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\title{Math 581 Assignment 3}
\author{Due Friday March 1}
\date{Winter 2013} % Activate to display a given date or no date
\begin{document}
\maketitle
\begin{enumerate}[1.]
\item
\begin{enumerate}[a)]
\item
Derive a formula for $\widehat{u\circ A}$, where $A$ is an $n\times n$ invertible matrix.
\item
There are (at least) two ways to define the Fourier transform on $L^2(\R^n)$.
\begin{itemize}
\item
Extend the Fourier transform from $\tstS$ to $L^2$ by using the density of $\tstS$ in $L^2$ (as well as the Plancherel bound).
\item
First define the Fourier transform on $\tstS'$ by duality, and then restrict it to $L^2$.
\end{itemize}
Show that these two approaches are consistent with each other.
\item
Show that the Fourier transform acting on $L^1$ is not onto $\mathscr{C}_0$.
\item
Give an example of $u\in C(\R^n)$ such that $\varphi\mapsto\int u\varphi$ is a tempered distribution and that there is no polynomial $p$ satisfying
$|u(x)|\leq|p(x)|$ for all $x\in\R^n$.
\end{enumerate}
\item
For each of the following functions,
determine if it is a tempered distribution,
and if so compute its Fourier transform.
\begin{enumerate}[a)]
\item
$x\sin x$,
\item
$\frac1x\sin x$,
\item
$e^{i|x|^2}$,
\item
$x\vartheta(x)$, where $\vartheta$ is the Heaviside step function,
\item
$\mathrm{sgn}(x)=\vartheta(x)-\vartheta(-x)$.
\end{enumerate}
\item
Prove that a distribution $u\in\tstD'$ is tempered if and only if $u=\partial^\alpha f$ for some continuous function $f$
satisfying $|f(x)|\leq C(1+|x|)^{m}$ with some constants $C$ and $m$.
That is, tempered distributions are derivatives of functions of polynomial growth.
\item
\begin{enumerate}[a)]
\item
Let $a\in\mathscr{E}(\R^n)$.
Prove that the pointwise multiplication $u\mapsto au:\tstS'\to\tstS'$ is well-defined and continuous if and only if
$a\in\mathscr{O}_M$,
that is, for every multi-index $\alpha$ there is a polynomial $p$ such that
$|\partial^\alpha a(x)|\leq p(x)$, $x\in\R^n$.
\item
Prove that if $p$ is a polynomial with no real zeroes, then there are constants $c>0$ and $m$ such that $|p(\xi)|\geq c(1+|\xi|)^m$ for all $\xi\in\R^n$.
Operators $p(D)$ with $p$ satisfying this condition are called {\em strictly elliptic}.
\item
Show that if $p(D)$ strictly elliptic, then the equation $p(D)u=f$ has a solution for each $f\in\tstS'$.
\end{enumerate}
\item
Prove the followings.
\begin{enumerate}[a)]
\item
For a compactly supported distribution $u\in\tstE'$,
its Fourier transform is equal to
$$
\hat{u}(\xi) = \langle u(x), e^{-i\xi\cdot x}\rangle,
$$
where the notation $u(x)$ is to indicate that the distribution $u$ acts on $e^{-i\xi\cdot x}$ as a function of $x$.
The above expression also makes sense for $\xi\in\C^n$,
defining an entire analytic function $\hat{u}$.
(This is called the {\em Fourier-Laplace transform} of $u$.)
\item
The {\em Paley-Wiener-Schwartz theorem}:
Let $K\subset\R^n$ be a compact convex set, and let $\psi\in\tstS'$.
Then a necessary and sufficient condition for $\psi$ to be the Fourier transform of a distribution supported in $K$
is that $\psi$ is entire and satisfies the growth estimate
$$
|\psi(\zeta)| \leq C(1+|\zeta|)^{N} e^{I_K(\eta)},
\qquad \zeta=\xi+i\eta\in\C^n,
$$
with some constants $C$ and $N$.
Hence the Fourier-Laplace transform of a compactly supported distribution is an entire function of growth order at most $1$.
Recall that the indicator function $I_K$ is defined as
$$
I_K(\eta) = \sup_{x\in K} \eta\cdot x.
$$
\item
If the set of real zeroes of $p$ is bounded, then every tempered distribution solution of $p(D)u=0$ is an entire function of growth order at most $1$.
\end{enumerate}
\item
Let $p$ be a nonzero polynomial. Show the followings.
\begin{enumerate}[a)]
\item
The equation $p(D)u=f$ has at least one smooth solution for every $f\in\tstD$.
\item
If all solutions of $p(D)u=0$ are smooth,
then $\singsupp\,u\subset\singsupp\,p(D)u$ for any $u\in\tstD'$.
So hypoelliptic operators can be defined as those $p(D)$ such that all solutions of $p(D)u=0$ are smooth.
\item
If $p(D)$ admits a fundamental solution that is smooth outside some ball of finite radius (centred at the origin),
then $p(D)$ is hypoelliptic.
\end{enumerate}
\item
Recall that by H\"ormander's theorem, $p(D)$ is hypoelliptic if and only if
for any $\eta\in\R^n$ one has $p(\xi+i\eta)\neq0$ for all sufficiently large $\xi\in\R$.
\begin{enumerate}[a)]
\item
Construct a non-hypoelliptic polynomial $p$ in dimension $n>1$ such that
$|p(\xi)|\to\infty$ as $|\xi|\to\infty$ for $\xi\in\R^n$.
\item
For any given $c>0$, construct a non-hypoelliptic polynomial $p$ in dimension $n>1$ such that
$|p(\xi+i\eta)|\to\infty$ uniformly in $\{|\eta|\leq c\}$ as $|\xi|\to\infty$ for $\xi\in\R^n$.
\end{enumerate}
\end{enumerate}
\end{document}