# Convergence of linear combinations of iterates of an inner function

Artur Nicolau
Departament de Matemàtiques Universitat Autònoma de Barcelona and Centre de
Recerca Matemàtica 08193 Barcelona<EMAIL_ADDRESS>Supported in part by the
Generalitat de Catalunya (grant 2017 SGR 395) and the Spanish Ministerio de
Ciencia e Innovación (project MTM2017-85666-P)

###### Abstract

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let
$f^{n}$ be its $n$-th iterate. Let $\\{a_{n}\\}$ be a sequence of complex
numbers. We prove that the series $\sum a_{n}f^{n}(\xi)$ converges at almost
every point $\xi$ of the unit circle if and only if $\sum|a_{n}|^{2}<\infty$.
The main step in the proof is to show that under this assumption, the function
$F=\sum a_{n}f^{n}$ has bounded mean oscillation. We also prove that $F$ is
bounded on the unit disc if and only if $\sum|a_{n}|<\infty$. Finally we
describe the sequences of coefficients $\\{a_{n}\\}$ such that $F$ belongs to
other classical function spaces, as the disc algebra and the Dirichlet class.

##  1 Introduction

Inner functions are analytic mappings from the unit disc $\mathbb{D}$ of the
complex plane into itself whose radial limits are of modulus one at almost
every point of the unit circle $\partial\mathbb{D}$. Inner functions are a
central notion in Complex and Functional Analysis. See for instance [Ga]. Any
inner function $f$ induces a map from $\partial\mathbb{D}$ into itself defined
at almost every point $\xi\in\partial\mathbb{D}$ by $f(\xi)=\lim\limits_{r\to
1}f(r\xi)$. Let $f^{n}$ denote the $n$-th iterate of $f$, which is again an
inner function. The main purpose of this paper is to study the convergence of
linear combinations of iterates of an inner function and to show that in a
certain sense, the iterates
$f^{n}\colon\partial\mathbb{D}\to\partial\mathbb{D}$ behave as independent
random variables. This phenomena has been extensively studied in the context
of random series of functions ([Ka]) but it also occurs in other settings
where independence is not present, as in the theory of lacunary series (see
Section 6 of Chapter V and Section 5 of Chapter XVI of [Zy]).

We start discussing pointwise convergence. Let $\\{X_{n}\\}$ be a sequence of
independent random variables with mean $0$ and finite variances $V(X_{n})$.
The classical Khintchine–Kolmogorov Convergence Theorem asserts that $\sum
X_{n}$ converges almost surely if and only if $\sum V(X_{n})<\infty$. In our
context we have the following analogous result.

######  Theorem 1.1.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let
$\\{a_{n}\\}$ be a sequence of complex numbers. Then the following conditions
are equivalent:

  1. (a)

The series $\sum\limits_{n=1}^{\infty}a_{n}f^{n}(\xi)$ converges at almost
every point $\xi\in\partial\mathbb{D}$.

  2. (b)

The set
$\Bigl{\\{}\xi\in\partial\mathbb{D}:\sup\limits_{N}\Bigl{|}\sum\limits_{n=1}^{N}a_{n}f^{n}(\xi)\Bigr{|}<\infty\Bigr{\\}}$
has positive Lebesgue measure.

  3. (c)

The complex numbers $\\{a_{n}\\}$ satisfy
$\sum\limits_{n=1}^{\infty}|a_{n}|^{2}<\infty$.

Let $m$ denote normalized Lebesgue measure in $\partial\mathbb{D}$. For
$0<p<\infty$ let $\mathbb{H}^{p}(\mathbb{D})$ be the classical Hardy space of
analytic functions $F$ in $\mathbb{D}$ such that

$\|F\|_{p}^{p}=\sup_{r<1}\int_{\partial\mathbb{D}}|F(r\xi)|^{p}\,dm(\xi)<\infty.$

Let $\mathbb{H}^{\infty}(\mathbb{D})$ be the algebra of bounded analytic
functions $F$ in $\mathbb{D}$ and
$\|F\|_{\infty}=\sup\\{|F(z)|:z\in\mathbb{D}\\}$. Let $\operatorname{BMOA}$ be
the space of analytic functions $F$ in $\mathbb{D}$ such that

$\|F\|^{2}_{\operatorname{BMOA}(\mathbb{D})}=\sup_{z\in\mathbb{D}}\int_{\partial\mathbb{D}}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)<\infty.$

Here $P(z,\xi)=(1-|z|^{2})|\xi-z|^{-2}$ is the Poisson kernel. The subspace
$\operatorname{VMOA}$ is formed by the functions $F\in\operatorname{BMOA}$
such that the integral above tends to $0$ as $|z|$ tends to $1$. Any function
$F\in\mathbb{H}^{p}(\mathbb{D})$, $0<p\leq\infty$, has non-tangential limit,
denoted by $F(\xi)$, at almost every point $\xi\in\partial\mathbb{D}$.
Moreover if $p\geq 1$, then

$F(z)=\int_{\partial\mathbb{D}}F(\xi)P(z,\xi)\,dm(\xi),\quad z\in\mathbb{D}.$

We have
$\mathbb{H}^{\infty}(\mathbb{D})\subseteq\operatorname{BMOA}(\mathbb{D})\subseteq\mathbb{H}^{p}(\mathbb{D})$
for any $0<p<\infty$. Actually $\operatorname{BMOA}(\mathbb{D})$ is the
natural substitute of $\mathbb{H}^{\infty}(\mathbb{D})$ in several results of
the theory of Hardy spaces. See [Ga] for all these well known results.

The proof of Theorem 1.1 is based in the following $\operatorname{BMOA}$ type
estimate.

######  Theorem 1.2.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Then there
exists a constant $C=C(f)>0$ such that for any positive integer $N$, any set
$\\{a_{n}:n=1,\dots,N\\}$ of complex numbers and any point $z\in\mathbb{D}$,
we have

$\int_{\partial\mathbb{D}}\left|\sum_{n=1}^{N}a_{n}(f^{n}(\xi)-f^{n}(z))\right|^{2}P(z,\xi)\,dm(\xi)\leq
C\sum^{N}_{n=1}|a_{n}|^{2}(1-|f^{n}(z)|^{2}).$

Let $f$ be an analytic selfmapping of $\mathbb{D}$ with $f(0)=0$ which is not
a rotation. The classical Denjoy–Wolff Theorem asserts that $f^{n}$ tends to
$0$ uniformly on compact sets of $\mathbb{D}$. See Chapter V of [Sh]. The
proof of Theorem 1.2 uses the interplay between those dynamical properties of
the inner function $f$ as a self mapping of $\partial\mathbb{D}$ and those as
a self mapping of $\mathbb{D}$. In particular we use a result due to
Pommerenke ([Po]) which provides an exponential decay in the Denjoy–Wolff
Theorem.

It was proved in [NS2] that $\sum a_{n}f^{n}$ converges in
$\mathbb{H}^{2}(\mathbb{D})$ if and only if $\sum|a_{n}|^{2}<\infty$. As a
consequence of Theorem 1.2 we will show that under this last condition, we
actually have $\sum a_{n}f^{n}\in\operatorname{BMOA}(\mathbb{D})$.

######  Corollary 1.3.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let
$\\{a_{n}\\}$ be a sequence of complex numbers. Assume
$\sum\limits^{\infty}_{n=1}|a_{n}|^{2}<\infty$.

  1. (a)

Then
$F=\sum\limits_{n=1}^{\infty}a_{n}f^{n}\in\operatorname{BMOA}(\mathbb{D})$ and
for almost every $\xi\in\partial\mathbb{D}$ we have

(1.1) $\lim_{r\to 1}F(r\xi)=\sum^{\infty}_{n=1}a_{n}f^{n}(\xi).$

Moreover there exists a constant $C=C(f)>0$ only depending on $f$, such that

$C^{-1}\sum^{\infty}_{n=1}|a_{n}|^{2}\leq\|F\|^{2}_{\operatorname{BMOA}(\mathbb{D})}\leq
C\sum^{\infty}_{n=1}|a_{n}|^{2}.$

  2. (b)

If $f$ is a finite Blaschke product, then
$F=\sum\limits^{\infty}_{n=1}a_{n}f^{n}\in\operatorname{VMOA}$.

The following Khintchine type estimate follows easily from the previous
result.

######  Corollary 1.4.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let
$\\{a_{n}\\}$ be a sequence of complex numbers. Assume
$\sum\limits^{\infty}_{n=1}|a_{n}|^{2}<\infty$. Then
$\sum\limits^{\infty}_{n=1}a_{n}f^{n}\in\mathbb{H}^{p}(\mathbb{D})$ for any
$0<p<\infty$. Moreover for any $0<p<\infty$, there exists a constant
$C(p,f)>0$ such that for any sequence of complex numbers $\\{a_{n}\\}$ we have

$C(p,f)^{-1}\left(\sum^{\infty}_{n=1}|a_{n}|^{2}\right)^{1/2}\leq\left\|\sum_{n=1}^{\infty}a_{n}f^{n}\right\|_{p}\leq
C(p,f)\left(\sum^{\infty}_{n=1}|a_{n}|^{2}\right)^{1/2}.$

By Theorem 1.1, if $\sum|a_{n}|^{2}<\infty$, the series $\sum a_{n}f^{n}(\xi)$
converges at almost every $\xi\in\partial\mathbb{D}$, while if
$\sum|a_{n}|^{2}=\infty$, the series $\sum a_{n}f^{n}(\xi)$ converges at
almost no point $\xi\in\partial\mathbb{D}$. If the coefficients satisfy the
stronger condition $\sum|a_{n}|<\infty$, it is clear that $\sum
a_{n}f^{n}\in\mathbb{H}^{\infty}(\mathbb{D})$. Our next result says that this
condition is also necessary.

######  Theorem 1.5.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Let
$\\{a_{n}\\}$ be a sequence of complex numbers with
$\sum\limits_{n=1}^{\infty}|a_{n}|^{2}<\infty$ and
$F=\sum\limits_{n=1}^{\infty}a_{n}f^{n}$. Assume there exists an arc
$I\subset\partial\mathbb{D}$ such that $\sup\\{|F(\xi)|:\xi\in I\\}<\infty$.
Then $\sum\limits_{n=1}^{\infty}|a_{n}|<\infty$.

