\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{compactly supported distribution} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis} [[!include functional analysis - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{as_continuous_linear_duals_to_smooth_functions}{As continuous linear duals to smooth functions}\dotfill \pageref*{as_continuous_linear_duals_to_smooth_functions} \linebreak \noindent\hyperlink{AsSmoothLinearDuals}{As smooth linear duals to smooth functions}\dotfill \pageref*{AsSmoothLinearDuals} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{fourierlaplace_transform}{Fourier-Laplace transform}\dotfill \pageref*{fourierlaplace_transform} \linebreak \noindent\hyperlink{SingularSupportAndWaveFrontSet}{Singular support and Wave front set}\dotfill \pageref*{SingularSupportAndWaveFrontSet} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{traditional}{Traditional}\dotfill \pageref*{traditional} \linebreak \noindent\hyperlink{in_terms_of_smooth_toposes}{In terms of smooth toposes}\dotfill \pageref*{in_terms_of_smooth_toposes} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{compactly supported distribution} is a [[distribution]] whose [[support of a distribution]] is a [[compact topological space|compact]] [[subset]]. This implies that compactly supported distributions may be evaluated not just on [[bump functions]], but in fact on the larger space of all [[smooth functions]]. Indeed it turns out that the compactly supported distributions exhaust the [[continuous linear functionals]] on the space of smooth functions (\hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 2.3.1}). Therefore and since Schwartz's notation for the space $C^\infty(X)$ of all smooth functions on the given [[smooth manifold]] is $\mathcal{E}(X)$, the space of compactly supported distributions is often denoted by $\mathcal{E}'(X)$ (\hyperlink{Hoermander90}{H\"o{}rmander 90, p. 45}). \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{as_continuous_linear_duals_to_smooth_functions}{}\subsubsection*{{As continuous linear duals to smooth functions}}\label{as_continuous_linear_duals_to_smooth_functions} \begin{defn} \label{TVSOfCompactlySupportedFunctions}\hypertarget{TVSOfCompactlySupportedFunctions}{} \textbf{([[topological vector space]] of [[smooth functions]] on a [[Cartesian space]])} For $n \in \mathbb{R}^n$, write $C^\infty(\mathbb{R}^n)$ for the $\mathbb{R}$-[[vector space]] of [[smooth functions]] $\mathbb{R}^n \to \mathbb{R}$. Then the vector space $C^\infty(\mathbb{R}^n)$ becomes a [[Fréchet vector space]] induced by the family of [[seminorms]] \begin{displaymath} \itexarray{ C^\infty_c(\mathbb{R}^n) &\overset{p_{K}^\alpha}{\longrightarrow}& [0,\infty) \\ \Phi &\mapsto& \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x) \vert} } \,, \end{displaymath} indexed by $K \subset \mathbb{R}^n$ a [[compact subset]] and $\alpha \in \mathbb{N}^n$ a [[tuple]] encoding the degrees of the [[partial derivative]] $\partial^\alpha$. Hence the seminorm $p_K^\alpha$ sends a [[bump function]] $\Phi$ to the [[supremum]] over the points $x \in K$ of the [[absolute value]] of the [[partial derivative]] $\partial^\alpha \Phi$; and the [[open subsets]] defined thereby are the [[unions]] of translations of the [[neighbourhood base|base]] [[open balls]] \begin{displaymath} B^\circ_{K,\alpha,\epsilon}(0) = \left\{ v \in \mathbb{R}^n \;\vert\; p_K^\alpha(v) \lt \epsilon \right\} \end{displaymath} for $\epsilon \in (0,\infty)$. We write \begin{displaymath} \mathcal{E}(\mathbb{R}^n) \in TopVect_{\mathbb{R}} \end{displaymath} for the resulting [[Fréchet