove that for all $G \in \mathsf{L}^{\infty}(\Rn)$ we have $G(D_x) \in \mathcal{L} (L^2 (\R^n))$ and the mapping 
    \[ 
    \Phi\colon \mathsf{L}^{\infty} (\Rn) \ni G \mapsto G(D_x) \in \mathcal{L} (L^2 (\R^n ))
    \]
    is linear and bounded. Moreover, show that for every $G_j \in \mathsf{L}^{\infty} (\Rn)$, $j=1,2$, it holds that  
    \begin{equation}\label{composition}
     G_1(D_x) \circ G_2(D_x) = (G_1 \cdot G_2) (D_x).
    \end{equation}

    \item Prove that if $G\in C^\infty_{\rm poly}(\Rn)$, then $G(D_x)\colon\SD(\Rn)\to\SD(\Rn)$ (defined similarly as in \eqref{DefG(Dx)}), is a bounded operator, and for $G_j \in C^\infty_{\rm poly}(\Rn)$, $j=1,2$, Equation \eqref{composition} holds as well.

  \item Let $p\in C^\infty_{\rm poly}(\Rn)$. Prove that for all $\lambda \in \C \setminus\, \overline{p(\Rn)}  $ we have $ (\lambda - p(D_x))^{-1} \in \mathcal{L} (L^2 (\R^n))$ and 
    \[
     (\lambda - p(D_x)) (\lambda - p(D_x))^{-1} f = (\lambda - p(D_x))^{-1} (\lambda - p(D_x)) f =f
    \] 
    for all $f \in \mathcal{S}(\R^n)$, where $p(D_x ) f = \mathcal{F}^{-1} [p(\xi) \hat{f} (\xi)]$  for all $f \in \mathcal{S}(\R^n)$.
   \item For which $\lambda \in \C$ there exists  $(\lambda - \Delta)^{-1}$ in the sense above?
\end{enumerate}
%$f  $ eine beschr\"ankte reelle Funktion auf dem Bild $M$ von $p$. Zeigen Sie, dass dann der zu $f \circ p$ geh\"orige Operator beschr\"ankt (und nat\"urlich linear) ist. Zeigen Sie weiterhin, dass diese Korrespondenz Multiplikation bzw. Addition von Funktionen auf $M$ in die Multiplikation bzw. Addition von Operatoren \"ubersetzt! Folgern Sie, dass $Id - \Delta $ auf $S$ invertierbar ist und dass $(Id - \Delta)^{-1}$ beschr\"ankt ist!

\end{enumerate}

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