The proof of Theorem 1.5 is the most technical part of the paper. The main
idea is to construct by induction a sequence of arcs $I_{k}\subset I$ and a
sequence $M_{k}$ of positive integers tending to infinity such that

(1.2) $\frac{1}{m(I_{k})}\int_{I_{k}}\operatorname{Re}F\,dm\geq
A\sum^{M_{k}}_{j=1}|a_{j}|-B,\quad k=1,2,\dotsc,$

where $A$ and $B$ are positive constants independent of $k$. We start showing
that if $|f(z)|$ is sufficiently small, the two terms in the conclusion of
Theorem 1.2 are comparable. This provides a way of finding arcs $I_{k}\subset
I$ and of splitting $F=\sum\limits_{j}F_{j}$ into blocks $F_{k}$ such that

$\frac{1}{m(I_{k})}\int_{I_{k}}\sum_{j=1}^{k}\operatorname{Re}F_{j}\,dm\geq
C\sum^{k}_{j=1}\|\operatorname{Re}F_{j}\|_{2},\quad k=1,2,\dotsc$

Since the blocks $F_{j}$ have a uniformly bounded number of terms,
$\|\operatorname{Re}F_{j}\|_{2}$ can be estimated from below by the sum of the
modulus of the corresponding coefficients and (1.2) follows easily.

Let $\overline{\mathbb{H}^{\infty}(\mathbb{D})}$ denote the closure of
$\mathbb{H}^{\infty}(\mathbb{D})$ in $\operatorname{BMOA}(\mathbb{D})$. This
space was studied in [GJ]. See also [NS1] and [SS]. Theorems 1.1, 1.5 and
Corollary 1.3 lead to the following result.

######  Corollary 1.6.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Let
$\\{a_{n}\\}$ be a sequence of complex numbers with
$\sum\limits_{n=1}^{\infty}|a_{n}|^{2}<\infty$ but
$\sum\limits_{n=1}^{\infty}|a_{n}|=\infty$. Then
$F=\sum\limits^{\infty}_{n=1}a_{n}f^{n}\in\overline{\mathbb{H}^{\infty}(\mathbb{D})}$.
Moreover

$F(\xi)=\lim_{r\to 1}F(r\xi)=\sum^{\infty}_{n=1}a_{n}f^{n}(\xi)$

at almost every $\xi\in\partial\mathbb{D}$ but $\sup\\{|F(\xi)|:\xi\in
I\\}=\infty$ for any arc $I\subset\partial\mathbb{D}$.

Theorems 1.1 and 1.5 and Corollaries 1.3 and 1.4 can be understood as
analogues to classical results in the theory of lacunary series (see Section 6
of Chapters V and VI of [Zy]). However it should be noted that no lacunarity
assumption is needed in our results.

The paper is organized as follows. Next section is devoted to the proof of
Theorems 1.1, 1.2 and Corollaries 1.3 and 1.4. In Section 3 we start
collecting several auxiliary results which are used in the proof of Theorem
1.5. In Section 4 we describe in terms of the coefficients, those linear
combinations of iterates which belong to certain classical function spaces,
such as the disc algebra, the Dirichlet space or the Bloch space. Finally last
section is devoted to state some related open questions we have not explored.

It is a pleasure to thank Odí Soler i Gibert for some very helpful comments.

##  2 BMO estimates and pointwise convergence

Let $\rho(z,w)$ denote the pseudohyperbolic distance between the points
$z,w\in\mathbb{D}$ defined as $\rho(z,w)=|z-w||1-\overline{w}z|^{-1}$.
Schwarz’s Lemma asserts that any analytic selfmapping $f$ of $\mathbb{D}$
contracts hyperbolic distances, that is, $\rho(f(z),f(w))\leq\rho(z,w)$ for
any $z,w\in\mathbb{D}$. Equivalently ${D}_{h}(f)(z)\leq 1$, $z\in\mathbb{D}$,
where

(2.1) $D_{h}(f)(z)=\frac{(1-|z|^{2})|f^{\prime}(z)|}{1-|f(z)|^{2}},\quad
z\in\mathbb{D},$

is the hyperbolic derivative of $f$ at the point $z$. Our first auxiliary
result is a quantitative version of the Denjoy–Wolff Theorem which is
essentially due to Pommerenke ([Po]). Its short proof is included for the sake
of completeness.

######  Lemma 2.1.

Let $f\in H^{\infty}(\mathbb{D})$, $\|f\|_{\infty}\leq 1$ with $f(0)=0$ which
is not a rotation.

  1. (a)

Then, for any $0<r<1$, there exists $0<c=c(r,f)<1$ such that $1-|z|\leq
c(1-|f(z)|)$ if $|z|\geq r$.

  2. (b)

Assume $f^{\prime}(0)\neq 0$. Then there exists $0<r_{0}=r_{0}(f)<1$ such that

$|f^{n}(z)|\leq r_{0}^{-1}|f^{\prime}(0)|^{n}|z|,\quad n\geq
1,\quad\text{if}\quad|z|\leq r_{0}.$

  3. (c)

Assume $f^{\prime}(0)=0$. Then $|f^{n}(z)|\leq|z|^{2^{n}}$, $z\in\mathbb{D}$.

###### Proof.

Since $f(0)=0$, Schwarz’s Lemma gives

$\rho\left(\frac{f(z)}{z},f^{\prime}(0)\right)\leq|z|,\quad z\in\mathbb{D}.$

Denote $\psi(z)=z(z+|f^{\prime}(0)|)(1+|f^{\prime}(0)|z)^{-1}$,
$z\in\mathbb{D}$, to obtain

(2.2) $|f(z)|\leq\psi(|z|),\quad z\in\mathbb{D}.$

Then

$1-|f(z)|\geq 1-\psi(|z|)=(1-|z|)\frac{1+|z|}{1+|f^{\prime}(0)||z|}.$

Given $0<r<1$, taking $c=(1+r)^{-1}(1+|f^{\prime}(0)|r)$ the estimate in (a)
follows.

We now prove (b). For $n=1,2,\dotsc$, consider the function
$g_{n}(z)=\psi^{n}(z)|f^{\prime}(0)|^{-n}$. It is known that $\\{g_{n}\\}$
converges uniformly on compact sets of $\mathbb{D}$ to an analytic function
$g$ on $\mathbb{D}$, known as the Königs function of $\psi$, which satisfies
$g(\psi(z))=|f^{\prime}(0)|g(z)$, $z\in\mathbb{D}$. See [Sh, pp. 89–93].
Moreover for $0\leq x\leq 1$ we have

$g_{n+1}(x)=\frac{\psi^{n+1}(x)}{|f^{\prime}(0)|^{n+1}}=g_{n}(x)\frac{1+|f^{\prime}(0)|^{n-1}g_{n}(x)}{1+|f^{\prime}(0)|^{n+1}g_{n}(x)}\geq
g_{n}(x).$

Hence $g_{n}(x)\leq g(x)$, $n=1,2,\dotsc,0<x<1$. Note that there exists a
constant $\delta=\delta(|f^{\prime}(0)|)>0$ such that $\psi$ is univalent in
$\\{z\in\mathbb{D}:|z|<\delta\\}$. Hence $g_{n}$ is also univalent in
$\\{z\in\mathbb{D}:|z|<\delta\\}$. By Koebe Distortion Theorem, there exists
$0<r_{0}=r_{0}(f)<1$ such that $|g(w)|<1$ if $|w|<r_{0}$. By Schwarz’s Lemma
we deduce $|g(w)|\leq r_{0}^{-1}|w|$ if $|w|<r_{0}$. Applying (2.2), for any
$|z|<r_{0}$ we have

$|f^{n}(z)|\leq\psi^{n}(|z|)\leq|f^{\prime}(0)|^{n}g(|z|)\leq
r_{0}^{-1}|f^{\prime}(0)|^{n}|z|,\quad n=1,2,\dotsc$

This proves (b). Assume now $f^{\prime}(0)=0$. Note that (2.2) gives
$|f(z)|\leq|z|^{2}$, $z\in\mathbb{D}$. Iterating we obtain (c). ∎

For future reference we state the following easy consequence of Lemma 2.1.

######  Corollary 2.2.

Let $f\in\mathbb{H}^{\infty}(\mathbb{D})$, $\|f\|_{\infty}\leq 1$ with
$f(0)=0$ which is not a rotation. Then there exist constants
$0<r_{0}=r_{0}(f)<1$ and $0<c_{0}=c_{0}(f)<1$ such that

$|f^{n}(z)|\leq r_{0}^{-1}c_{0}^{n}|z|\quad\text{if}\quad|z|\leq r_{0}.$

###### Proof.

If $f^{\prime}(0)\neq 0$ let $r_{0}=r_{0}(f)$ be the constant given by part
(b) of Lemma 2.1 and $c_{0}=|f^{\prime}(0)|$. If $f^{\prime}(0)=0$, pick any
$0<r_{0}<1$. Part (c) of Lemma 2.1 gives $|f^{n}(z)|\leq r_{0}^{2^{n}-1}|z|$,
$|z|\leq r_{0}$. Since $2^{n}-1\geq n$, $n\geq 1$, we can take $c_{0}=r_{0}$.
∎

For future reference we also state the following auxiliary result whose proof
can be found in [NS2, Theorem 9].

######  Lemma 2.3.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Let
$\\{a_{n}\\}$ be a sequence of complex numbers. Then

$\frac{1-|f^{\prime}(0)|}{1+|f^{\prime}(0)|}\sum^{N}_{n=1}|a_{n}|^{2}\leq\left\|\sum^{N}_{n=1}a_{n}f^{n}\right\|_{2}^{2}\leq\frac{1+|f^{\prime}(0)|}{1-|f^{\prime}(0)|}\sum^{N}_{n=1}|a_{n}|^{2},\quad
N=1,2,\dotsc$

We are now ready to prove Theorem 1.2.