space|Fréchet]] [[topological vector space]]. \end{defn} \begin{defn} \label{CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions}\hypertarget{CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions}{} \textbf{(compactly supported distibutions as continuous linear duals to bump functions)} The space $\mathbb{E}'(\mathbb{R}^n)$ of \emph{compactly supported distributions} on $\mathbb{R}^n$ is the [[dual space]] \begin{displaymath} \mathcal{E}'(\mathbb{R}^n) \;\coloneqq\; \left(\mathcal{E}(\mathbb{R}^n)\right)^\ast \end{displaymath} to the [[topological vector space]] of [[bump functions]] from def. \ref{TVSOfCompactlySupportedFunctions}. \end{defn} e.g. (\hyperlink{Klainerman08}{Klainerman 08, p. 9}) This means the following \begin{prop} \label{CharacterizationOfContinuityBySeminormBounds}\hypertarget{CharacterizationOfContinuityBySeminormBounds}{} \textbf{(characterization of continuity for compactly supported distributions)} A [[linear function]] \begin{displaymath} u \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R} \end{displaymath} is [[continuous function|continuous]] with respect to the [[topological space|topology]] of def. \ref{TVSOfCompactlySupportedFunctions}, hence is a compactly supported distribution (def. \ref{CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions}) \begin{displaymath} \mathcal{E}(\mathbb{R}^n) \longrightarrow \mathbb{R} \end{displaymath} precisely if the following condition holds \begin{itemize}% \item there exists a compact subset $K \subset \mathbb{R}^n$ and $k \in \mathbb{N}$ and $C \in (0,\infty)$ such that \begin{equation} \underset{\Phi \in C^\infty(\mathbb{R}^n)}{\forall} \left( \vert u(\Phi) \vert \;\leq\; C \underset{ {\vert \alpha \vert \leq k} }{\sum} \underset{x \in K}{sup} \vert \partial^\alpha K \vert \right) \label{BoundBySumOfNormOfDerivatives}\end{equation} \end{itemize} \end{prop} (see also \hyperlink{Hoermander90}{H\"o{}rmander 90, (2.3.2) and theorem 2.3.1}) \begin{proof} By \href{locally+convex+topological+vector+space#AlternativeCharacterizationOfContinuityForLinearFunctionals}{this prop.} the continuity of $L$ is equivalent to there being an [[inhabited set|inhabited]] [[finite set]] $\{ (K_1, \alpha_1), \cdots, (K_r, \alpha_r) \}$ and $C \in (0,\infty)$ such that \begin{equation} {\vert u(\Phi)\vert} \;\leq\; C \underset{i = 1, \cdots, n}{max} \left( \underset{x \in K_i}{sup} {\vert \partial^{\alpha_i} \Phi \vert} \right) \label{BoundByMaximumOfSeminormsOverFiniteSet}\end{equation} We need to see that this is equivalent to a bound of the form \eqref{BoundBySumOfNormOfDerivatives}. In one direction, assume that \eqref{BoundByMaximumOfSeminormsOverFiniteSet} holds. The union $K \coloneqq \underset{i = 1, \cdots, n}{\cup} K_i$ is still a compact subset (\href{compact+space#UnionsAndIntersectionOfCompactSubspaces}{this prop.}). Hence \eqref{BoundByMaximumOfSeminormsOverFiniteSet} implies that \begin{displaymath} \begin{aligned} {\vert u(\Phi)\vert} & \leq C \underset{i = 1, \cdots, n}{max} \underset{x \in K}{sup} {\vert \partial^{\alpha_i} \Phi \vert} \end{aligned} \end{displaymath} and hence with $k \coloneqq \underset{i = 1, \cdots, n}{max} {\vert \alpha_i\vert}$ that \begin{displaymath} {\vert u(\Phi)\vert} \;\leq\; C \underset{{\vert \alpha\vert} \leq k}{\sum} \underset{x \in K}{sup} {\vert \partial^{\alpha} \Phi \vert} \,, \end{displaymath} which is of form \eqref{BoundBySumOfNormOfDerivatives}. Conversely, assume that a bound of the form \eqref{BoundBySumOfNormOfDerivatives} holds. Then take the finite set of pairs $(K_i, \alpha_i)$ where all $K_i \coloneqq K$ and with all ${\vert\alpha_i\vert} \leq