\documentclass{hjm}
\ExecuteOptions{newlfont}\RequirePackage{newlfont}
%\newtheorem{thm}{Theorem}[section]
%\newtheorem{lemma}{Lemma}[section]
%\newtheorem{claim}{Claim}
%\newtheorem*{theorem}{Theorem}
%\newtheorem{Theorem}{Theorem 1}
%\newtheorem{ex}{Example}
%\renewcommand{\theTheorem}{}
%\renewcommand{\thetheorem}{}
%\newtheorem{prop}{Proposition}[section]
%\def\pfclm{\noindent {\em Proof of claim}\quad}
%\def\proof{\noindent{\em Proof}\quad}
%\def\pflm{\noindent {\em Proof of lemma}\quad}
%\def\pfthm{\noindent {\em Proof of theorem}\quad}
%\def\rse#1{\mbox{$^{\raisebox{.3ex}{$\scriptstyle{#1}$}}$}}
%\def\b{\begin{equation}}
%\def\e{\end{equation}}
\begin{document}
\title{Random walk and boundary behavior of functions in the disk.}
\author{Michael D. O'Neill}
\address{ Department of Mathematics, UTEP\\El Paso, TX 79968}
\email{michael@math.utep.edu}
\begin{abstract}
Simple martingale proofs of some results of Rohde~\cite{R1,R2} on the
boundary behavior of Bloch functions are presented, making clear their
connection with random walk in the plane.
\end{abstract}
\maketitle
\begin{center}
1991 mathematics subject classification: 30D45, 30D50.
\end{center}
\section{Introduction }
A function $f$, defined and analytic in the unit disk, is called a Bloch
function if
$$
\|f\| \sb {\cal B}=\sup \limits \sb {z\in \Bbb D}(1-|z|^2)|f'(z)| < \infty.
$$
We write $f \in \cal B$. The function is said to be in the little Bloch
space $\cal B \sb 0$ if
$$
(1-|z|^2)|f'(z)| \to 0 \qquad |z| \to 1 \quad z\in \Bbb D.
$$
The following proposition, which establishes a close connection between
Bloch functions and conformal mappings, is well known.(See \cite{Be,BeP})
\begin{prop}
If $g$ is a univalent function in $\Bbb D$ and $f=\log{g'}$ then $f \in \cal B$
and $\|f\|\sb {\cal B} \le 6$. Conversely, if $\|f\|\sb {\cal B} \le 1$
then there exists a univalent function $g$ such that $f=\log{g'}$ .
\end{prop}
In this note we use the device of Bloch martingales, developed
by Makarov in \cite{Mak} and briefly outlined below, to give
simple proofs of the following two theorems of S. Rohde.
\begin{thm}[Rohde]An inner function in the little Bloch space which is
not a finite Blaschke product has, for each $\delta \in \Bbb D$, the
non-tangential limit $\delta$ on a set $E\sb {\delta} \subset \Bbb T$ with
Hausdorff dimension 1.
\end{thm}
\begin{thm}[Rohde]Let $\{Q\sb n\}$ be a sequence of squares in the plane
with pairwise disjoint interiors, edges parallel to the
coordinate axes and of length $a>0$, and such that $Q\sb n$ is adjacent to
$Q\sb {n+1}$ for all $n$.
There is a universal constant $K>0$ such that if
$g\in \cal B$ has nontangential limits almost nowhere then there exists
a set $E$ with
$$
\text{dim } E \ge 1-a^{-1}K\|g\| \sb {\cal B}
$$
such that, for each $\zeta \in E$ we can find $r\sb n \to 1$ with
$$
g(r\zeta)\subset Q\sb n \cup Q\sb {n+1} \qquad \text{for } r\sb n \le r \le
r\sb {n+1}.
$$
\end{thm}
Given an arc $I\subset \Bbb D$ let $z\sb I= \tau (0)$ where $\tau$ is the
conformal self mapping of the disk which maps $\partial \Bbb D \cup \{\text
{Re }z>0\}$ onto $I$.
