\documentclass[12pt]{article}
\usepackage{amssymb,amsfonts,amsmath,amsbsy,theorem,amscd,dina42e}
%../..//Styles
%\usepackage[a4]{../../Styles/hiwi}
\pagestyle{empty}
\usepackage[german]{babel}
\parindent0pt
%\resize{10mm}
\newcommand{\C}{\ensuremath{\mathbb{C}}}
\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\Cn}{\ensuremath{\mathbb{C}^n}}
\newcommand{\Rn}{\ensuremath{\mathbb{R}^n}}
\newcommand{\D}{\ensuremath{\mathcal{D}}}
\newcommand{\numR}{\ensuremath{\overline{\mathbb{R}}}}
\newcommand{\N}{\ensuremath{\mathbb{N}}}
\newcommand{\Z}{\ensuremath{\mathbb{Z}}}
\newcommand{\K}{\ensuremath{\mathbb{K}}}
\newcommand{\F}{\ensuremath{\mathcal{F}}}
\newcommand{\SD}{\ensuremath{\mathcal{S}}}
\newcommand{\Torus}{\ensuremath{\mathbb{T}}}
%\renewcommand{\appendixname}{Anhang}
\newcommand{\vol}{\operatorname{vol}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\diam}{\operatorname{diam}}
\newcommand{\im}{\operatorname{Im}}
\newcommand{\re}{\operatorname{Re}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\ind}{\operatorname{ind}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\T}{\operatorname{T}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\U}{\operatorname{U}}
\newcommand{\SU}{\operatorname{SU}}
\newcommand{\On}{\operatorname{O}}
\newcommand{\SO}{\operatorname{SO}}
\newcommand{\SI}{\ensuremath{\mathbb{S}^{1}}}
\newcommand{\Spur}{\operatorname{Spur}}
\newcommand{\metrik}{\operatorname{d}}
\newcommand{\slim}{\operatorname{s-lim}}
\newcommand{\wlim}{\operatorname{w-lim}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\lbm}{\ensuremath{\lambda^{n}}}
\newcommand{\eps}{\ensuremath{\varepsilon}}
\newcommand{\dlangle}{\ensuremath{\langle\langle}}
\newcommand{\drangle}{\ensuremath{\rangle\rangle}}
\newcommand{\sigalgebra}{\ensuremath{\sigma}-Algebra\ }
\newcommand{\supnorm}[1]{\| #1\|_{\infty}}
\newcommand{\weight}[1]{\langle #1\rangle}
\newcommand{\enumalpha}{\renewcommand{\theenumi}{\alph{enumi}}}
\newcommand{\wiener}{\ensuremath{\mathcal{A}(\Torus)}}
\newcommand{\anmerkung}[1]{\emph{\bf #1}}
\newcommand{\ol}[1]{\overline{ #1}}
\newcommand{\ul}[1]{\underline{ #1}}
\newcommand{\Div}{\operatorname{div}}
\newcommand{\OP}{\operatorname{OP}}
\newcommand{\h}{\ensuremath{\mathcal{H}}}
\newtheorem{thm}{THEOREM}[section]
%\newtheorem{hisa}[thm]{Hilfssatz}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\theorembodyfont{\rmfamily}
\newtheorem{rem}[thm]{Remark}
\newtheorem{remarks}[thm]{Remarks}
\theorembodyfont{\sffamily}
\newtheorem{theorem}[thm]{Theorem}
\theorembodyfont{\upshape}
\newtheorem{defn}[thm]{Definition}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{claim}[thm]{Claim}
\newtheorem{ex1}[thm]{Example}
\newtheorem{examples}[thm]{Examples}
\newtheorem{example}[thm]{Example}
\newtheorem{assumption}[thm]{Assumption}
%\newtheorem{anwendung}[thm]{Anwendung}
\newenvironment{proof*}[1]{{\bf Proof
#1:}}{\hspace*{\fill}\rule{1.2ex}{1.2ex}\\ } 
\newenvironment{proof}{{\bf
Proof:\,}}{\hspace*{\fill}\rule{1.2ex}{1.2ex}\\ }
%\newenvironment{bewskizze}{{\bf
%    Beweisskizze\,}}{\hspace*{\fill}\rule{1.2ex}{1.2ex}} 
\newcommand{\sign}{\operatorname{sign}}
\newcommand{\pv}{\operatorname{p.v.}}
\newcommand{\diag}{\operatorname{diag}}
%\newcommand{\dd}{d\hspace{-4pt} \widetilde{\ }}  %\dbar ?


