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\begin{document}
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\large



\title{Math 533, Spring 2020, HW8} 
\author{Due in class, April 13}
\maketitle

\begin{problem}
 A distribution $F$ is called harmonic if $\sum_1^n \partial_j^2 F = 0$.  If $F$ is harmonic and tempered, prove that $F$ is a polynomial.
\end{problem}

\begin{problem}
 Let $f$ be a continuous function on $\bR^n \setminus \{0\}$ which is homogeneous of degree $-n$, $f(rx) = r^{-n}f(x)$ for $r > 0$, and has mean 0 on the unit sphere.  Show that $f$ is not locally integrable near 0 unless $f = 0$, but $f$ defines a tempered distribution $PV(f)$ by
 \[
  \langle PV(f), \phi \rangle = \lim_{\epsilon \to 0} \int_{|x|>\epsilon} f(x) \phi(x) dx.
 \]
The limit equals
\[
 \int_{|x| \leq 1} f(x)[\phi(x)-\phi(0)]dx + \int_{|x|>1} f(x)\phi(x)dx,
\]
and these integrals converge absolutely.

\end{problem}


\begin{problem}
 On $\bR$, let $F = PV((\pi x)^{-1})$. Check that 
 \begin{enumerate}
  \item $\hat{F}(\xi) = -i \sgn \xi$.
  \item The map $\phi \to F*\phi$, initially defined on Schwartz functions, extends to a unitary operator on $L^2$.
 \end{enumerate}
This is called the Hilbert transform.
\end{problem}

%\begin{problem}
% The following construction gives a distribution which behaves like $|x|^{-\lambda}$ on $\bR^n \setminus \{0\}$. 
% \begin{enumerate}
%  \item If $\phi \in C_c^\infty$, let $P_\phi^k$ be the Taylor polynomial of $\phi$ about $x=0$ of degree $k$.  If $k > \lambda-n-1$ and $a>0$, define
%  \[
%  \langle F_a^k, \phi \rangle = \int_{|x| \leq a} \left[ \phi(x) - P_\phi^k(x)\right]|x|^{-\lambda} dx + \int_{|x|>a} \phi(x)|x|^{-\lambda} dx.
%  \]
% \item If $\lambda$ is not an integer and we choose $k$ to satisfy $\lambda-n-1 <k< \lambda-n$, we can let $a \to \infty$ and define 
% \[
%  \langle F, \phi \rangle = \int [\phi(x)-P_\phi^k(x)]|x|^{-\lambda} dx.
% \]

% \end{enumerate}

%\end{problem}

\begin{problem}
 Give $C^\infty(\bT^n)$ the Fr\'echet space topology defined by the seminorms $\|\phi\|_{(\alpha)} = \|\partial^\alpha \phi\|_\infty$.  The space $\sD'(\bT^n)$ of distributions on $\bT^n$ is the space of continuous linear functionals on $C^\infty(\bT^n)$, with the weak * topology. Prove the following.
 \begin{enumerate}
  \item Distributions on $\bT^n$ can be translated, differentiated, and multiplied by $C^\infty$ functions, just as on $\bR^n$.
  \item If $F \in \sD'(\bT^n)$, its Fourier transform is the function $\hat{F}$ on $\zed^n$ defined by $\hat{F}(\kappa) = \langle F, E_\kappa\rangle$ where $E_\kappa(x) = e^{-2\pi i \kappa \cdot x}$.  Prove that a function $g$ on $\zed^n$ is the Fourier transform of a distribution on $\bT^n$ iff $|g(\kappa)|\leq C(1 + |\kappa|)^N$ for some $C, N>0$.
  \item If $F \in \sD'(\bT^n)$ and $\phi \in C^\infty(\bT^n)$, prove $\langle F, \overline \phi \rangle = \sum_\kappa \hat{F}(\kappa)\overline{\hat{\phi}(\kappa)}.$
 \end{enumerate}

\end{problem}

%\begin{problem}
% Show that there exist constants $C_1, C_2 > 0$ such that for all $\xi \in \bR^n$,
% \[
%  C_1 (1 + |\xi|^2)^k \leq \sum_{|\alpha|\leq k}|\xi^\alpha|^2 \leq C_2 (1 + |\xi|^2)^k.
% \]

%\end{problem}


\begin{problem}
 If $k \in \bN$, $H_k$ is the space of $L^2$ functions whose $L^2$ derivatives up to order $k$ exist.  Show that these strong derivatives coincide with the distribution derivatives.
\end{problem}

%\begin{problem}
% Suppose $r < s < t$. Show that for any $\epsilon>0$ there exists $C>0$ such that $\|f\|_{(s)} \leq \epsilon\|f\|_{(t)} + C \|f\|_{(r)}$.
%\end{problem}

%\begin{problem}
% Check that if $H_s \subset C_0^k$, then $s > k + (n/2)$.
%\end{problem}

%\begin{problem}
% Suppose $s_0 \leq s_1$, $t_0 \leq t_1$.  If $T$ is a bounded linear map from $H_{s_0}$ to $H_{t_0}$ whose restriction to $H_{s_1}$ is bounded from $H_{s_1}$ to $H_{t_1}$, then the restriction of $T$ to $H_{s_\lambda}$ is bounded from $H_{s_\lambda}$ to $H_{t_\lambda}$ for $0 < \lambda < 1$, where $s_\lambda = (1-\lambda)s_0 + \lambda s_1$ and $t_\lambda = (1-\lambda)t_0 + \lambda t_1$.
%\end{problem}



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