k$. With $N$ denoting the number of $n$-tuples $\alpha$ with ${\vert \alpha \vert} \leq k$ we then get a bound as in \eqref{BoundByMaximumOfSeminormsOverFiniteSet} with coefficient $N C$. \end{proof} \hypertarget{AsSmoothLinearDuals}{}\subsubsection*{{As smooth linear duals to smooth functions}}\label{AsSmoothLinearDuals} Given that [[distributions]] are concerned with \emph{[[smooth functions]]} it is sometimes more natural to regard them not as [[continuous linear functionals]] as in def. \ref{CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions}, but as \emph{smooth linear functionals}. Indeed this turns out to be equivalent (prop. \ref{CompactlySupportedDistributionsAreTheSmoothLinearFunctionals} below), if one considers an ambient context of suitably [[generalized smooth spaces]], namely [[diffeological spaces]] or more generally [[smooth sets]] or [[formal smooth sets]]. We will write $\mathbf{H}$ for any of these [[categories]] of [[generalized smooth spaces]]. We may canonially regard any [[smooth manifold]] such as the [[Cartesian space]] $\mathbb{R}^n$ as an object of $\mathbf{H}$. For $X \in \mathbf{H}$ any object, we write $[X,\mathbb{R}]$ for the [[mapping space]] (the [[internal hom]]). The underlying set is $C^\infty(X)$. If $X$ itself has $\mathbb{R}$-linear structure, we write \begin{displaymath} [X,\mathbb{R}]_{\mathbb{R}} \hookrightarrow [[X,\mathbb{R}], \mathbb{R}] \end{displaymath} for the [[subobject]] of $\mathbb{R}$-linear maps. Concretely, for $U$ a [[smooth manifold]] (or just a [[Cartesian space]]), then the [[sheaf]] $[X,\mathbb{R}]$ assigns (see at \emph{[[closed monoidal structure on presheaves]]} for details) \begin{displaymath} [X,\mathbb{R}](U) = C^\infty(U \times X) \end{displaymath} and $[X,\mathbb{R}](U) \subset C^\infty(U \times X)$ is the subset of those functions $\Phi_{(-)}(-)$ such that for all $u \in U$ the function $\Phi_u \colon X \to \mathbb{R}$ is $\mathbb{R}$-linear. The [[global elements]] $\Gamma(-)$ of the [[mapping space]] constitute the ordinary [[hom set]] \begin{displaymath} \Gamma [X,\mathbb{R}] \simeq \mathbf{H}(X,\mathbb{R}) \,. \end{displaymath} \begin{prop} \label{CompactlySupportedDistributionsAreTheSmoothLinearFunctionals}\hypertarget{CompactlySupportedDistributionsAreTheSmoothLinearFunctionals}{} \textbf{([[compactly supported distributions are the smooth linear functionals]])} For $n \in \mathbb{N}$, there is a [[natural bijection]] between the underlying sets of compactly supported distributions on $\mathbb{R}^n$ (def. \ref{CompactlySupportedDistributionsAsContinuousLinearDualsToBumpFunctions}) and the $\mathbb{R}$-linear [[mapping space]] formed in the [[category]] $\mathbf{H}$ of either [[diffeological space]] or [[smooth sets]] or [[formal smooth sets]]: \begin{displaymath} \widetilde{(-)} \;\colon\; \mathcal{E}'(\mathbb{R}^n) \overset{\simeq}{\longrightarrow} \mathbf{H}([\mathbb{R}^n, \mathbb{R}], \mathbb{R})_{\mathbb{R}} \end{displaymath} given by sending $\mu \in \mathcal{E}'(\mathbb{R}^n)$ to the [[natural transformation]] which on a test space $U \in$ [[CartSp]] takes a smoothly $U$-parameterized function $\Phi_{(-)}(-) \colon U \times \mathbb{R}^n \to \mathbb{R}$ to its evaluation in $\mu$ pointwise in $U$: \begin{displaymath} \tilde \mu(\Phi_{(-)})(u) \;\coloneqq\; \langle \mu, \Phi_{u}\rangle \,. \end{displaymath} \end{prop} (\hyperlink{MoerdijkReyes91}{Moerdijk-Reyes 91, chapter II, prop. 3.6}) \begin{proof} First consider this for the case that $\mathbf{H} =$ [[SmoothSet]] (which immediately subsumes the case that $\mathbf{H} =$ [[diffeological space|DiffelogicalSpace]]). To see that $\widetilde{(-)}$ is well defined, we need to check that the function \begin{displaymath} \itexarray{ U &\overset{ \tilde \mu(\Phi_{(-)})}{\longrightarrow}& \mathbb{R} \\ u &\mapsto& \langle \mu, \Phi_u\rangle } \end{displaymath} is [[smooth function|smooth]]. But this follows immediately since $\langle \mu,-\rangle$ by definition is [[linear function|linear]] and [[continuous function|continuous]] (\hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 2.1.3}). To see that $\widetilde{(-)}$ is indeed a [[bijection]] for each $U$ it remains that every $\mathbb{R}$-linear smooth functional (morphisms of [[smooth sets]]) of the form \begin{displaymath} A \;\colon\; [\mathbb{R}^n,\mathbb{R}] \longrightarrow \mathbb{R} \end{displaymath} when restricted on [[global elements]] to a [[function]] of sets \begin{displaymath} A(\ast) \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R} \end{displaymath} is [[continuous function|continuous]] with respect to the [[topological vector space]] structure from def. \ref{TVSOfCompactlySupportedFunctions} on the left. Now by definition of the [[internal hom]] $A$ is actually ``path-smooth'' (\href{Frechet+space#PathSmoothFunction}{this def.}) and hence the statement is implied by \href{Frechet+space#PathSmoothLinearFunctionsOnFrechetSpaceAreContinuous}{this prop}. Finally to see that this argument generalizes to $\mathbf{H} =$ [[formal smooth set|FormalSmoothSet]] observe that the [[Weil algebra]] of every [[infinitesimally thickened point]] is a [[quotient ring]] of an algebra of smooth functions on some [[Cartesian space]] (by the [[Hadamard lemma]]). The previous argument now applies to representatives under this quotient coprojection and one checks that it is independent of the representative chosen. \end{proof} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{fourierlaplace_transform}{}\subsubsection*{{Fourier-Laplace transform}}\label{fourierlaplace_transform} \begin{defn} \label{FourierTransformOfCompactlySupportedDistribution}\hypertarget{FourierTransformOfCompactlySupportedDistribution}{} For $n \in \mathbb{N}$, let $u \in \mathcal{E}'(\mathbb{R}^n)$ be a compactly supported distribution on [[Cartesian space]] $\mathbb{R}^n$. Then its \emph{[[Fourier transform of distributions]]} is the function \begin{displaymath} \itexarray{ \mathbb{R}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R} \\ \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right) } \end{displaymath} where on the right we have the application of $u$, regarded as a [[linear function]] $u \colon C^\infty(\mathbb{R}^n) \to \mathbb{R}$, to the [[exponential function]] applied to the canonical [[inner product]] $\langle -,-\rangle$ on $\mathbb{R}^n$. This same formula makes sense more generally for [[complex numbers]] $\zeta \in \mathbb{C}^n$. This is then called the \emph{[[Fourier-Laplace transform]]} of $u$, still denoted by the same symbol: \begin{displaymath} \itexarray{ \mathbb{C}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R} \\ \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right) } \end{displaymath} This is an [[entire analytic function]] on $\mathbb{C}^n$. \end{defn} (\hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 7.1.14}) \begin{theorem} \label{}\hypertarget{}{} \textbf{([[Paley-Wiener-Schwartz theorem]])} For $n \in \mathbb{N}$ the vector space $C^\infty_c(\mathbb{R}^n)$ of [[compact support|compactly supported]] [[smooth functions]] ([[bump functions]]) on [[Euclidean space]] $\mathbb{R}^n$ is (algebraically and topologically) [[isomorphism|isomorphic]], via the [[Fourier-Laplace