###### Proof of Theorem 1.2.

Without loss of generality we can assume $f^{n}(z)\neq 0$ for any $n\geq 1$.
Denote $F=\sum\limits^{N}_{n=1}a_{n}f^{n}$. Then

$\int_{\partial\mathbb{D}}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)=\sum^{N}_{n=1}|a_{n}|^{2}c_{n,n}+2\operatorname{Re}\sum^{N-1}_{n=1}\sum^{N}_{k>n}\overline{a}_{n}a_{k}c_{k,n},$

where

$c_{k,n}=\int_{\partial\mathbb{D}}\overline{(f^{n}(\xi)-f^{n}(z))}(f^{k}(\xi)-f^{k}(z))P(z,\xi)\,dm(\xi),\quad
k,n=1,\dotsc,N.$

For $k\geq n$, the function
$\overline{f^{n}(\xi)}f^{k}(\xi)=f^{k}(\xi)/f^{n}(\xi)$,
$\xi\in\partial\mathbb{D}$, has an analytic extension to $\mathbb{D}$. Then

$c_{k,n}=\frac{f^{k}(z)}{f^{n}(z)}-f^{k}(z)\overline{f^{n}(z)}=\frac{f^{k}(z)(1-|f^{n}(z)|^{2})}{f^{n}(z)},\quad
k\geq n.$

Hence

(2.3)
$\int_{\partial\mathbb{D}}|F(\xi)-F(z)|^{2}P(z,\xi)=A+2\operatorname{Re}B,$

where

$\displaystyle A$ $\displaystyle=\sum^{N}_{n=1}|a_{n}|^{2}(1-|f^{n}(z)|^{2}),$
$\displaystyle B$
$\displaystyle=\sum^{N-1}_{n=1}\sum^{N}_{k>n}\overline{a}_{n}a_{k}\frac{f^{k}(z)(1-|f^{n}(z)|^{2})}{f^{n}(z)}.$

The idea is that the cross term $B$ can be estimated by the diagonal term $A$.
Let $0<r_{0}=r_{0}(f)<1$ and $0<c_{0}=c_{0}(f)<1$ be the constants appearing
in Corollary 2.2. The Denjoy-Wolff Theorem says that the iterates $f^{n}$
converge to $0$ uniformly on compacts of $\mathbb{D}$. So we can consider the
smallest positive integer $\ell=\ell(z)$ such that $|f^{\ell}(z)|\leq r_{0}$.
Corollary 2.2 gives that $|f^{k}(z)|\leq r_{0}^{-1}c_{0}^{k-n}|f^{n}(z)|$ if
$k\geq n>\ell$. Assume $\ell<N$. We have

$\sum^{N-1}_{n\geq\ell}|a_{n}|\frac{1-|f^{n}(z)|^{2}}{|f^{n}(z)|}\sum^{N}_{k>n}|a_{k}||f^{k}(z)|\leq
r_{0}^{-1}\sum^{N-1}_{n\geq\ell}|a_{n}|\sum^{N}_{k>n}|a_{k}|c_{0}^{k-n}.$

Writting $j=k-n$ and applying Cauchy–Schwarz’s inequality, last double sum can
be bounded by

$\sum^{N-\ell}_{j=1}c_{0}^{j}\sum^{N-j}_{n\geq\ell}|a_{n}||a_{n+j}|\leq(1-c_{0})^{-1}\sum^{N}_{n\geq\ell}|a_{n}|^{2}.$

Since $|f^{n}(z)|\leq|f^{\ell}(z)|\leq r_{0}$ for $n\geq\ell$, last expression
above is bounded by

$(1-r_{0}^{2})^{-1}(1-c_{0})^{-1}\sum^{N}_{n\geq\ell}|a_{n}|^{2}(1-|f^{n}(z)|^{2}).$

Hence we only need to estimate $B_{1}+B_{2}$ where

$\displaystyle B_{1}$
$\displaystyle=\sum^{\ell-1}_{n=1}|a_{n}|\frac{1-|f^{n}(z)|^{2}}{|f^{n}(z)|}\sum^{\ell}_{k>n}|a_{k}||f^{k}(z)|,$
$\displaystyle B_{2}$
$\displaystyle=\sum^{\ell-1}_{n=1}|a_{n}|\frac{1-|f^{n}(z)|^{2}}{|f^{n}(z)|}\sum^{N}_{k>\ell}|a_{k}||f^{k}(z)|.$

Since $|f^{k}(z)|\leq|f^{n}(z)|$ for $k\geq n$, writting $j=k-n$ we obtain

$B_{1}\leq\sum^{\ell-1}_{n=1}|a_{n}|(1-|f^{n}(z)|^{2})\sum^{\ell}_{k>n}|a_{k}|\leq\sum^{\ell}_{j=1}\sum^{\ell-j}_{n=1}|a_{n}||a_{n+j}|(1-|f^{n}(z)|^{2}).$

Since $|f^{n}(z)|\geq r_{0}$ for $n<\ell$, part (a) of Lemma 2.1 provides a
constant $0<c_{1}=c_{1}(f)<1$ such that $1-|f^{n}(z)|\leq
c_{1}^{j}(1-|f^{n+j}(z)|)$ for any $n<n+j\leq\ell$. Hence

$B_{1}\leq
2\sum^{\ell}_{j=1}c_{1}^{j/2}\sum^{\ell-j}_{n=1}|a_{n}|(1-|f^{n}(z)|)^{1/2}|a_{n+j}|(1-|f^{n+j}(z)|)^{1/2}.$

Applying Cauchy–Schwarz’s inequality, we deduce

$B_{1}\leq 2(1-c_{1}^{1/2})^{-1}\sum^{\ell}_{n=1}|a_{n}|^{2}(1-|f^{n}(z)|).$

Finally we estimate $B_{2}$. Note that Corollary 2.2 gives $|f^{k}(z)|\leq
r_{0}^{-1}c_{0}^{k-\ell}|f^{\ell}(z)|$ if $k\geq\ell$. Since
$|f^{n}(z)|\geq|f^{\ell}(z)|$ for $n\leq\ell$, we have

$B_{2}\leq
r_{0}^{-1}\sum^{\ell-1}_{n=1}|a_{n}|(1-|f^{n}(z)|^{2})\sum^{N}_{k>\ell}|a_{k}|c_{0}^{k-\ell}.$

Cauchy–Schwarz’s inequality gives

(2.4)
$\sum^{N}_{k>\ell}|a_{k}|c_{0}^{k-\ell}\leq(1-c_{0}^{2})^{-1/2}\left(\sum^{N}_{k>\ell}|a_{k}|^{2}\right)^{1/2}$

and

$\sum^{\ell-1}_{n=1}|a_{n}|(1-|f^{n}(z)|)\leq\left(\sum^{\ell-1}_{n=1}|a_{n}|^{2}(1-|f^{n}(z)|)\right)^{1/2}\left(\sum^{\ell-1}_{n=1}(1-|f^{n}(z)|)\right)^{1/2}.$

Since $1-|f^{n}(z)|\leq c_{1}^{\ell-n}$ for $1\leq n<\ell$, last sum is
bounded by $(1-c_{1})^{-1}$ and we deduce

(2.5)
$\sum^{\ell-1}_{n=1}|a_{n}|(1-|f^{n}(z)|)\leq(1-c_{1})^{-1/2}\left(\sum^{\ell-1}_{n=1}|a_{n}|^{2}(1-|f^{n}(z)|)\right)^{1/2}.$

Now applying (2.4) and (2.5) we deduce

$\begin{split}B_{2}&\leq
c_{2}\left(\sum_{n=1}^{\ell-1}|a_{n}|^{2}(1-|f^{n}(z)|^{2})\right)^{1/2}\left(\sum^{N}_{k>\ell}|a_{k}|^{2}\right)^{1/2}\\\\[5.0pt]
&\leq\frac{c_{2}}{2}\left(\sum^{\ell-1}_{n=1}|a_{n}|^{2}(1-|f^{n}(z)|^{2})+\sum^{N}_{k>\ell}|a_{k}|^{2}\right),\end{split}$

where $c_{2}=2r_{0}^{-1}(1-c_{0}^{2})^{-1/2}(1-c_{1})^{-1/2}$. Finally observe
that $|f^{k}(z)|\leq r_{0}$ for $k>\ell$ to deduce

$\sum^{N}_{k>\ell}|a_{k}|^{2}\leq(1-r_{0}^{2})^{-1}\sum^{N}_{k>\ell}|a_{k}|^{2}(1-|f^{k}(z)|^{2}).$

This finishes the proof if $\ell<N$. Assume now $\ell\geq N$, that is
$|f^{k}(z)|>r_{0}$ for any $k<N$. Then we argue as in the estimate of $B_{1}$
replacing $\ell$ by $N$, to obtain

$\sum^{N-1}_{n=1}|a_{n}|\frac{1-|f^{n}(z)|^{2}}{|f^{n}(z)|}\sum^{N}_{k>n}|a_{k}||f^{k}(z)|\leq
2(1-c_{1}^{1/2})^{-1}\sum_{n=1}^{N}|a_{n}|^{2}(1-|f^{n}(z)|).$

This finishes the proof. ∎

The following well known auxiliary result plays a fundamental role in the
classical work of Paley and Zygmund, as well as in the study of pointwise
convergence of random series of functions ([Ka]). Its short proof is included
for the sake of completeness.

######  Lemma 2.4 (Paley–Zygmund inequality).

Let $(X,\Omega,d\mu)$ be a probability space and $Z\colon X\to[0,\infty)$ be a
positive square integrable random variable. Then for any $0<\lambda<1$, we
have

$\mu\left\\{x\in
X:Z(x)>\lambda\int_{X}Z\,d\mu\right\\}\geq(1-\lambda)^{2}\frac{\left(\displaystyle\int_{X}Z\,d\mu\right)^{2}}{\displaystyle\int_{X}Z^{2}\,d\mu}.$

###### Proof.

We can assume $\displaystyle\int_{X}Z\,d\mu=1$. Let $W_{\lambda}$ denote the
indicator function of the set $\\{x\in X:Z(x)\leq\lambda\\}$. Cauchy–Schwarz’s
inequality gives

$1=\int_{X}ZW_{\lambda}\,d\mu+\int_{X}Z(1-W_{\lambda})\,d\mu\leq\lambda+\left(\int_{X}Z^{2}\,d\mu\right)^{1/2}\mu\\{x\in
X:Z(x)>\lambda\\}^{1/2}.$

∎

We are now ready to prove Theorem 1.1 stated in the Introduction.