The theorems in section 2 will follow from the next lemma of
Makarov. (See
\cite{Mak}.)
\begin{lem}[Makarov]\label{mainlemma}
If $b\in \cal B$ and $I$ is an arc on $\partial \Bbb D$ then we have
$$
(*)\qquad \left | \frac 1{|I|}\int \limits \sb I (b(\zeta)-b(z\sb I))^n \,d
\zeta \right| \le Cn!\|b\|\sb {\cal B}^n \qquad n\ge1.
$$
Here, the integral is defined as the limit of the integrals of
$$
(b(r\xi) - b(z_{I}))^n
$$
as $r\to1$.
\end{lem}
\begin{proof}
We have
$$
\int \limits \sb 0 ^1 \left(\log{\frac{1+t}{1-t}}\right)^n \,dt
=
2\int \limits \sb {-\infty}^0 x^n \frac{e^x}{(1+e^x)^2}\,dx
\le
2\int \limits \sb {-\infty}^0 x^n e^x\,dx
\le 2n!\quad .
$$
If $b(0)=0$ and $I=[e^{-i\frac{\pi}2},e^{i\frac{\pi}2}]\subset \Bbb T$
then $(*)$ follows by twice integrating the inequality
$$
|b'(z)|\le \frac{\|b\|\sb{\cal B}}{1-|z|^2}
$$
on the imaginary axis from $-i$ to $i$ and applying the Cauchy
integral theorem. In general, let
$$
g(z)=b(\tau (z))-b(\tau(0))
$$
where $\tau$ is the self mapping of $\Bbb D$ used to define the point $z\sb I$.
We then have
$$
\left | \int \limits \sb I (b(\zeta)-b(z\sb I))^n \,d
\zeta \right| =\left|\int \limits \sb {-1}^1 (g(iy))^n\tau ' (iy)\,dy\right|
$$
and
$$
(1-|z|^2)|g'(z)|=(1-|\zeta|^2)|b'(\zeta)| \qquad \zeta=\tau(z).
$$
Now since
$$
|\tau ' (z)|=\frac{1-|z\sb I|^2}{|1+\overline{z\sb I}z|^2}\le C|I|,
$$
$(*)$ follows from the first case considered, and the proof is
complete.
\end{proof}
We remark that if $b \in \cal B \sb 0$ then we may replace the inequality
$$
|b'(z)|\le \frac{\|b\|\sb{\cal B}}{1-|z|^2}
$$
in the above proof by
$$
|b'(z)|\le \frac{\beta(1-|z|)}{1-|z|}
$$
for some function $\beta = \beta (\delta)$, determined by $b$, for which
$\beta \to 0$ as $\delta \to 0$ and obtain the inequalities
$$
\left | \frac 1{|I|}\int \limits \sb I (b(\zeta)-b(z\sb I))^n \,d
\zeta \right| \le Cn! \beta^n(|I|)\|b\|\sb {\cal B}^n \qquad n\ge1.
$$
For any $b\in \cal B$ let $b\sb r (z)=b(rz)$ for
all $01$, $E\sb n =\cup I\sb {n,k}$
where $I\sb {n,k}$ are disjoint closed arcs such that
for each $I\sb {n,k}$ there is a unique $I\sb {n-1,j}$ with
\begin{enumerate}
\item $I\sb {n,k}\subset I\sb {n-1,j}$
\item $|I\sb {n,k}|\le \epsilon|I\sb {n-1,j}|$
\item $\sum \limits \sb {i(j)} |I\sb {n,i}| \ge c|I\sb {n-1,j}| $, where
$i(j)$ runs through all indices such that $I\sb {n,i} \subset I\sb {n-1,j}$.
\end{enumerate}
Let $E=\bigcap \limits \sb n E\sb n $. Then with $\text{dim}E$ denoting the
Hausdorff dimension of $E$, we have
$$
\text{dim}E \ge 1-\frac{\log c}{\log{\epsilon}}.
$$
\end{lem}
Proofs appear in \cite{H} and \cite{P}.
\section{Martingale proofs of some results of Rohde}
The behavior of Bloch functions at the boundary of the disk is explained
by the following lemma, which is a slight refinement of lemma[5.6] in
\cite{Mak}.
\begin{lem} \label{randomwalklemma}
Let $M=\{M\sb n\}$ be a complex dyadic martingale determined by a Bloch function
$b$ as explained in section 1. Assume that $M\sb 0 = 0$ and that $|\Delta
M\sb n| \le 1$ for all $n$. Given $0<\alpha <\frac{\pi}2$, there exist
$00$ such that for all $a>a\sb 0$ we have
$$
m\left(\left \{|\text{arg}\, M\sb {\tau}|<\frac {\pi}2 -\alpha \right\}\right)
>C\sb {\alpha}
$$
where $\tau = \tau \sb a = \inf\{n:|M\sb n|\ge a\}$.
