% LaTeX file -- H^p Spaces, Fall 2017, Assignment 1

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 950 $H^p$ Spaces \hfill Assignment 1 \hfill Due Thursday October 5.}} 

\vskip1em 

\begin{Ex}


\item 
\begin{parts}
\item Show that if $u \in h_p$ for $p\ge1$, then 
$|u(z)| \le \Big(\dfrac{1+|z|}{1-|z|}\Big)^{1/p} \|u\|_p$.
\item Show that there is a universal constant $C_{1,p}$ so that 
if $f\in H^p$ for $p\ge 1$, then
\[ |f'(z)| \le C_{1,p} (1-|z|)^{-1-1/p}\|f\|_p .\]
\hint consider $g(z) = \frac{f(z) - f(z_0)}{z-z_0}$.
\item Generalize to find a similar estimate for the $n$th derivative of $f \in H^p$.
\end{parts}

\item 
\begin{parts}
\item Show that if $K \subset \bC$ is compact and $v:K\to [-\infty,+\infty)$ is u.s.c.,
then there is a decreasing sequence $u_n$ of continuous functions decreasing to $v$.
\hint $u_n(x) = \sup_{y\in K} v(y) - n |x-y|$.
\item Show that is $v$ is subharmonic on a connected open set $\Omega$ and is
not constantly $-\infty$, then for every $\ol{D_r(z_0)} \subset \Omega$,
\[ \frac1{2\pi} \int_{-\pi}^\pi v(z_0+re^{it}) \,dt > - \infty .\]
\item Show that if $v_1$ and $v_2$ are subharmonic on $\Omega$, then so is
$\max\{v_1,v_2\}$.
\item Prove that if $v$ is $C^2$ and $\Delta v \ge 0$ on $\Omega$, then $v$ is subharmonic.
\end{parts}

\item \begin{parts}
\item Rewrite the formula for the conjugate function of $u=f*P$ for $f\in L^1$ as follows:
\[
 \tilde u(re^{i\theta}) = 
 \frac1{2\pi} \int_0^\pi \Big( \frac{4r\sin^2(t/2)}{(1-r)^2 +4r\sin^2(t/2)} \Big) 
 \frac{f(\theta - t) - f(\theta+t)}{\tan(t/2)} \,dt .
\] 
Hence show that if 
$\dfrac1{2\pi} \dint_0^\pi \Big| \frac{f(\theta - t) - f(\theta+t)}{\tan(t/2)} \Big| \,dt < \infty$,
then $\tilde u$ has a radial limit $\dlim_{r\to 1^-}  \tilde u(re^{i\theta})$.

\item  Hence show that if $f$ is $C^1$ on an arc $I \subset \bT$, then for every
compact arc $K \subset I$, $ \tilde u(re^{i\theta})$ converges
to $\frac1{2\pi} \int_0^\pi \frac{f(\theta - t) - f(\theta+t)}{\tan(t/2)} \,dt$
\textit{uniformly} as $z \in \ol{\bD}$ approaches $e^{i\theta} \in K$.

\item In particular, show that $\tilde u$ extends to be continuous on $\bD \cup I$.
\end{parts}

\item Let $K$ be a compact subset of $\bT$ of Lebesgue measure zero.
\begin{parts}
\item Find a positive function $f\in L^1$ such that $f$ is $C^\infty$ on $\bT\setminus K$ and
$\dlim_{t\to t_0} f(t) = +\infty$ for every $t_0\in K$.

\item Define $u=f*P$ and let $\tilde u$ be its harmonic conjugate.
Set $g(z) = \dfrac{u+i\tilde u}{1+u+i\tilde u}$.\\[.3ex] 
 Prove that $g\in\AD$ (i.e. $g$ extends to be
continuous on the closed disc $\ol{\bD}$),\\
that $g|_K=1$ and $|g|<1$ on $\ol{\bD}\setminus K$.

\item If $\mu \in \MT$ is of analytic type, evaluate ${\dlim_{k\to\infty}} \int g^k \,d\mu$ in two ways.
Hence provide another proof of the F.\ and M.\ Riesz Theorem.
\end{parts}

\item  
\begin{parts}
\item Show that if $f \in H^p$ and $w$ is an analytic function from $\bD$ into itself,
then $f(w(z))$ belongs to $H^p$. \hint use the least harmonic majorant. \vspace{-.4ex}
\item Show that $f(z) = (1-z)^{-1}$ is in $H^p$ for $0 < p < 1$, but not $p=1$.
\item Show that if $f$ is analytic on $\bD$ and $\re f(z) > 0$, then $f \in H^p$
for $0 < p < 1$.
\item Generalize to analytic functions with values in a sector 
$S=\{z : \alpha < \arg f(z) < \beta \}$..
\end{parts}

\end{Ex}
\end{document}