###### Proof of Theorem 1.1.

It is obvious that (a) implies (b). We start proving that (c) implies (a). By
Lemma 2.3, the series $\sum a_{n}f^{n}$ converges in
$\mathbb{H}^{2}(\mathbb{D})$. Let
$F(z)=\sum\limits_{n=1}^{\infty}a_{n}f^{n}(z)$, $z\in\mathbb{D}$. Hence the
non-tangential limit

$F(\xi)=\lim_{{z{\to}_{\nless}\xi}}F(z)$

exists for almost every $\xi\in\partial\mathbb{D}$. For almost every
$\xi\in\partial\mathbb{D}$ we will construct a sequence of points
$\\{z_{N}=z_{N}(\xi):N=1,2,\dotsc\\}$ tending non-tangentially to $\xi$, such
that for any $\varepsilon>0$ there exists an integer
$N_{0}=N_{0}(\varepsilon,\xi)>0$ satisfying

(2.6) $\left|\sum^{N}_{n=1}a_{n}f^{n}(\xi)-F(z_{N})\right|<\varepsilon,\quad
N\geq N_{0}.$

It is clear that (2.6) implies the statement in (a). By Corollary 2.2, there
exist constants $0<r_{0}=r_{0}(f)<1$, $0<c_{0}=c_{0}(f)<1$ such that

(2.7) $|f^{n}(z)|\leq r_{0}^{-1}c_{0}^{n}|z|,\quad|z|\leq r_{0},\quad
n=1,2,\dotsc$

Fix $N\geq 1$. Write $F_{N}=\sum\limits^{N}_{n=1}a_{n}f^{n}$. Since $f(0)=0$
and $f^{N}$ is inner, there exists a subset $S=S_{N}\subset\partial\mathbb{D}$
with $m(\partial\mathbb{D}\backslash S)=0$ such that for any $\xi\in S$ there
exists $0<r_{N}=r_{N}(\xi)<1$ such that $|f^{N}(r_{N}\xi)|=r_{0}$. Since
$f^{N}$ tends to $0$ uniformly on compact sets of $\mathbb{D}$ as
$N\to\infty$, we deduce that $r_{N}\to 1$ as $N\to\infty$. Note also that
$|f^{j}(r_{N}\xi)|\geq|f^{N}(r_{N}\xi)|=r_{0}$ for $1\leq j\leq N$. For
$\xi\in S$ consider the arc
$I(N,\xi)=\\{z\in\partial\mathbb{D}:|z-\xi|<1-r_{N}\\}$. Apply Vitali’s
Covering Lemma (see [EG, p. 27]) to obtain a subcollection
$\\{I(N,\xi_{k}):k=1,2,\dotsc\\}$ of pairwise disjoint arcs such that
$m(\partial\mathbb{D}\backslash\cup 5I(N,\xi_{k}))=0$. Here $5I$ denotes the
arc in the unit circle having the same center than $I$ and with $m(5I)=5m(I)$.
Given $\varepsilon>0$ consider the set

$E(N)=\bigcup_{k}\\{\xi\in
5I(N,\xi_{k}):|F_{N}(\xi)-F_{N}(\xi_{k})|\geq\varepsilon\\}$

and

$E=\bigcap_{\ell\geq 1}\bigcup_{N\geq\ell}E(N).$

We first show that $m(E)=0$. By the classical Borel–Cantelli Lemma, it is
sufficient to show

(2.8) $\sum^{\infty}_{N=1}m(E(N))<\infty.$

Observe that Theorem 1.2 gives that $F\in\operatorname{BMOA}$. Since
$P(r_{N}\xi_{k},\xi)$ is comparable to $m(I(N,\xi_{k}))^{-1}$ for $\xi\in
5I(N,\xi_{k})$, there exists a constant $c_{1}=c_{1}(f)>0$ such that for any
$k,N\geq 1$, we have

$\frac{1}{m(I(N,\xi_{k}))}\int_{5I(N,\xi_{k})}|F_{N}(\xi)-F_{N}(r_{N}\xi_{k})|^{2}\,dm(\xi)\leq
c_{1}\sum^{N}_{n=1}|a_{n}|^{2}(1-|f^{n}(r_{N}\xi_{k})|).$

By part (a) of Lemma 2.1 and the choice of $r_{N}$, there exists a constant
$0<c<1$ such that $1-|f^{n}(r_{N}\xi_{k})|\leq c^{N-n}(1-r_{0})$,
$n=1,\dotsc,N$. Hence

$\frac{1}{m(I(N,\xi_{k}))}\int_{5I(N,\xi_{k})}|F_{N}(\xi)-F_{N}(r_{N}\xi_{k})|^{2}\,dm(\xi)\leq
c_{1}(1-r_{0})\sum^{N}_{n=1}|a_{n}|^{2}c^{N-n},\quad k,N\geq 1.$

Given $\varepsilon>0$, we deduce that for any $k,N\geq 1$, we have

$\frac{1}{m(I(N,\xi_{k}))}m(\\{\xi\in
5I(N,\xi_{k}):|F_{N}(\xi)-F_{N}(r_{N}\xi_{k})|>\varepsilon\\})\leq\frac{c_{1}(1-r_{0})}{\varepsilon^{2}}\sum^{N}_{n=1}|a_{n}|^{2}c^{N-n}.$

Since $\\{I(N,\xi_{k}):k=1,2,\dotsc\\}$ are pairwise disjoint, we deduce

$m(E(N))\leq\frac{c_{1}(1-r_{0})}{\varepsilon^{2}}\sum^{N}_{n=1}|a_{n}|^{2}c^{N-n}.$

Then

$\sum^{\infty}_{N=1}m(E(N))\leq\frac{c_{1}(1-r_{0})}{\varepsilon^{2}}\sum^{\infty}_{N=1}\sum^{N}_{n=1}|a_{n}|^{2}c^{N-n}=\frac{c_{1}(1-r_{0})}{\varepsilon^{2}(1-c)}\sum^{\infty}_{n=1}|a_{n}|^{2}.$

This proves (2.8) and shows that $m(E)=0$. Finally for almost every
$\xi\in\partial\mathbb{D}\backslash E$ we will construct the sequence
$\\{z_{N}=z_{N}(\xi)\\}$ for which (2.6) holds. Since
$m(\partial\mathbb{D}\backslash\cup 5I(N,\xi_{k}))=0$, for almost every
$\xi\in\partial\mathbb{D}\backslash E$ there exists $N_{0}=N_{0}(\xi)>0$ such
that for any $N>N_{0}$ there exists $\xi_{k}\in\partial\mathbb{D}$ with
$\xi\in 5I(N,\xi_{k})$ and

(2.9) $|F_{N}(\xi)-F_{N}(r_{N}\xi_{k})|<\varepsilon.$

The choice of $r_{N}$ and (2.7) gives that

$|f^{n}(r_{N}\xi_{k})|\leq c_{0}^{n-N},\quad n\geq N.$

Hence

(2.10)
$\sum_{n>N}|a_{n}||f^{n}(r_{N}\xi_{k})|\leq\sum_{n>N}|a_{n}|c_{0}^{n-N}\leq\left(1-c_{0}^{2}\right)^{-1/2}\left(\sum_{n>N}|a_{n}|^{2}\right)^{1/2}.$

Pick $z_{N}=z_{N}(\xi)=r_{N}\xi_{k}$, where $\xi_{k}$ is chosen such that
$\xi\in 5I(N,\xi_{k})$. Now (2.9) and (2.10) imply (2.6). Note that since
$\xi\in 5I(N,\xi_{k})$ and $r_{N}\to 1$, the points $z_{N}$ converge non-
tangentially to $\xi$ as $N$ tends to $\infty$.

We now prove that (b) implies (c). Given two sets
$A,B\subset\partial\mathbb{D}$ we use the notation
$A\overset{\text{a.e.}}{=}B$ if $A$ and $B$ differ at most on a set of
Lebesgue measure zero. Consider the set

$A=\left\\{\xi\in\partial\mathbb{D}:\sup_{N}\left|\sum^{N}_{n=1}a_{n}f^{n}(\xi)\right|<\infty\right\\}.$

Note that $f^{-k}(f^{k}(A))\overset{\text{a.e.}}{=}A$ for any $k=1,2,\dotsc$
Since the mapping $f\colon\partial\mathbb{D}\to\partial\mathbb{D}$ is exact
([Ne]) we deduce that $m(A)=0$ or $m(A)=1$. Our assumption gives $m(A)=1$.
Write

$F_{N}=\sum^{N}_{n=1}a_{n}f^{n}$

and $s_{N}=\|F_{N}\|_{2}$. By Lemma 2.3 there exists a constant $c=c(f)>1$
such that

(2.11) $c^{-1}\sum^{N}_{n=1}|a_{n}|^{2}\leq s^{2}_{N}\leq
c\sum^{N}_{n=1}|a_{n}|^{2},\quad N=1,2,\dotsc$

Now the Paley–Zygmund inequality stated in Lemma 2.4 gives that for any
$0<\lambda<1$ we have

$m\\{\xi\in\partial\mathbb{D}:|F_{N}(\xi)|^{2}\geq\lambda
s_{N}^{2}\\}\geq(1-\lambda)^{2}\frac{s_{N}^{4}}{\displaystyle\int_{\partial\mathbb{D}}|F_{N}|^{4}\,dm}.$

It was proved in [NS2, part (c) of Theorem 9] that there exists a constant
$c_{1}=c_{1}(f)>0$ such that

$\int_{\partial\mathbb{D}}|F_{N}|^{4}\,dm\leq c_{1}s_{N}^{4}.$

Hence

$m\\{\xi\in\partial\mathbb{D}:|F_{N}(\xi)|^{2}\geq\lambda s_{N}^{2}\\}\geq
c_{1}^{-1}(1-\lambda)^{2},\quad 0<\lambda<1.$

Since $m(A)=1$, we deduce that $s_{N}$ is bounded. Applying (2.11) we deduce
$\sum\limits_{n}|a_{n}|^{2}<\infty$. ∎

We are now ready to prove Corollaries 1.3 and 1.4.