\renewcommand{\T}{\mathbb{T}}
\begin{document}


\newenvironment{aufzaehl}[1]{\begin{list}{}{\settowidth{\labelwidth}{#1}
\labelsep=1ex \leftmargin=\labelwidth \advance\leftmargin by \labelsep}}%
{\end{list}}
\def\RR{{\mathbb R}}

\bigskip

Prof.\ Dr.\ H.\ Abels, Prof.\ Dr.\ B.\ Ammann \hfill SoSe 2012 \\
Dr.\ C.\ Neira Jimenez  \hfill 17.04.2012 \\
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 1} 
\end{center}

%\vskip 15pt


\bigskip

\begin{enumerate}
 \item {\bf (Concrete Fourier Transformations)}
\begin{enumerate}
  \item Compute the Fourier transformation of the following functions:
\begin{eqnarray*}
  f_1(x)&=&
  \begin{cases}
    e^{-ax}& \text{for}\ x\geq 0,\\
    0 &\text{else},
  \end{cases}\\
  f_2(x) &=& e^{-a|x|},\\
  f_3(x) &=& \chi_{[-a,a]}(x),
\end{eqnarray*}
where $a>0$.
\item Compare the properties of the functions $f_j$ (continuity, differentiability, analyticity, and the decay for $|x|\to\infty$) with the corresponding properties of $\widehat{f_j}$.  
  \end{enumerate}

\item In the following, for $f\in\SD$ and $m\in N$ let 
\[
 |f|_{m,\SD}:=\sup_{|\alpha|+|\beta|\leq m}\sup_{x\in\Rn}\left|x^\alpha\partial_x^\beta f(x)\right|.
\]

Prove that for every $\alpha\in\N_0^n$ and $m\in\N$ there are constants $C_{m,\alpha}, C_{m,\alpha}'>0$ such that
    \begin{eqnarray*}
      |x^\alpha f|_{m,\SD} \leq C_{m,\alpha}|f|_{m+|\alpha|,\SD} \text{\qquad and\qquad}
      |\partial_x^\alpha f|_{m,\SD} \leq C'_{m,\alpha}|f|_{m+|\alpha|,\SD} 
    \end{eqnarray*}
    uniformly in $f\in\SD(\Rn)$.

\item Let $C^\infty_{\rm poly}(\Rn)$ be the set of all smooth functions $m\colon\Rn\to\C$ of \emph{polynomial growth}, i.e., for every $\alpha\in\N_0^n$ there exist a $k(\alpha)\in \N$ and $C_\alpha>0$ with
  \begin{equation*}
    |\partial_x^\alpha m(x)|\leq C_\alpha(1+|x|)^{k(\alpha)},\qquad \text{ for all }x\in\Rn.
  \end{equation*} 
  Moreover, let $m\in C^\infty_{\rm poly}(\Rn)$ and let $(Mf)(x):= m(x) f(x)$ for all $f\in\SD(\Rn)$. Prove that $M\colon\SD(\Rn)\to\SD(\Rn)$ is a bounded operator, i.e., for all $k\in\N$ there exist $n(k)\in\N$ and $C>0$ such that
\[
 |Mf|_{k,\SD}\leq C|f|_{n(k),\SD}.
\]

As a corollary prove the following:\\ For any pair of functions $f,g\in\SD(\R^n)$ the product $fg$ lies in $\SD(\R^n)$.

