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Problems

$\begin{problem} Prove that $u_+$, defined by \eqref{L2.5} is linear. \end{problem}$

$\begin{problem} % latex2html id marker 4649Prove Lemma \ref{L2.13}. \par Hint(... ...satisfies the inductive assumptions. \end{enumerate}\end{enumerate}\end{problem}$

$\begin{problem} Show that $\sigma$-algebras are closed under countable intersections. \end{problem}$

$\begin{problem} (Easy) Show that if $\mu$ is a complete measure and $E\subset F$ where $F$ is measurable and has measure $0$ then $\mu(E)=0.$\end{problem}$

$\begin{problem} Show that compact subsets are measurable for any Borel measure. ... ...ts are Borel sets if you follow through the tortuous terminology.) \end{problem}$

$\begin{problem} Show that the smallest $\sigma$-algebra containing the sets \beg... ...is called above the \lq Borel' $\sigma$-algebra on $[-\infty,\infty].$\end{problem}$

$\begin{problem} % latex2html id marker 4680Write down a careful proof of Proposition~\ref{L1.2}. \end{problem}$

$\begin{problem} % latex2html id marker 4683Write down a careful proof of Proposition~\ref{L1.4}. \end{problem}$

$\begin{problem} Let $X$ be the metric space \begin{equation*} X= \left\{ 0 \r... ...al space in terms of sequences; at least \emph{guess} the answer. \end{problem}$

$\begin{problem} For the space $Y=\mathbb{N}= \left\{ 1,2, \ldots \right\} \subse... ... (Y)$ and guess a description of its dual in terms of sequences. \end{problem}$

$\begin{problem} Let $(X,\mathcal{M},\mu)$ be any measure space (so $\mu$ is a ... ...in{equation*} \Vert f\Vert=\int_X\vert f\vert d\mu. \end{equation*}\end{problem}$

$\begin{problem} Let $(X,\mathcal{M})$ be a set with a $\sigma$-algebra. Let $\m... ...position of $A_j$. \par\textbf{Hint 2.} See \cite{Rudin1} p. 117! \end{problem}$

$\begin{problem} % latex2html id marker 4745 (Hahn Decomposition) \par With assum... ...1/m.$\ Show that $E=\bigcup_m G_m$\ is as required. \end{enumerate}\end{problem}$

$\begin{problem} \par Now suppose that $\mu$ is a finite, positive Radon measure... ...' \backslash E$ with $V$ open and look at $K = K' \backslash V$. \end{problem}$

$\begin{problem} % latex2html id marker 4774Using Problem~\ref{L7.P3} show that... ...$\ and $\mu_-$\ are finite positive Radon measures. \end{enumerate}\end{problem}$

$\begin{problem} Let $\Vert \Vert$ be a norm on a vector space $V$. Show that ... ...{displaymath}both are easy to prove using $\Vert a\Vert^2 = (a,a).$\end{problem}$

$\begin{problem} Show (Rudin does it) that if $u: \mathbb{R}^n \to \mathbb{C}$\ ... ... it is differentiable at each point in the sense of \eqref{L9.6}. \end{problem}$

$\begin{problem} Consider the function $f(x) = \langle x \rangle^{-1} = (1+ \left... ...ngle^{-1} \in \mathcal{C}^k_0 (\mathbb{R}^n)$\ for \emph{all} $k$. \end{problem}$

$\begin{problem} Show that $\exp (- \left\vert x \right\vert^2) \in \mathcal{S} (\mathbb{R}^n)$. \end{problem}$

$\begin{problem} Prove \eqref{L10.8}, probably by induction over $k$. \end{problem}$

$\begin{problem} % latex2html id marker 4816Prove Lemma \ref{L10.12}. \end{problem}$

Hint. Show that a set $U \ni 0$ in $\mathcal{S}(\mathbb{R}^n)$ is a neighbourhood of 0 if and only if for some and $\epsilon >0$ it contains a set of the form

$\displaystyle \left\{ \varphi \in \mathcal{S} (\mathbb{R}^n) ; \sum_{\ove... ...up \left\vert x^{\alpha} D^{\beta} \varphi \right\vert < \epsilon \right\} .$

$\begin{problem} Prove \eqref{L11.7}, by estimating the integrals. \end{problem}$

$\begin{problem} Prove \eqref{L11.9} where \begin{equation*} \psi_j(z;x') = \int... ...c{\partial \psi}{\partial z_j} (z+ tx') \, dt \, . \end{equation*}\end{problem}$

$\begin{problem} Prove \eqref{L11.25}. You will probably have to go back to first... ...{R}^n \, ; \, p' +t \, , \, p' \in E \right\} \, . \end{equation*}\end{problem}$

$\begin{problem} Prove Leibniz' formula \begin{equation*} {D^{\alpha}}_x (\varph... ...\binom{\alpha_j}{\beta_j} . \end{equation*}I suggest induction! \end{problem}$

$\begin{problem} % latex2html id marker 4872Prove the generalization of Propos... ... such that $\varphi_j \to \varphi$\ in the $\mathcal{C}^m$\ norm. \end{problem}$