###### Proof of Corollary 1.3.

By Lemma 2.3, the series $\sum a_{n}f^{n}$ converges in
$\mathbb{H}^{2}(\mathbb{D})$. Let $F=\sum\limits_{n\geq 1}a_{n}f^{n}$. Theorem
1.2 gives that $F\in\operatorname{BMOA}(\mathbb{D})$ and
$\|F\|^{2}_{\operatorname{BMOA}(\mathbb{D})}\leq C\sum\limits_{n\geq
1}|a_{n}|^{2}$. The converse estimate follows easily from Lemma 2.3 which
gives

$\|F\|_{\operatorname{BMOA}(\mathbb{D})}^{2}\geq\|F\|_{2}^{2}\geq\frac{1-|f^{\prime}(0)|}{1+|f^{\prime}(0)|}\sum\limits_{n=1}^{\infty}|a_{n}|^{2}.$

The identity (1.1) follows from (2.6). This finishes the proof of (a). Assume
now that $f$ is a finite Blaschke product. Fix a number $0<r<1$ and let
$0<c=c(r,f)<1$ be the constant appearing in part (a) of Lemma 2.1. For
$z\in\mathbb{D}$ let $N=N(z)$ be the smallest positive integer such that
$|f^{N}(z)|\leq r$. Note that $|f^{n}(z)|\leq r$ for $n\geq N$ and
$1-|f^{n}(z)|\leq c^{N-n}$ for $1\leq n\leq N$. Since $f$ is a finite Blaschke
product, we have

(2.12) $\lim_{|z|\to 1}N(z)=+\infty.$

Now

$\sum^{\infty}_{n=1}|a_{n}|^{2}(1-|f^{n}(z)|)\leq\sum^{N}_{n=1}|a_{n}|^{2}c^{N-n}+\sum^{\infty}_{n=N+1}|a_{n}|^{2}$

and (2.12) gives

$\lim_{|z|\to 1}\sum^{\infty}_{n=1}|a_{n}|^{2}(1-|f^{n}(z)|^{2})=0.$

Theorem 1.2 finishes the proof. ∎

###### Proof of Corollary 1.4.

Let $F=\sum\limits_{n=1}^{\infty}a_{n}f^{n}$ and
$s^{2}=\sum\limits_{n=1}^{\infty}|a_{n}|^{2}$. We first prove the upper
estimate. Consider the distribution function
$\Phi(\lambda)=m\\{\xi\in\partial\mathbb{D}:|F(\xi)|>\lambda\\}$, $\lambda>0$.
Corollary 1.3 and the John–Nirenberg Theorem give that there exist universal
constants $A,B>0$ such that

$\Phi(\lambda)\leq Ae^{-B\lambda/s},\quad\lambda>0.$

Then

$\|F\|_{p}^{p}=\int_{0}^{\infty}p\lambda^{p-1}\Phi(\lambda)\,d\lambda\leq
C(p)s^{p},$

where $C(p)$ is a constant depending on $A$, $B$ and $p$. The lower estimate
follows from the following standard duality argument. We can assume $p<2$. By
Hölder’s inequality

$\int_{\partial\mathbb{D}}|F|^{2}\,dm\leq\left(\int_{\partial\mathbb{D}}|F|^{p}\,dm\right)^{1/p}\left(\int_{\partial\mathbb{D}}|F|^{q}\,dm\right)^{1/q},$

where $p^{-1}+q^{-1}=1$. By Lemma 2.3 and the upper estimate we have already
proved, there exists a constant $c=c(f,p)>0$ such that

$\sum^{\infty}_{n=1}|a_{n}|^{2}\leq
c\left(\int_{\partial\mathbb{D}}|F|^{p}\,dm\right)^{1/p}\left(\sum^{\infty}_{n=1}|a_{n}|^{2}\right)^{1/2}.$

This finishes the proof. ∎

##  3 Uniform estimates

Given $F\in L^{1}(\partial\mathbb{D})$ and $z\in\mathbb{D}$, let $F(z)$ denote
the value of its harmonic extension at $z$, that is,

$F(z)=\int_{\partial\mathbb{D}}F(\xi)P(z,\xi)\,dm(\xi).$

If $F$ is the characteristic function of a measurable set
$E\subset\partial\mathbb{D}$, the corresponding function is called the
harmonic measure of the set $E$ from the point $z\in\mathbb{D}$ and will be
denoted by $F(z)=w(z,E)$. A function $F\in L^{1}(\partial\mathbb{D})$ is in
the space $\operatorname{BMO}(\partial\mathbb{D})$ if

$\|F\|_{\operatorname{BMO}(\partial\mathbb{D})}^{2}=\sup_{z\in\mathbb{D}}\int_{\partial\mathbb{D}}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)<\infty.$

Equivalently, $F\in L^{1}(\partial\mathbb{D})$ is in
$\operatorname{BMO}(\partial\mathbb{D})$ if and only if there exists a
constant $C=C(F)>0$ such that for any arc $I\subset\partial\mathbb{D}$, we
have

$\frac{1}{m(I)}\int_{I}|F-F_{I}|^{2}\,dm\leq C.$

Here $F_{I}=m(I)^{-1}\int_{I}F\,dm$ denotes the mean of $F$ on $I$. See
Chapter VI of [Ga]. For $z\in\mathbb{D}$ let $\tau_{z}$ be the automorphism of
$\mathbb{D}$ given by $\tau_{z}(w)=(w-z)(1-\overline{z}w)^{-1}$,
$w\in\mathbb{D}$. The John–Nirenberg Theorem applied to $F\circ\tau_{z}-F(z)$
provides two universal constants $A,B>0$ such that

(3.1) $w(z,\\{\xi\in\partial\mathbb{D}:|F(\xi)-F(z)|>\lambda\\})\leq
Ae^{-B\lambda/\|F\|_{\operatorname{BMO}(\partial\mathbb{D})}},\quad\lambda>0.$

See [Ba]. The proof of Theorem 1.5 is based on a careful study of the
oscillation of the partial sums of $\sum a_{n}f^{n}$. We start with an
auxiliary result which holds for any $BMO$ function.

######  Lemma 3.1.

For any $0<c<1$ there exists a constant $0<\delta=\delta(c)<1$ such that the
following statement holds. Let $F$ be a real valued function defined on
$\partial\mathbb{D}$ with $\|F\|_{\operatorname{BMO}(\partial\mathbb{D})}=1$.
Let $z\in\mathbb{D}$ such that

(3.2) $\int_{\partial\mathbb{D}}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)\geq c^{2}.$

Then $w(z,\\{\xi\in\partial\mathbb{D}:F(\xi)-F(z)\geq\delta\\})\geq\delta$.

###### Proof.

Given $0\leq a<b$, let $E(a,b)$ denote the set of points
$\xi\in\partial\mathbb{D}$ such that $a\leq|F(\xi)-F(z)|\leq b$. Apply the
John–Nirenberg estimate (3.1) to find a constant $c_{1}=c_{1}(c)>0$, such that

$\int_{\partial\mathbb{D}\backslash
E(0,c_{1})}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)\leq\frac{c^{2}}{4}.$

Since

$\int_{E(0,c/2)}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)\leq\frac{c^{2}}{4},$

we deduce that

$\int_{E(c/2,c_{1})}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)\geq\frac{c^{2}}{2}.$

Hence

(3.3)
$w\left(z,\left\\{\xi\in\partial\mathbb{D}:|F(\xi)-F(z)|\geq\frac{c}{2}\right\\}\right)\geq\frac{c^{2}}{2c_{1}^{2}}.$

Apply the John–Nirenberg estimate (3.1) to find a constant $c_{2}=c_{2}(c)>0$
such that

(3.4) $\int_{\partial\mathbb{D}\backslash
E(0,c_{2})}|F(\xi)-F(z)|P(z,\xi)\,dm(\xi)\leq\frac{c^{3}}{16c_{1}^{2}}.$

Let $0<\delta=\delta(c)<c/2$ be a small constant to be fixed later. We will
show that if $\delta>0$ is conveniently chosen, then

(3.5)
$w(z,\\{\xi:F(\xi)-F(z)>\delta\\})\geq\min\left\\{\frac{c^{3}}{16c_{1}^{2}c_{2}},\frac{c^{2}}{16c_{1}^{2}}\right\\}.$

We argue by contradiction. Assume

(3.6)
$w(z,\\{\xi:F(\xi)-F(z)>\delta\\})\leq\min\left\\{\frac{c^{3}}{16c_{1}^{2}c_{2}},\frac{c^{2}}{16c_{1}^{2}}\right\\}.$

Apply (3.3) to deduce that $w(z,\\{\xi:F(\xi)-F(z)<-c/2\\})\geq
7c^{2}/16c_{1}^{2}$. Hence

(3.7) $\int_{\\{\xi:F(\xi)-F(z)\leq
0\\}}(F(\xi)-F(z))P(z,\xi)\,dm(\xi)\leq\frac{-7c^{3}}{32c_{1}^{2}}.$

For $0<\gamma<\tau$ consider the set
$G(\gamma,\tau)=\\{\xi\in\partial\mathbb{D}:\gamma\leq
F(\xi)-F(z)\leq\tau\\}$. We have

$\begin{split}\int_{\\{\xi:F(\xi)-F(z)>0\\}}(F(\xi)-F(z))P(z,\xi)\,dm(\xi)&\leq\delta+\int_{G(\delta,c_{2})}(F(\xi)-F(z))P(z,\xi)\,dm(\xi)\\\
&\quad+\int_{\\{\xi:F(\xi)-F(z)>c_{2}\\}}(F(\xi)-F(z))P(z,\xi)\,dm(\xi).\end{split}$

The choice (3.4) of $c_{2}$ gives that last integral is bounded by
$c^{3}/16c_{1}^{2}$. Moreover by (3.6), we have

$\int_{G(\delta,c_{2})}(F(\xi)-F(z))P(z,\xi)\,dm(\xi)\leq
c_{2}w(z,\\{\xi:F(\xi)-F(z)>\delta\\})\leq\frac{c^{3}}{16c_{1}^{2}}.$

We deduce

(3.8)
$\int_{\\{\xi:F(\xi)-F(z)>0\\}}(F(\xi)-F(z))P(z,\xi)\,dm(\xi)\leq\delta+\frac{c^{3}}{8c_{1}^{2}}.$

Choosing $0<\delta<c^{3}/16c_{1}^{2}$ we observe that (3.7) and (3.8)
contradict the identity

$\int_{\partial\mathbb{D}}(F(\xi)-F(z))P(z,\xi)\,dm(\xi)=0.$

Hence choosing $0<\delta<\min\\{c/2,c^{3}/16c_{1}^{2}\\}$, estimate (3.5)
holds. This finishes the proof. ∎

Our next auxiliary result says that $F=\sum\limits_{n=M}^{N}a_{n}f^{n}$
satisfies condition (3.2) in Lemma 3.1 if $|f^{M}(z)|$ is sufficiently small.