\item {\bf (Supplementary exercise)}\\ 
Let $(Mf)(x):= m(x) f(x)$ for $f\in\SD(\Rn)$, where $m\colon \Rn\to \C$ is a smooth function. Prove that $m\in C^\infty_{\rm poly}(\Rn)$ if $M\colon \SD(\Rn)\to \SD(\Rn)$ is a bounded operator.\\
  {\bf Hint:} First of all
  \begin{equation*}
    \sup_{x\in\Rn} |m(x) f(x)|\leq C |f|_{k,\SD},\qquad f\in\SD(\Rn)
  \end{equation*}
  for some $k\in\N$. Then consider $f(x)= (1+|x|^2)^{-k/2} e^{-\eps |x|^2/2}$, $\eps >0$.\\

\end{enumerate}





\end{document}
% erste Aufgabe ---------------------------------------------------------------
\item[{\bf \"U1}]{\bf (Fourier-Transformation und lineare Abbildungen)}\\
 Zeigen Sie, dass f"ur alle invertierbaren Matrizen $A\in \R^{n\times n}$ und $f\in L^1(\Rn)$ gilt:
  \begin{equation*}
    \F[f\circ A](\xi)= \frac1{|\det A|} \hat{f}(A^{-T}\xi),
  \end{equation*}
wobei $A$ mit der Abbildung $x\mapsto Ax$ identifiziert wird.
\item[{\bf \"U2}]{\bf (Konkrete Fourier-Transformierte)}
  \begin{enumerate}
  \item 
Berechnen Sie die Fourier-Transformierten der folgenden Funktionen:
\begin{eqnarray*}
  f_1(x)&=&
  \begin{cases}
    e^{-ax}& \text{f"ur}\ x\geq 0,\\
    0 &\text{sonst},
  \end{cases}\\
  f_2(x) &=& e^{-a|x|},\\
  f_3(x) &=& \chi_{[-a,a]}(x),
\end{eqnarray*}
wobei $a>0$.
\item Vergleichen Sie die Eigenschaften der Funktionen $f_j$ (Stetigkeit, Differenzierbarkeit, Analytizit"at, und deren Abfall f"ur $|x|\to\infty$) mit den entsprechenden Eigenschaften von $\hat{f}_j$.  
  \end{enumerate}
\item[{\bf \"U3}] {\bf (Multiindizes und verwandete Identit"aten)}
  \begin{enumerate}
  \item Es sei $e_j$, $j=1,\ldots,n$, der $j$-te kanonische Einheitsvektor im  $\Rn$, d.h., die $j$-te Komponente von $e_j$ ist $1$ und die anderen Komponenten sind $0$. Zeigen Sie, dass
\begin{equation*}
  \binom{\alpha+e_j}{\beta} = \binom{\alpha}{\beta-e_j} + \binom{\alpha}{\beta}
\end{equation*}
f"ur alle $\alpha,\beta \in \N_0^n$.
\item Zeige sie die allgemeine {\bf Binomialformel} mithilfe von vollst"andiger Induktion:
\begin{alignat*}{2}
(x+y)^\alpha &= \sum_{\beta\leq \alpha} \binom{\alpha}{\beta} x^\beta y^{\alpha-\beta}
\end{alignat*}
f"ur $x,y\in \Rn$ und $\alpha\in \N_0^n$.
\item Zeigen Sie die {\bf Leibniz-Formel}:
\begin{equation*}
  \partial^\alpha_x (uv)(x)= \sum_{\beta\leq \alpha} \binom{\alpha}{\beta} (\partial^\beta_x u)(x)(\partial^{\alpha-\beta}_xv)(x)
\end{equation*}
f"ur alle $\alpha\in\N_0^n$ und $u,v\in C^{|\alpha|}(\Rn)$.
\end{enumerate}

\end{enumerate}


\newpage



%\newenvironment{aufzaehl}[1]{\begin{list}{}{\settowidth{\labelwidth}{#1}\labelsep=1ex \leftmargin=\labelwidth \advance\leftmargin by \labelsep}}{\end{list}}
%\def\RR{{\mathbb R}}
%\NeueListe{Exercises}{{\bfseries Exercises}}{E}

%\Veranstaltung%
  {Pseudodifferential Operators WS 2003/04}%
    {Helmut Abels}%