$\begin{problem} If $m \in \mathbb{N}$, $m' >0$ show that $u \in H^m (\mathbb{R}... ...m$ implies $u \in H^{m+m'} (\mathbb{R}^n)$. Is the converse true? \end{problem}$

$\begin{problem} Show that every element $u \in L^2 (\mathbb{R}^n)$ can be writt... ...n H^1 (\mathbb{R}^n) , j=0 , \ldots , n . \end{equation*}\end{problem}$

$\begin{problem} Consider for $n=1$, the locally integrable function (the Heavisi... ...align*}Show that $D_x H (x) = c \delta$; what is the constant $c$? \end{problem}$

$\begin{problem} For what range of orders $m$ is it true that $\delta \in H^m (\mathbb{R}^n) , \delta ( \varphi ) = \varphi (0)$? \end{problem}$

$\begin{problem} Try to write the Dirac measure explicitly (as possible) in the form \eqref{L14.13}. How many derivatives do you think are necessary? \end{problem}$

$\begin{problem} Go through the computation of $\overline{\partial} E$ again, bu... ... out a disk $\left\{ x^2 + y^2 \leq \epsilon^2 \right\}$ instead. \end{problem}$

$\begin{problem} Consider the Laplacian, \eqref{L15.6}, for $n=3$. Show that $E=c(x^2 + y^2)^{-1/2}$\ is a fundamental solution for some value of $c$. \end{problem}$

$\begin{problem} Recall that a topology on a set $X$\ is a collection $\mathcal{F... ...n set $U \ni u \ \exists N$\ st. $u_j \in U \ \forall\ j \geq N$. \end{problem}$

$\begin{problem} Prove \eqref{L16.6} where $u \in \mathcal{S}' (\mathbb{R}^n)$\ and $\varphi , \psi \in \mathcal{S} (\mathbb{R}^n)$. \end{problem}$

$\begin{problem} Show that for fixed $v \in \mathcal{S}' (\mathbb{R}^n)$ with co... ...thcal{S} (\mathbb{R}^n) \end{equation*}is a continuous linear map. \end{problem}$

$\begin{problem} % latex2html id marker 4981Prove the ?? to properties in Theor... ...(\mathbb{R}^n)$\ with at least one of them having compact support. \end{problem}$

$\begin{problem} % latex2html id marker 4989Use Theorem~\ref{L16.15} to show th... ...{R}^n)$\ has $\operatorname{sing\,supp} (F) = \left\{ 0 \right\}$. \end{problem}$

$\begin{problem} Show that if $P(D)$ is an ellipitic differential operator of o... ... and $P(D)u\in L^2(\mathbb{R}^n)$ then $u\in H^m(\mathbb{R}^n).$\end{problem}$

$\begin{problem}[Taylor's theorem]. Let $u:\bbR^n\longrightarrow \bbR$be a real-v... ...ownarrow0}\frac{\vert v(x)\vert}{\vert x\vert^k}=0. \end{equation*}\end{problem}$

$\begin{problem} Let $\mathcal{C}(\bbB^n)$ be the space of continuous functions ... ...ctive and the image of the first is the null space of the second.) \end{problem}$

$\begin{problem}[Measures] A measure on the ball is a continuous linear functiona... ...ther space so that this can be extended to a short exact sequence? \end{problem}$

$\begin{problem} Show that the Riemann integral defines a measure \begin{equatio... ...hcal{C}(\bbB^n)\ni f\longmapsto \int_{\bbB^n}f(x)dx. \end{equation}\end{problem}$

$\begin{problem} If $g\in\mathcal{C}(\bbB^n)$ and $\mu\in M(\bbB^n)$show that $g... ...equation*} x_j\mu=0\Min M(\bbB^n)\Mfor j=1,\dots,n. \end{equation*}\end{problem}$

$\begin{problem}[H\uml ormander, Theorem 3.1.4] Let $I\subset\bbR$ be an open, n... ...stant $c,$ show that $u=cx+d$ for some $d\in\bbC.$\end{enumerate}\end{problem}$

$\begin{problem}[H\uml ormander Theorem 3.1.16] \begin{enumerate} \item [i)] Use ... ...R).$\item [v)] Compute $\frac{d}{dx}H\in\CmI(\bbR).$\end{enumerate}\end{problem}$

$\begin{problem} % latex2html id marker 5096Using Problems~\ref{HW1.15} and \re... ...uation \begin{equation*} x\frac{du}{dx}=0\Min\bbR. \end{equation*}\end{problem}$

These three problems are all about homogeneous distributions on the line, extending various things using the fact that

$\displaystyle x_+^z=\begin{cases}\exp(z\log x)&x>0\ 0&x\le0\end{cases}$

is a continuous function on $\bbR$ if $\Re z>0$ and is differentiable if $\Re z>1$ and then satisfies