######  Lemma 3.2.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Then there
exists a constant $0<\varepsilon=\varepsilon(f)<1$ such that if $M<N$ are
positive integers, $\\{a_{n}:M\leq n\leq N\\}$ are complex numbers and
$z\in\mathbb{D}$ satisfies $|f^{M}(z)|\leq\varepsilon$, we have

$\int_{\partial\mathbb{D}}\left|\sum^{N}_{n=M}a_{n}(f^{n}(\xi)-f^{n}(z))\right|^{2}P(z,\xi)\,dm(\xi)\geq\frac{1}{2}\,\frac{1-|f^{\prime}(0)|}{1+|f^{\prime}(0)|}\sum^{N}_{n=M}|a_{n}|^{2}.$

###### Proof.

By continuity we can assume that $f^{n}(z)\neq 0$ for any $n$. Write
$F=\sum\limits_{n=M}^{N}a_{n}f^{n}$. In (2.3) we already noted that

(3.9)
$\begin{split}\int_{\partial\mathbb{D}}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)&=\sum^{N}_{n=M}|a_{n}|^{2}(1-|f^{n}(z)|^{2})\\\
&\quad+2\operatorname{Re}\sum^{N-1}_{n=M}\sum^{N}_{k>n}\overline{a}_{n}a_{k}(1-|f^{n}(z)|^{2})\frac{f^{k}(z)}{f^{n}(z)}.\end{split}$

The idea is to compare the expression above with

(3.10)
$\int_{\partial\mathbb{D}}|F(\xi)|^{2}\,dm(\xi)=\sum^{N}_{n=M}|a_{n}|^{2}+2\operatorname{Re}\sum^{N-1}_{n=M}\sum^{N}_{k>n}\overline{a}_{n}a_{k}f^{\prime}(0)^{k-n}.$

Assume $f^{\prime}(0)\neq 0$. For $k>n\geq M$ consider the function
$g=g_{k,n}$ defined by $g(w)=f^{k-n}(w)$, $w\in\mathbb{D}$. By part (b) of
Lemma 2.1, there exists a constant $0<r_{0}=r_{0}(f)<1$ such that $|g(w)|\leq
r_{0}^{-1}|f^{\prime}(0)|^{k-n}|w|$ if $|w|<r_{0}$. Hence there exists a
constant $c(r_{0})>0$ such that

(3.11) $\left|\frac{g(w)}{w}-g^{\prime}(0)\right|\leq
c(r_{0})|f^{\prime}(0)|^{k-n}|w|,\quad|w|\leq\frac{r_{0}}{2}.$

Let $0<\varepsilon<r_{0}/2$ be a constant to be fixed later and assume
$|f^{M}(z)|\leq\varepsilon$. Taking $w=f^{n}(z)$ in (3.11) we obtain

$\left|\frac{f^{k}(z)}{f^{n}(z)}-f^{\prime}(0)^{k-n}\right|\leq
c(r_{0})|f^{\prime}(0)|^{k-n}|f^{n}(z)|\leq\varepsilon
c(r_{0})|f^{\prime}(0)|^{k-n},\quad k>n\geq M.$

By part (b) of Lemma 2.1, we also have $|f^{k}(z)|\leq
r_{0}^{-1}|f^{\prime}(0)|^{k-M}\varepsilon$ and $|f^{n}(z)|\leq
r_{0}^{-1}|f^{\prime}(0)|^{n-M}\varepsilon$ if $k\geq M$ and $n\geq M$. Then

$\begin{split}&\left|\sum^{N-1}_{n=M}\sum^{N}_{k>n}\overline{a}_{n}a_{k}\left((1-|f^{n}(z)|^{2})\frac{f^{k}(z)}{f^{n}(z)}-f^{\prime}(0)^{k-n}\right)\right|\\\\[5.0pt]
\leq&\sum^{N-1}_{n=M}\sum^{N}_{k>n}|a_{n}||a_{k}|(\varepsilon
c(r_{0})|f^{\prime}(0)|^{k-n}+r_{0}^{-2}\varepsilon^{2}|f^{\prime}(0)|^{k+n-2M}).\end{split}$

As in the proof of Theorem 1.2, writting $j=k-n$ and applying Cauchy-Schwarz
inequality, we find a constant $c(f)>0$ such that

$\displaystyle\sum^{N-1}_{n=M}\sum^{N}_{k>n}|a_{n}||a_{k}||f^{\prime}(0)|^{k-n}\leq
c(f)\sum^{N}_{n=M}|a_{n}|^{2},$
$\displaystyle\sum^{N-1}_{n=M}\sum^{N}_{k>n}|a_{n}||a_{k}||f^{\prime}(0)|^{k+n-2M}\leq
c(f)\sum^{N}_{n=M}|a_{n}|^{2}.$

We deduce that there exists a constant $c(f,r_{0})>0$ such that

$\left|\sum^{N-1}_{n=M}\sum^{N}_{k>n}\overline{a}_{n}a_{k}\left((1-|f^{n}(z)|^{2})\frac{f^{k}(z)}{f^{n}(z)}-f^{\prime}(0)^{k-n}\right)\right|\leq
c(f,r_{0})\varepsilon\sum^{N}_{n=M}|a_{n}|^{2}.$

Applying (3.9) and (3.10), we deduce

$\left|\int_{\partial\mathbb{D}}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)-\int_{\partial\mathbb{D}}|F(\xi)|^{2}\,dm(\xi)\right|\leq\varepsilon^{2}\sum^{N}_{n=M}|a_{n}|^{2}+2c(f,r_{0})\varepsilon\sum^{N}_{n=M}|a_{n}|^{2}.$

Recall that

$\int_{\partial\mathbb{D}}|F(\xi)|^{2}\,dm(\xi)\geq\frac{1-|f^{\prime}(0)|}{1+|f^{\prime}(0)|}\sum^{N}_{n=M}|a_{n}|^{2}.$

Choose $0<\varepsilon<r_{0}/2$ small enough so that
$\varepsilon^{2}+2c(f,r_{0})\varepsilon<(1-|f^{\prime}(0)|)(1+|f^{\prime}(0)|)^{-1}/2$
and deduce

$\int_{\partial\mathbb{D}}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)\geq\frac{1}{2}\,\frac{1-|f^{\prime}(0)|}{1+|f^{\prime}(0)|}\sum^{N}_{n=M}|a_{n}|^{2}.$

This finishes the proof if $f^{\prime}(0)\neq 0$. Assume now
$f^{\prime}(0)=0$. Apply part (c) of Lemma 2.1 to the inner function $f^{k-n}$
to obtain $|f^{k}(z)|\leq|f^{n}(z)|^{2^{k-n}}$, $k>n$. Let $0<\varepsilon<1$
be a (small) constant to be fixed later and assume
$|f^{M}(z)|\leq\varepsilon$. If $k>n\geq M$, we have

$\frac{|f^{k}(z)|}{|f^{n}(z)|}\leq|f^{n}(z)|^{2^{k-n}-1}\leq\varepsilon^{2^{k-n}-1}\leq\varepsilon^{k-n}.$

Writing $j=k-n$ and applying Cauchy-Schwarz’s inequality, we obtain

$\left|\sum^{N-1}_{n=M}\overline{a}_{n}\sum^{N}_{k>n}a_{k}(1-|f^{n}(z)|^{2})\frac{f^{k}(z)}{f^{n}(z)}\right|\leq\sum^{N-1}_{n=M}\sum^{N}_{k>n}|a_{n}||a_{k}|\varepsilon^{k-n}\leq\varepsilon(1-\varepsilon)^{-1}\sum^{N}_{n=M}|a_{n}|^{2}.$

As before, applying (3.9) and (3.10) we deduce

$\left|\int_{\partial\mathbb{D}}|F(\xi)-F(z)|^{2}P(z,\xi)\,dm(\xi)-\int_{\partial\mathbb{D}}|F(\xi)|^{2}\,dm(\xi)\right|\leq(\varepsilon^{2}+2\varepsilon(1-\varepsilon)^{-1})\sum^{N}_{n=M}|a_{n}|^{2}.$

Since

$\int_{\partial\mathbb{D}}|F(\xi)|^{2}\,dm(\xi)=\sum^{N}_{n=M}|a_{n}|^{2}$

we only need to pick $\varepsilon>0$ small enough so that
$\varepsilon^{2}+2\varepsilon(1-\varepsilon)^{-1}<1/2$. ∎

Next auxiliary result is an easy consequence of Lemmas 3.1 and 3.2.

######  Lemma 3.3.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Then there
exist two constants $0<\varepsilon=\varepsilon(f)<1$ and $0<c=c(f)<1$ such
that the following statement holds. Let $M<N$ be positive integers,
$z\in\mathbb{D}$ satisfying $|f^{M}(z)|<\varepsilon$ and $\\{a_{n}:M\leq n\leq
N\\}$ a set of complex numbers. Then there exists a set
$E=E(z,\\{a_{n}\\},f)\subset\partial\mathbb{D}$ with $w(z,E)\geq c$ such that

$\operatorname{Re}\sum^{N}_{n=M}a_{n}f^{n}(\xi)\geq
c\left(\sum_{n=M}^{N}|a_{n}|^{2}\right)^{1/2},\quad\xi\in E.$

###### Proof.

Note that for any analytic function $G\in\mathbb{H}^{2}(\mathbb{D})$ with
$G(z)=0$, we have

(3.12)
$\int_{\partial\mathbb{D}}(\operatorname{Re}G(\xi))^{2}P(z,\xi)\,dm(\xi)=\frac{1}{2}\int_{\partial\mathbb{D}}|G(\xi)|^{2}P(z,\xi)\,dm(\xi).$

Let $0<r_{0}=r_{0}(f)<1$ and $0<c_{0}=c_{0}(f)<1$ be the constants given in
Corollary 2.2 and let $\varepsilon_{0}=\varepsilon_{0}(f)>0$ be the constant
appearing in Lemma 3.2. Let $0<\varepsilon<\min\\{\varepsilon_{0},r_{0}\\}$ be
a (small) constant to be fixed later. Lemma 3.2 and the identity (3.12)
applied to the function $G=F-F(z)$, where $F=\sum\limits_{n=M}^{N}a_{n}f^{n}$,
give that

$\int_{\partial\mathbb{D}}(\operatorname{Re}F(\xi)-\operatorname{Re}F(z))^{2}P(z,\xi)\,dm(\xi)\geq
c(f)\sum^{N}_{n=M}|a_{n}|^{2},$

where $c(f)=4^{-1}(1-|f^{\prime}(0)|)(1+|f^{\prime}(0)|)^{-1}$. Applying Lemma
3.1 to the function