transform]] (prop. \ref{FourierTransformOfCompactlySupportedDistribution}), to the space of [[entire functions]] $F$ on $\mathbb{C}^n$ which satisfy the following estimate: there is a [[positive number|positive]] [[real number]] $B$ such that for every [[integer]] $N \gt 0$ there is a [[real number]] $C_N$ such that: \begin{displaymath} \underset{\xi \in \mathbb{C}^n}{\forall} \left( {\Vert F(\xi) \Vert} \le C_N \left( 1 + {\vert \xi\vert } \right)^{-N} e^{B \, |Im(\xi)|} \right) \,. \end{displaymath} More generally, the space of [[compactly supported distributions]] on $\mathbb{R}^n$ of [[order of a distribution|order]] $N$ is isomorphic via [[Fourier transform of distributions]] to those [[entire functions]] on $\mathbb{C}^n$ for which there exists positive [[real numbers]] $C$ and $B$ such that \begin{displaymath} \underset{\xi \in \mathbb{C}^n}{\forall} \left( {\Vert F(\xi) \Vert} \le C_N (1 + {\vert \xi\vert })^{N} e^{ B \; |Im(\xi)|} \right) \,. \end{displaymath} (Notice that the [[Fourier transform of distributions|Fourier transform]] of a [[compactly supported distribution]] is guaranteed to be a [[smooth function]], by \href{Fourier+transform+of+distributions#FourierTransformOfCompactlySupportedDistributions}{this prop.}.) \end{theorem} (e.g. \hyperlink{Hoermander90}{Hoermander 90, theorem 7.3.1}) \begin{prop} \label{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}\hypertarget{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}{} \textbf{(decay of [[Fourier transform of distributions|Fourier transform]] of [[compactly supported functions]])} A [[compactly supported distribution]] $u \in \mathcal{E}'(\mathbb{R}^n)$ is a [[non-singular distribution]] (def. \ref{NonSingularCompactlySupportedDistributions}), hence given by a [[compactly supported function]] $b \in C^\infty_{cp}(\mathbb{R}^n)$ via $u(f) = \int b(x) f(x) dvol(x)$, precisely if its [[Fourier transform of a distribution|Fourier transform]] $\hat u$ (\href{compactly+supported+distribution#FourierTransformOfCompactlySupportedDistribution}{this def.}) satisfies the following decay property: For all $N \in \mathbb{N}$ there exists $C_N \in \mathbb{R}_+$ such that for all $\xi \in \mathbb{R}^n$ we have that the [[absolute value]] ${\vert \hat v(\xi)\vert}$ of the Fourier transform at that point is bounded by \begin{equation} {\vert \hat v(\xi)\vert} \;\leq\; C_N \left( 1 + {\vert \xi\vert} \right)^{-N} \,. \label{DecayEstimateForFourierTransformOfNonSingularDistribution}\end{equation} \end{prop} (e.g. \hyperlink{Hoermander90}{Hoermander 90, around (8.1.1)}) \hypertarget{SingularSupportAndWaveFrontSet}{}\subsubsection*{{Singular support and Wave front set}}\label{SingularSupportAndWaveFrontSet} \begin{defn} \label{SingularSupportOfCompactlySupportedDistribution}\hypertarget{SingularSupportOfCompactlySupportedDistribution}{} \textbf{([[singular support]] of a [[compactly supported distribution]])} For $n \in \mathbb{N}$ and $u \in \mathcal{E}'(\mathbb{R}^n)$ a [[compactly supported distribution]], its \emph{[[singular support]]} is the [[subset]] of the [[Cartesian space]] $\mathbb{R}^n$ of those points which have no [[neighbourhood]] on which $u$ [[restriction of distributions|restricts]] to a [[non-singular distribution]]: \begin{displaymath} supp_{sing}(u) \;\coloneqq\; \left\{ u \in \mathbb{R}^n \,\vert\, \not \left( \underset{U \underset{\text{nbhd}}{\supset} \{x\}}{\exists} \left( u\vert_U \in C^\infty_{cp}(U) \right) \right) \right\} \,. \end{displaymath} \end{defn} By prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions} the [[singular support of a distribution]] (def. \ref{SingularSupportOfCompactlySupportedDistribution}) consists