$\displaystyle \frac{d}{dx}x_+^z=zx_+^{z-1}.$

We used this to define

$\displaystyle x_+^z=\frac1{z+k}\frac1{z+k-1}\cdots\frac1{z+1}\frac{d^k}{dx^k}x_+^{z+k}\Mif z\in\bbC\setminus-\bbN.$

(165)

$\begin{problem}[Hadamard regularization] \begin{enumerate} \item [i)]Show that \... ...x_+^z=0$ in $x<0$ for all $z\notin-\bbN.$ (Duh.) \end{enumerate}\end{problem}$

$\begin{problem}[Null space of $x\frac{d}{dx}-z$] \begin{enumerate} \item [i)] Sh... ...\} \end{equation}is a two-dimensional vector space. \end{enumerate}\end{problem}$

$\begin{problem}[Negative integral order]To do the same thing for negative integr... ...[There are still some things to prove to get this.] \end{enumerate}\end{problem}$

$\begin{problem} Show that for any set $G\subset\bbR^n$\begin{equation*} v^*(G)=\... ...ver coverings of $G$ by rectangular sets (products of intervals). \end{problem}$

$\begin{problem} Show that a $\sigma$-algebra is closed under countable intersections. \end{problem}$

$\begin{problem} Show that compact sets are Lebesgue measurable and have finite v... ...n} v(E)=\sup\{ v(K);K\subset E, K\text{ compact}\}. \end{equation}\end{problem}$

$\begin{problem} Show that a set $B\subset\bbR^n$ is Lebesgue measurable if and ... .... \end{equation*}[The definition is this for all $E\subset\bbR^n.]$\end{problem}$

$\begin{problem} Show that a real-valued continuous function $f:U\longrightarrow ... ...-1}(I)\subset U\subset\bbR^n$ is measurable for each interval $I.$\end{problem}$

$\begin{problem} Hilbert space and the Riesz representation theorem. If you need ... ... \hbox{ for a unique } f \in H . \end{displaymath}\end{enumerate}\end{problem}$

$\begin{problem} Density of $\mathcal{C}^\infty_c (\RR^n)$ in $L^p (\RR^n).$ \b... ...ense in $L^p (\RR^n)$ for any $1 \leq p < \infty.$ \end{enumerate}\end{problem}$

$\begin{problem} Schwartz representation theorem. Here we (well you) come to grip... ...and a bounded continuous function }U \end{multline*}\end{enumerate}\end{problem}$

$\begin{problem} Distributions of compact support. \begin{enumerate} \item[i)] Re... ...ote the space of distributions of compact support by $\CmIc(\bbR).$\end{problem}$

$\begin{problem} Hypoellipticity of the heat operator $H=iD_t+\Lap=iD_t+\sum\limi... ...\item Show that $iD_t-\Lap$ is also hypoelliptic. \end{enumerate}\end{problem}$

$\begin{problem} Wavefront set computations and more - all pretty easy, especial... ... f)$ tell us about $\WF(u)$ in terms of $\WF(f)$? \end{enumerate}\end{problem}$

$\begin{problem} The wave equation in two variables (or one spatial variable). \b... ...*}\item[viii)] Bound $\WF(u)$ in terms of $\WF(f).$\end{enumerate}\end{problem}$

$\begin{problem} A little uniqueness theorems. Suppose $u\in\CmIc(\bbR^n)$ recal... ...s $P(D)u=0$ for some non-trivial polynomial $P,$ show that $u=0.$\end{problem}$

$\begin{problem} (Poisson summation formula) As in class, let $L\subset\bbR^n$ ... ...I(\bbT_L)$ if and only if $\hat f=0$ on $L^\circ.$\end{enumerate}\end{problem}$

$\begin{problem} For a measurable set $\Omega \subset\bbR^n,$ with non-zero measu... ...perp\text{ are finite dimensional.} \end{equation*}\end{enumerate}\end{problem}$

$\begin{problem}[Separable Hilbert spaces] \begin{enumerate} \item [i)] (Gramm-Sc... ...j=1}^\infty\langle u,e_j\rangle e_j. \end{equation*}\end{enumerate}\end{problem}$

$\begin{problem}[Compactness]Let's agree that a compact set in a metric space is... ...here $e_k$ is the complete orthonormal sequence. \end{enumerate}\end{problem}$

$\begin{problem}[Spectral theorem, compact case]Recall that a bounded operator $... ...Be careful about the null space - it could be big. \end{enumerate}\end{problem}$

$\begin{problem} % latex2html id marker 5677Show that a (complex-valued) squar... ...Then use Problem~\ref{HW.33} to show that it is true in this case. \end{problem}$

$\begin{problem}[Ascoli-Arzela]Recall the proof of the theorem of Ascoli and Arz... ...>1/\delta \Longrightarrow \vert u(x)\vert<\epsilon . \end{equation}\end{problem}$

$\begin{problem} % latex2html id marker 5696[Compactness of sets in $L^2(\math... ...strongly so $\bar B$\ is sequently compact, and hence is compact. \end{problem}$

Next: Bibliography Up: Lecture notes for 18.155, Previous: Differential operators. Contents

Richard B. Melrose 2003-02-18