\end{lem}
\begin{proof}
By familiar properties of the Fejer kernel, if we are given $\eta >0$ we
may choose $m(\eta)$ so that if $0\le \rho (t)\le 1$ is nondecreasing and
left continuous on $-\pi \le t \le \pi$ then
$$
\sum \limits \sb {\nu =-m}^m \frac{m+1-|\nu|}{m+1}\int \limits \sb {-\pi}
^{\pi} \rho (t)e^{-i \nu t}\, dt \ge 1- \eta
$$
$$
\implies \int \limits \sb {-\frac{\pi}2+\alpha}
^{\frac{\pi}2-\alpha} \rho (t)\, dt >C\sb {\alpha} >0.
$$
For each dyadic interval $I$ we have by lemma \ref{mainlemma},
$$
(*)\qquad \left [(b-b\sb I)^n\right]\sb I \le C(n) n!\|b\| \sb {\cal B}^n
$$
and
$$
\qquad\left [(b-b\sb I)^n\right]\sb I =
(b^n)\sb I +\sum \limits \sb {k=1}^{n}(-1)^k\binom nk (b^{n-k})\sb I
b\sb I^k
$$
$$
(**)\qquad = (b^n)\sb I - b \sb I ^n + \sum \limits \sb {k=1}^{n-2}(-1)^k
\binom nk
\left[(b^{n-k})\sb I-b\sb I^{n-k}\right]b\sb I ^k
$$
where the last equality follows from the identity
$$
\sum \limits \sb {k=0}^n (-1)^k\binom nk =0.
$$
Letting $M^n$ denote the martingale determined by $b^n$ and with
$0\le n\le m(\alpha)$, we have
$$
\int M\sb {\tau}\, dm = 0
$$
and
\begin{eqnarray}
\int M\sb {\tau}^n\, dm&=&-\int\left[(M-M\sb {\tau})^n\right]\sb {\tau}\, dm
\nonumber\\ &+&\int \sum \limits \sb {k=1}^{n-2}(-1)^k \binom nk
\left[(M^{n-k})\sb {\tau}-M\sb {\tau}^{n-k}\right]M\sb {\tau} ^k\, dm .
\nonumber
\end{eqnarray}
Applying $(**)$ and $(*)$ to the terms in square brackets, we have by induction
that
$$
\left|\int M\sb {\tau}^n\, dm \right|
\le C(m) +C'(m)a^{n-2}\qquad 0\le n \le m(\alpha).
$$
Let $\rho (t),\quad-\pi \le t\le \pi$ be the distribution density of
$\text{arg}\,M\sb {\tau}$. From the above argument we have
$$
\left|\int \limits \sb {-\pi}^{\pi} e^{int}\rho (t) \, dt \right|
\le C(m)a^{-2} \qquad
0\le n \le m(\alpha)
$$
so that
$$
\sum \limits \sb {\nu =-m}^m \frac{m+1-|\nu|}{m+1}\int \limits \sb {-\pi}
^{\pi} \rho (t)e^{-i \nu t}\, dt \ge 1-C(m)a^{-2} \ge 1- \eta
$$
if $a$ is large enough.
By the first paragraph of the proof this implies that
$$
m\left(\left \{|\text{arg}\, M\sb {\tau}|<\frac {\pi}2 -\alpha \right\}\right)
>C\sb {\alpha}
$$
and completes the proof of the lemma.\end{proof}
We now use lemma \ref{randomwalklemma} to prove the theorems mentioned in the
introduction.
Recall that there are singular inner functions $S$ in the little Bloch
space
since there are singular measures with integrals in the little Zygmund
class,
(\cite{Piranian},\cite{Kahane},\cite{Sarason}). By a theorem of
Frostman
(see \cite{Garnett}), the functions
$$
\frac{S-\lambda}{1-\overline{\lambda}S} \qquad \lambda \in \mathbb D
$$
are non-finite Blaschke products except for a set of $\lambda
\in\mathbb D$ with logarithmic capacity zero.