$(\operatorname{Re}F)\|\mathrm{Re}\,F\|^{-1}_{\operatorname{BMO}(\partial\mathbb{D})},$

we find a constant $\delta=\delta(f)>0$ and a set
$E=E(z,\\{a_{n}\\},f)\subset\partial\mathbb{D}$ with $w(z,E)\geq\delta$ such
that

$\operatorname{Re}F(\xi)-\operatorname{Re}F(z)\geq\delta\|\mathrm{Re}\,F\|_{\operatorname{BMO}(\partial\mathbb{D})},\quad\xi\in
E.$

By Corollary 1.3,
$\|\mathrm{Re}\,F\|^{2}_{\operatorname{BMO}(\partial\mathbb{D})}$ is
comparable to $\sum\limits_{n=M}^{N}|a_{n}|^{2}$. Reducing $\delta>0$ if
necessary, we can assume

(3.13)
$\operatorname{Re}F(\xi)-\operatorname{Re}F(z)\geq\delta\left(\sum^{N}_{n=M}|a_{n}|^{2}\right)^{1/2},\quad\xi\in
E.$

We have $|f^{M}(z)|<\varepsilon$. Note that by Corollary 2.2, we obtain
$|f^{n}(z)|\leq r_{0}^{-1}c_{0}^{n-M}\varepsilon$, $n\geq M$. Cauchy-Schwarz
inequality yields

$|F(z)|\leq\varepsilon
c(c_{0},r_{0})\left(\sum^{N}_{n=M}|a_{n}|^{2}\right)^{1/2},$

where $c(c_{0},r_{0})$ is a constant depending on $c_{0}$ and $r_{0}$. Pick
$0<\varepsilon<\min\\{\varepsilon_{0},r_{0}\\}$ sufficiently small such that
$\varepsilon c(c_{0},r_{0})\leq\delta/2$. Taking $c=\delta/2$ the proof is
completed. ∎

The next easy auxiliary result is stated for future reference.

######  Lemma 3.4.

Let $F\in\operatorname{BMO}(\partial\mathbb{D})$ and $0<c<1$. Then there
exists a constant $\varepsilon=\varepsilon(c)>0$ such that if
$I\subset\partial\mathbb{D}$ is an arc with $m(\\{\xi\in
I:\operatorname{Re}F(\xi)\geq
c\|F\|_{\operatorname{BMO}(\partial\mathbb{D})}\\})\geq cm(I)$ and

(3.14)
$\frac{1}{m(I)}\int_{I}|F-F_{I}|^{2}\,dm\leq\varepsilon\|F\|^{2}_{\operatorname{BMO}(\partial\mathbb{D})},$

then $\operatorname{Re}(F_{I})\geq
2^{-1}c\|F\|_{\operatorname{BMO}(\partial\mathbb{D})}$.

###### Proof.

Denote $E=\\{\xi\in I:\operatorname{Re}F(\xi)\geq
c\|F\|_{\operatorname{BMO}(\partial\mathbb{D})}\\}$. We argue by
contradiction. Assume $\operatorname{Re}(F_{I})\leq
2^{-1}c\|F\|_{\operatorname{BMO}(\partial\mathbb{D})}$. Then the estimate
(3.14) gives

$\frac{c^{2}}{4}\|F\|^{2}_{\operatorname{BMO}(\partial\mathbb{D})}\frac{m(E)}{m(I)}\leq\varepsilon\|F\|^{2}_{\operatorname{BMO}(\partial\mathbb{D})}$

and we deduce $m(E)\leq 4\varepsilon c^{-2}m(I)$. Since $m(E)\geq cm(I)$,
picking $0<\varepsilon<c^{3}/4$ one gets a contradiction. ∎

Given an arc $I$ centered at $\xi\in\partial\mathbb{D}$ we denote by $z(I)$
the point $z(I)=(1-m(I))\xi$. Conversely, given a point
$z\in\mathbb{D}\backslash\\{0\\}$ let $I(z)$ be the arc in the unit circle
such that $z(I(z))=z$. Given an arc $I\subset\partial\mathbb{D}$ and a number
$0<c<1/m(I)$, we denote by $cI$ the arc which has the same center as $I$ and
with $m(cI)=cm(I)$. Given an arc $I\subset\partial\mathbb{D}$ we consider its
dyadic decomposition $\mathcal{D}(I)=\bigcup\limits_{n\geq
0}\mathcal{D}_{n}(I)$ where $\mathcal{D}_{n}(I)$ is the set of the $2^{n}$
pairwise disjoint subarcs of $I$ of Lebesgue measure $2^{-n}m(I)$.

######  Lemma 3.5.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Then there
exist constants $0<\varepsilon=\varepsilon(f)<1$ and $0<c=c(f)<1$ such that
the following statement holds. Let $0<\gamma<1$ be a constant with
$1-\gamma\leq\varepsilon$, let $M<N$ be positive integers, $\\{a_{n}:M\leq
n\leq N\\}$ be complex numbers, and $z\in\mathbb{D}$ with
$|f^{M}(z)|<\varepsilon$. Then there exist pairwise disjoint arcs
$\\{I_{k}\\}$ with $c^{-1}I_{k}\subset c^{-1}I(z)$ for any $k$, such that

  1. (a)

For any $k=1,2,\dotsc$, we have $\tau(1-\gamma)\leq 1-|f^{N}(z(I_{k}))|\leq
1-\gamma,$ where $\tau>0$ is a universal constant.

  2. (b)

$\sum m(I_{k})\geq cm(I(z))$.

  3. (c)

For any $k=1,2,\dotsc$, we have

$\frac{1}{|I_{k}|}\int_{I_{k}}\operatorname{Re}\sum^{N}_{n=M}a_{n}f^{n}\,dm\geq
c\left(\sum^{N}_{n=M}|a_{n}|^{2}\right)^{1/2}.$

###### Proof.

Denote $F=\sum\limits_{n=M}^{N}a_{n}f^{n}$. Let
$0<\varepsilon_{0}=\varepsilon_{0}(f)<1$, $0<c_{0}=c_{0}(f)<1$ be the
constants given by Lemma 3.3. Let $0<\varepsilon<\varepsilon_{0}(f)/2$ be a
(small) constant to be fixed later. Assume $|f^{M}(z)|<\varepsilon$. Apply
Lemma 3.3 to find a set $E\subset\partial\mathbb{D}$ with $w(z,E)\geq c_{0}$
such that

(3.15) $\operatorname{Re}F(\xi)\geq
c_{0}\left(\sum^{N}_{n=M}|a_{n}|^{2}\right)^{1/2},\quad\xi\in E.$

Reducing $E$ if necessary, we can assume that there exists a constant
$c_{1}=c_{1}(c_{0})>0$ such that $E\subset c_{1}^{-1}I(z)$ and $m(E)\geq
c_{1}m(I(z))$. Let $I_{0}=c_{1}^{-1}I(z)$ and consider its dyadic
decomposition $\mathcal{D}(I_{0})$. Fix $0<\gamma<1$ with
$1-\gamma<\varepsilon$. Let $\mathcal{G}_{0}=\\{I_{k}:k\geq 1\\}$ be the
collection of maximal dyadic arcs in $\mathcal{D}(I_{0})$ such that

(3.16) $|f^{N}(z_{k})|\geq\gamma.$

Here $z_{k}=z(I_{k})$, $k\geq 1$. By maximality, the arcs in the collection
$\mathcal{G}_{0}$ are pairwise disjoint. Note also that by maximality and
Schwarz’s Lemma, there exists a universal constant $\tau>0$ such that

(3.17) $\tau(1-\gamma)\leq 1-|f^{N}(z_{k})|\leq 1-\gamma,\quad k\geq 1.$

This is the statement in (a). Since $f^{N}$ is inner, the union of
$\\{I_{k}:k\geq 1\\}$ covers almost every point of $I_{0}$. Consider the
subcollection $\mathcal{G}_{1}\subset\mathcal{G}_{0}$ of those dyadic arcs
$I_{k}\in\mathcal{G}_{0}$ such that

(3.18) $m(E\cap I_{k})\geq c_{2}m(I_{k}).$

Here $c_{2}>0$ is a small constant to be fixed later. We will show that the
subcollection $\mathcal{G}_{1}$ satisfies the conditions (b) and (c) in the
statement. Let $\mathcal{L}$ be the collection of indices $k$ such that
$I_{k}\in\mathcal{G}_{1}$. Note that

$c_{1}m(I(z))\leq m(E)\leq\sum_{k\notin\mathcal{L}}m(E\cap
I_{k})+\sum_{k\in\mathcal{L}}m(I_{k}).$

If $k\notin\mathcal{L}$, we have $m(E\cap I_{k})\leq c_{2}m(I_{k})$. Hence the
first sum is bounded by $c_{2}c_{1}^{-1}m(I(z))$. Pick $c_{2}>0$ sufficiently
small so that $c_{1}-c_{2}c_{1}^{-1}>c_{1}/2$ to deduce

(3.19) $\sum_{k\in\mathcal{L}}m(I_{k})\geq\frac{c_{1}}{2}m(I(z)).$

Choosing $0<c<c_{1}/2$, the estimate (3.19) gives the statement in (b). Note
that $|f^{N}(z)|\leq|f^{M}(z)|<\varepsilon$ and $|f^{N}(z_{k})|\geq\gamma$,
for any positive integer $k$. By Schwarz’s Lemma
$\rho(z_{k},z)\geq\rho(\gamma,\varepsilon)$. Since $0<1-\gamma<\varepsilon$,
taking $\varepsilon>0$ sufficiently small we deduce that
$\rho(\gamma,\varepsilon)$ is close to $1$ and then $m(I_{k})\leq
c_{3}(\varepsilon)m(I_{0})$ for any $k\geq 1$, where $c_{3}(\varepsilon)$ is
tends to $0$ as $\varepsilon$ tends to $0$. If $c>0$ and $\varepsilon>0$ are
taken sufficiently small we have $c^{-1}I_{k}\subset c^{-1}I(z)$ for any
$k\geq 1$.