of those points around which the [[Fourier transform of distributions|Fourier transform of the distribution]] receives large high-[[frequency]] (``UV'') contributions. But in fact prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions} allows to say more precisely \emph{which} high frequency Fourier modes make the distribution singular at a given point. These are said to be part of the \emph{wave front} of the distribution, and the collection of all of them is called the \emph{[[wave front set]]} of the distribution: \begin{defn} \label{WaveFrontSet}\hypertarget{WaveFrontSet}{} \textbf{([[wavefront set]])} For $n \in \mathbb{N}$ let $u \in \mathcal{E}'(\mathbb{R}^n)$ be a [[compactly supported distribution]]. For $b \in C^\infty_{cp}(\mathbb{R}^n)$ a [[compactly supported function|compactly supported]] [[smooth function]], write $b u \in \mathcal{E}'(\mathbb{R}^n)$ for the corresponding product (\href{product+of+distributions#ProductOfADistributionWithANonSingularDistribution}{this example}). For $x\in supp(b) \subset \mathbb{R}^n$, we say that a unit [[covector]] $\xi \in S((\mathbb{R}^n)^\ast)$ is \emph{regular} if there exists a [[neighbourhood]] $U \subset S((\mathbb{R}^n)^\ast)$ of $\xi$ in the [[unit sphere]] such that for all $c \xi' \in (\mathbb{R}^n)^\ast$ with $c \in \mathbb{R}_+$ and $\xi' \in U \subset S((\mathbb{R}^n)^\ast)$ the decay estimate \eqref{DecayEstimateForFourierTransformOfNonSingularDistribution} is valid for the [[Fourier transform of distributions|Fourier transform]] $\widehat{b u}$ of $b u$; at $c \xi'$. Otherwise $\xi$ is \emph{non-regular}. Write \begin{displaymath} \Sigma(b u) \;\coloneqq\; \left\{ \xi \in S((\mathbb{R}^n)^\ast) \;\vert\; \xi \, \text{non-regular} \right\} \end{displaymath} for the set of non-regular covectors of $b u$. The \emph{wave front set at $x$} is the [[intersection]] of these sets as $b$ ranges over [[bump functions]] whose [[support]] includes $x$: \begin{displaymath} \Sigma_x(u) \;\coloneqq\; \underset{ { b \in C^\infty_{cp}(\mathbb{R}^n) } \atop { x \in supp(b) } }{\cap} \Sigma(b u) \,. \end{displaymath} Finally the \emph{[[wave front set]]} of $u$ is the subset of the [[sphere bundle]] $S(T^\ast \mathbb{R}^n)$ which over $x \in \mathbb{R}^n$ consists of $\Sigma_x(U) \subset T^\ast_x \mathbb{R}^n$: \begin{displaymath} WF(u) \;\coloneqq\; \underset{x \in \mathbb{R}^n}{\cup} \Sigma_x(u) \;\subset\; S(T^\ast \mathbb{R}^n) \end{displaymath} Usually this is considered as the full [[conical set]] inside the [[cotangent bundle]] generated by the unit covectors under multiplication with [[positive number|positive]] [[real numbers]]. \end{defn} (\hyperlink{Hoermander90}{H\"o{}rmander 90, def. 8.1.2}) Hence the [[wave front set]] of a compactly supported distribution consists of all those [[direction of a vector|directions]] of [[wave vectors]] along which the [[Fourier transform of distributions|Fourier transform of the distribution]] is not a [[rapidly decreasing function]]. \begin{prop} \label{EmptyWaveFrontSetCorrespondsToOrdinaryFunction}\hypertarget{EmptyWaveFrontSetCorrespondsToOrdinaryFunction}{} \textbf{([[empty set|empty]] [[wave front set]] corresponds to ordinary [[smooth functions]])} The [[wave front set]] (def. \ref{WaveFrontSet}) of a [[compactly supported distribution]] is [[empty set|empty]] precisely if the distribution is [[non-singular distribution|non-singular]] (example \ref{NonSingularCompactlySupportedDistributions}). \end{prop} This is effectively the [[Paley-Wiener-Schwartz theorem]] (\hyperlink{Hoermander90}{H\"o{}rmander 90, below (8.1.1)}). \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{defn} \label{NonSingularCompactlySupportedDistributions}\hypertarget{NonSingularCompactlySupportedDistributions}{} \textbf{(non-singular compactly supported distributions)} For $n \in \mathbb{N}$, a [[compactly supported function|compactly supported]] [[smooth function]] $b \in C^\infty_{cp}(\mathbb{R}^n)$ (a [[bump function]]) induces a [[compactly supported distribution]] \begin{displaymath} \int_{\mathbb{R}^n} (-) b dvol_{\mathbb{R}^n} \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R} \end{displaymath} by [[integration]] of smooth functions against $b dvol$. This construction defines a linear inclusion \begin{displaymath} C^\infty_{cp}(\mathbb{R}^n) \hookrightarrow \mathcal{E}'(\mathbb{R}^n) \,. \end{displaymath} The compactly supported distributions arising this way are called the \emph{[[non-singular distributions]]}. \end{defn} \begin{itemize}% \item [[microcausal functional]] \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[compactly supported continuous function]] \item [[bump function]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{traditional}{}\subsubsection*{{Traditional}}\label{traditional} Textbook accounts include \begin{itemize}% \item [[Lars Hörmander]], section 2.3 of \emph{The analysis of linear partial differential operators}, vol. I, Springer 1983, 1990 \end{itemize} Lecture notes include \begin{itemize}% \item [[Sergiu Klainerman]], \emph{Analysis} 2008 (\href{https://web.math.princeton.edu/~seri/homepage/courses/Analysis2008.pdf}{pdf}) \end{itemize} \hypertarget{in_terms_of_smooth_toposes}{}\subsubsection*{{In terms of smooth toposes}}\label{in_terms_of_smooth_toposes} Discussion of compactly supported distributions in terms morphisms out of [[internal homs]] in a [[smooth topos]] is in \begin{itemize}% \item [[Ieke Moerdijk]], [[Gonzalo Reyes]], around prop. 3.6 in chapter II of \emph{[[Models for Smooth Infinitesimal Analysis]]}, Springer 1991 \end{itemize} and specifically for the [[Cahiers topos]] of [[formal smooth sets]] in \begin{itemize}% \item [[Anders Kock]], [[Gonzalo Reyes]], \emph{Some calculus with extensive quantities: wave equation}, Theory and Applications of Categories , Vol. 11, 2003, No. 14, pp 321-336 (\href{http://www.tac.mta.ca/tac/volumes/11/14/11-14abs.html}{TAC}) \item [[Anders Kock]], \emph{Commutative monads as a theory of distributions} (\href{http://arxiv.org/abs/1108.5952}{arxiv:1108.5952}) \end{itemize} using results of \begin{itemize}% \item [[Alfred Frölicher]], [[Andreas Kriegl]], section 5 of \emph{Linear spaces and differentiation theory}, Wiley 1988 (\href{http://www.fuw.edu.pl/~kostecki/scans/froelicherkriegl1988.pdf}{pdf}) \end{itemize} and following the general conception of ``[[intensive and extensive]]'' in \begin{itemize}% \item [[William Lawvere]], Introduction to \emph{[[Categories in Continuum Physics]], Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986} \end{itemize} Generalization to non-compactly supported distributions is in \begin{itemize}% \item [[Anders Kock]], [[Gonzalo Reyes]], \emph{Categorical distribution theory; heat equation} (\href{https://arxiv.org/abs/math/0407242}{arXiv:math/0407242}) \end{itemize} and [[sheaf theory|sheaf theoretic]] discussion of distributions as morphisms of [[smooth spaces]] is in \begin{itemize}% \item [[Frédéric Paugam]], section 3.2 of \emph{Towards the mathematics of quantum field theory}, 2012 (\href{https://webusers.imj-prg.fr/~frederic.paugam/documents/enseignement/master-mathematical-physics.pdf}{pdf}) \end{itemize} [[!redirects compactly supported distributions]] \end{document}