Hungerford showed in \cite{H} that the zeros of such products must always
accumulate on a set with Hausdorff dimension 1. This result was
strengthened by Rohde who proved the following theorem \cite{R2}.
\begin{thm}[Rohde]An inner function in the little Bloch space which is
not a finite Blaschke product has, for each $\delta \in \Bbb D$, the
non-tangential limit $\delta$ on a set $E\sb {\delta} \subset \Bbb T$ with
Hausdorff dimension 1.
\end{thm}
\begin{proof}
Fix $\delta \in \Bbb D$ and choose $N$ such that $2^{-N}<\frac12(1-|\delta|)$.
Let $D\sb n $ denote the disk of radius $2^{-(N+n)}$ centered at $\delta$.
Since $b$ is not a finite Blaschke product, given $\eta >0$ we can find
a dyadic arc $I$ with $|I|<\eta$, $|b\sb I - \delta|<\eta$ and $(1-|z|^2)|b'(z)|<
\eta$ for all $z$ with $|z| >1- \frac{|I|}{2\pi}$. Taking $\eta >0$
sufficiently small and applying lemma \ref{randomwalklemma} to the function
$\frac{b-b\sb I}{\eta}$, we can find a dyadic interval $E\sb 0 = I^ 0$ with
$b\sb {I^0} \in D\sb 2$ and $|I^0|<\eta$. Because $|b| \to 1$ nontangentially
almost
everywhere, the set of maximal dyadic subintervals $\{J\sb j\}$ with
$b\sb {J\sb j} \notin D\sb 0$ has
$$
\sum \limits \sb j |J\sb j| = |I^0|
$$
and given $\epsilon >0$, if $\eta >0$ is small enough then
since $\eta$ controls $(1-|z|^2)|b'(z)|$ which in turn controls
$\Delta M_n$, we have
$|J\sb j|<\epsilon |I^0| $ for all $j$.
We apply lemma \ref{randomwalklemma} in each $J\sb j $ to obtain $00$ and $C>0$. In general, for $n>1$, when
$N\sb n $ is sufficiently large we may change the construction by switching
to the disks $(D\sb n, D\sb {n+2})$ in place of $(D\sb {n-1}, D\sb {n+1})$,
keeping the same $\epsilon, C >0$. By lemma \ref{dimensionlemma} the set $E= \bigcap \limits
\sb k E\sb {N\sb k}$ has
$$
\text{dim}\,E \ge 1- \frac{\log{C}}{\log{\epsilon}}
$$
and by lemma \ref{mainlemma} and the remark following it, $ b\to \delta$ non-tangentially
at each point of $E$. Since $\epsilon>0$ may be chosen arbitrarily small
in the above argument, the theorem is proved.
\end{proof}
\begin{thm}[Rohde]Let $\{Q\sb n\}$ be a sequence of squares in the plane
with pairwise disjoint interiors, edges parallel to the
coordinate axes and of length $a>0$, and such that $Q\sb n$ is adjacent to
$Q\sb {n+1}$ for all $n$.
There is a universal constant $K>0$ such that if
$b\in \cal B$ has nontangential limits almost nowhere then there exists
a set $E$ with
$$
\text{dim } E \ge 1-a^{-1}K\|g\| \sb {\cal B}
$$
such that, for each $\zeta \in E$ we can find $r\sb n \to 1$ with
$$
b(r\zeta)\subset Q\sb n \cup Q\sb {n+1} \qquad \text{for } r\sb n \le r \le
r\sb {n+1}.
$$
\end{thm}
\begin{proof}
We may assume that $\|b\| \sb {\cal B} \le 1$.
The theorem is then interesting for large values of $a$.
Let $Q'\sb n$ denote the square
of edge length $\frac{a}2$ concentric with $Q\sb n$ and with parallel edges.
Let $D\sb n$ denote the disk of radius $\frac{a}{20}$ concentric with the
square $Q\sb n$. Consider any two adjacent squares $Q\sb n $ and $Q\sb {n+1}$.
Let $R\sb n$ denote the smallest rectangle containing $Q'\sb n\cup Q'\sb {n+1}$.
For sufficiently large $a>0$ independent of $b\in \cal B$, if
$b\sb I \in D\sb n$ then by the assumption on the function $b$, finitely
many applications of lemma \ref{randomwalklemma} show that there
exists a universal constant $0