  {Winter term 2003/04}

{November 6, 2003}{3}{1}{1} % Datum, Uebung, 1. Aufgabe, 1. Hausaufgabe
%\Uebungskopf
\bigskip
%\ohneLoesung
\begin{enumerate}
% erste Aufgabe ---------------------------------------------------------------
% \item   Let $\weight{\xi}=\sqrt{1+|\xi|^2}$. Prove that for every $\alpha\in \N_0^n$ there is a constant $C_\alpha>0$ such that
%   \begin{equation*}
%     |D_\xi^\alpha \weight{\xi}|\leq C_\alpha \weight{\xi}^{m-|\alpha|}.
%   \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 \emph{homogeneous of degree $m$}, i.e.,
%   $
%   f(ra,rx)= r^m f(a,x)
%   $
%   for all $r>0$, $a\in\R$, $x\in\Rn$. %% Uebung ueber Homogenitaet und elementare Aussagen
  \item Let $(Mf)(x) := \F^{-1}[m(\xi) \hat{f}]$, where $m\in L^\infty(\Rn)$.
    Prove that $M\colon L^2(\Rn) \to L^2(\Rn)$ is invertible if and only if
    $m^{-1}(\xi) \in L^\infty(\Rn)$.\\
{\bf Note:} $X=L^\infty(\Rn)$ and $X=\mathcal{L}(L^2(\Rn))$ are \emph{Banach algebras} with unit, i.e., $X$ is Banach space and a ring with unit such that $\lambda (xy)=(\lambda x)y= x(\lambda y)$, $\lambda\in\C$, and 
\[
\|xy\|_X \leq \|x\|_X\|y\|_X\qquad \text{for all\ } x,y\in X.
\]
% zweite Aufgabe --------------------------------------------------------------
\item  Prove that for every $\alpha\in\N_0^n$ and $m\in\N$ there are constants $C_{m,\alpha}, C_{m,\alpha}'>0$ such that
    \begin{eqnarray*}
      |x^\alpha f|_{m,\SD} \leq C_{m,\alpha}|f|_{m+|\alpha|,\SD} \qquad
      |\partial_x^\alpha f|_{m,\SD} \leq C'_{m,\alpha}|f|_{m+|\alpha|,\SD} 
    \end{eqnarray*}
    uniformly in $f\in\SD(\Rn)$.
  % dritte Aufgabe --------------------------------------------------------------
\item Let $\mathcal{P}$ be the set of all smooth functions $m\colon\Rn\to\C$ of \emph{polynomial growth}, i.e., for every $\alpha\in\N_0^n$ there exist a $k(\alpha)\in \N$ and $C_\alpha>0$ with
  \begin{equation*}
    |\partial_x^\alpha m(x)|\leq C_\alpha\weight{x}^{k(\alpha)},\qquad x\in\Rn.
  \end{equation*} 
  Moreover, let $m\in \mathcal{P}$ and let $(Mf)(x):= m(x) f(x)$ for all $f\in\SD(\Rn)$. Prove that $M\colon\SD(\Rn)\to\SD(\Rn)$ is a bounded operator. 
\item {\bf Supplementary exercise:} Let $(Mf)(x):= m(x) f(x)$ for $f\in\SD(\Rn)$, where $m\colon \Rn\to \C$ is a smooth function. Prove that $m\in\mathcal{P}$ if $M\colon \SD(\Rn)\to \SD(\Rn)$ is a bounded operator.\\
  {\bf Hint:} First of all
  \begin{equation*}
    \sup_{x\in\Rn} |m(x) f(x)|\leq C |f|_{k,\SD},\qquad f\in\SD(\Rn)
  \end{equation*}
  for some $k\in\N$. Then consider $f(x)= \weight{x}^{-k} e^{-\eps |x|^2/2}$, $\eps >0$.
% \item {\bf For students interested in Fr\'echet spaces} \\
%   \begin{enumerate}
%   \item Prove that $C_b^\infty(\Rn)$ is complete. {\bf Hint:} You may use that $C_b^k(\Rn)$ is complete.
%   \item Prove that $\SD(\Rn)$ is complete.
%   \end{enumerate}
\end{enumerate}


\end{document}