By Theorem 1.2, there exists a constant $c_{4}=c_{4}(f)>0$ such that for any
$k\geq 1$ we have

$\frac{1}{m(I_{k})}\int_{I_{k}}\left|F(\xi)-F_{I_{k}}\right|^{2}\,dm(\xi)\leq
c_{4}\sum^{N}_{n=M}|a_{n}|^{2}(1-|f^{n}(z_{k})|^{2}).$

By part (a) of Lemma 2.1, there exists a constant $0<c_{5}=c_{5}(\gamma,f)<1$
such that $1-|f^{n}(z_{k})|\leq c_{5}^{N-n}(1-|f^{N}(z_{k})|)$, $n\leq N$.
Since $1-|f^{N}(z_{k})|\leq 1-\gamma$, we obtain

$\frac{1}{m(I_{k})}\int_{I_{k}}\left|F(\xi)-F_{I_{k}}\right|^{2}\,dm(\xi)\leq
2c_{4}(1-\gamma)\sum^{N}_{n=M}|a_{n}|^{2}c_{5}^{N-n}.$

Recall that $m(E\cap I_{k})\geq c_{2}m(I_{k})$ if $k\in\mathcal{L}$. Then if
$1-\gamma$ is sufficiently small, by Lemma 3.4, the estimate (3.15) implies
that

$\frac{1}{m(I_{k})}\int_{I_{k}}\operatorname{Re}F\,dm\geq\frac{c_{0}}{2}\left(\sum^{N}_{n=M}|a_{n}|^{2}\right)^{1/2}.$

Choosing $0<c<c_{0}/2$, last estimate gives the statement in (c).This finishes
the proof. ∎

Our next auxiliary result is the building block of the main construction in
the proof of Theorem 1.5.

######  Lemma 3.6.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Then there
exist constants $0<\varepsilon=\varepsilon(f)<1$ and $0<c=c(f)<1$ such that
the following statement holds. Let $0<\gamma<1$ be a constant with
$1-\gamma\leq\varepsilon$, let $\\{a_{n}\\}$ be a sequence of complex numbers,
let $M_{1}<N_{1}<M_{2}<N_{2}$ be positive integers and

$F_{i}=\sum^{N_{i}}_{n=M_{i}}a_{n}f^{n},\quad i=1,2.$

Let $I\subset\partial\mathbb{D}$ be an arc such that
$c(1-\gamma)<1-|f^{N_{1}}(z(I))|\leq 1-\gamma$ and
$|f^{M_{2}}(z(I))|\leq\varepsilon$. Then there exists an arc $J$ with
$c^{-1}J\subset c^{-1}I$, such that

(3.20) $\displaystyle c(1-\gamma)\leq 1-|f^{N_{2}}(z(J))|\leq 1-\gamma,$
(3.21) $\displaystyle\frac{1}{|J|}\int_{J}\operatorname{Re}F_{2}\,dm\geq
c\left(\sum^{N_{2}}_{n=M_{2}}|a_{n}|^{2}\right)^{1/2},$ (3.22)
$\displaystyle\frac{1}{|J|}\int_{J}\operatorname{Re}F_{1}\,dm\geq\frac{1}{|I|}\int_{I}\operatorname{Re}F_{1}\,dm-c^{-1}(1-\gamma)^{1/2}\left(\sum^{N_{1}}_{n=M_{1}}|a_{n}|^{2}t^{N_{1}-n}\right)^{1/2},$

where $t=(1+\gamma)^{-1}(1+|f^{\prime}(0)|\gamma)<1$.

###### Proof.

Let $\varepsilon_{0}>0$ and $c_{0}>0$ be the constants appearing in Lemma 3.5.
Let $0<\varepsilon<\varepsilon_{0}$ be a constant to be fixed later. Let
$0<\gamma<1$ with $1-\gamma<\varepsilon$. Apply Lemma 3.5 to the function
$F_{2}$ to find arcs $J_{k}$ with $c_{0}^{-1}J_{k}\subset c_{0}^{-1}I$, such
that $\tau(1-\gamma)\leq 1-|f^{N_{2}}(z(J_{k}))|\leq 1-\gamma$, $\sum
m(J_{k})\geq c_{0}m(I)$ and

$\frac{1}{|J_{k}|}\int_{J_{k}}\operatorname{Re}F_{2}\,dm\geq
c_{0}\left(\sum^{N_{2}}_{n=M_{2}}|a_{n}|^{2}\right)^{1/2}.$

Here $\tau>0$ is the universal constant given by part (a) of Lemma 3.5. The
constant $c$ will satisfy $0<c<c_{0}$ and the arc $J$ will be one of the arcs
of the family $\\{J_{k}\\}$. Hence estimates (3.20) and (3.21) will follow
from the construction. Theorem 1.2 gives that there exists a constant
$c_{1}=c_{1}(f)>0$ such that

(3.23)
$\frac{1}{c_{0}^{-1}m(I)}\int_{c_{0}^{-1}I}|\mathrm{Re}\,F_{1}-\mathrm{Re}\,F_{1}(z(c_{0}^{-1}I))|^{2}\,dm\leq
c_{1}\sum^{N_{1}}_{n=M_{1}}|a_{n}|^{2}(1-|f^{n}(z(c_{0}^{-1}I))|^{2}).$

Note that

$\begin{split}|(\operatorname{Re}F_{1})_{I}-\mathrm{Re}\,F_{1}(z(c_{0}^{-1}I))|&\leq\frac{1}{m(I)}\int_{I}|\mathrm{Re}\,F_{1}-\mathrm{Re}\,F_{1}(z(c_{0}^{-1}I))|dm\\\
&\leq\left(\frac{1}{m(I)}\int_{I}|\mathrm{Re}\,F_{1}-\mathrm{Re}\,F_{1}(z(c_{0}^{-1}I))|^{2}dm\right)^{1/2}\end{split}$

Applying (3.23) we find a constant $c_{2}=c_{2}(f,c_{0})>0$ such that

(3.24)
$\frac{1}{m(I)}\int_{c_{0}^{-1}I}|\mathrm{Re}\,F_{1}-(\operatorname{Re}F_{1})_{I}|^{2}\,dm\leq
c_{2}\sum^{N_{1}}_{n=M_{1}}|a_{n}|^{2}(1-|f^{n}(z(c_{0}^{-1}I))|^{2}).$

By Schwarz’s Lemma, there exists a constant $c_{3}=c_{3}(c_{0})>0$ such that
$1-|f^{n}(z(c_{0}^{-1}I))|^{2}\leq c_{3}(1-|f^{n}(z(I))|^{2})$ for any $n\geq
1$. Since $1-|f^{N_{1}}(z(I))|\leq 1-\gamma$, part (a) of Lemma 2.1 provides a
constant $0<c_{4}<1$ such that $1-|f^{n}(z(I))|\leq
c_{4}^{N_{1}-n}(1-\gamma)$, $n\leq N_{1}$. Actually, according to Lemma 2.1
one can take $c_{4}=(1+\gamma)^{-1}(1+|f^{\prime}(0)|\gamma)$. Applying (3.24)
we obtain a constant $c_{5}=c_{5}(c_{0},f)>0$ such that

(3.25)
$\frac{1}{m(I)}\int_{c_{0}^{-1}I}|\mathrm{Re}\,F_{1}-(\operatorname{Re}F_{1})_{I}|^{2}\,dm\leq
c_{5}(1-\gamma)\sum^{N_{1}}_{n=M_{1}}|a_{n}|^{2}c_{4}^{N_{1}-n}.$

Let $K>0$ be a large constant to be fixed later. Let $\mathcal{G}$ be the
subcollection of those arcs $J_{k}$ such that

$(\operatorname{Re}F_{1})_{J_{k}}\leq(\operatorname{Re}F_{1})_{I}-K(1-\gamma)^{1/2}\left(\sum^{N_{1}}_{n=M_{1}}|a_{n}|^{2}c_{4}^{N_{1}-n}\right)^{1/2}.$

Note that for any $J_{k}\in\mathcal{G}$ we have

$\frac{1}{m(J_{k})}\int_{J_{k}}|\operatorname{Re}F_{1}-(\operatorname{Re}F_{1})_{I}|dm\geq|(\operatorname{Re}F_{1})_{J_{k}}-(\operatorname{Re}F_{1})_{I}|\geq
K(1-\gamma)^{1/2}\left(\sum^{N_{1}}_{n=M_{1}}|a_{n}|^{2}c_{4}^{N_{1}-n}\right)^{1/2}.$

Adding over $k$, we obtain

$\sum_{J_{k}\in\mathcal{G}}\int_{J_{k}}|\operatorname{Re}F_{1}-(\operatorname{Re}F_{1})_{I}|dm\geq
K(1-\gamma)^{1/2}\left(\sum^{N_{1}}_{n=M_{1}}|a_{n}|^{2}c_{4}^{N_{1}-n}\right)^{1/2}\sum_{J_{k}\in\mathcal{G}}m(J_{k}).$

Applying (3.25) we deduce

$\sum_{J_{k}\in\mathcal{G}}m(J_{k})\leq\frac{c_{5}^{1/2}}{K}m(I).$

Since $\sum m(J_{k})\geq c_{0}m(I)$, taking $K>0$ large enough so that
$c_{5}^{1/2}K^{-1}<c_{0}/2$, we deduce

$\sum_{J_{k}\notin\mathcal{G}}m(J_{k})\geq\frac{c_{0}}{2}m(I)$

and one can take as $J$ in the statement, any of the arcs
$J_{k}\notin\mathcal{G}$. ∎

We are now ready to prove Theorem1.5.

###### Proof of Theorem 1.5.

Since $\sum a_{n}f^{n}$ converges at a set of positive measure, Theorem 1.1
gives that $\sum|a_{n}|^{2}<\infty$. Hence the function $F=\sum a_{n}f^{n}$ is
in $\operatorname{BMOA}$. The plan of the proof is to find constants $A,B>0$,
a sequence of arcs $\\{I_{k}\\}$ contained in $I$ and a sequence of positive
integers $\\{M_{k}\\}$ tending to infinity such that

(3.26) $\frac{1}{|I_{k}|}\int_{I_{k}}\operatorname{Re}F\,dm\geq
A\sum^{M_{k}}_{n=1}|a_{n}|-B.$

It is clear that (3.26) finishes the proof. The construction of the arcs
$\\{I_{k}\\}$ and the sequence $\\{M_{k}\\}$ is performed inductively. We
start using an idea of Weiss ([We]). Let $T^{\prime}<T$ be two (large)
positive integers to be fixed later such that $T/T^{\prime}$ is integer. Split
the sum $F=\sum a_{n}f^{n}$ into blocks of length $T$, that is,
$F=\sum\limits_{k\geq 0}G_{k}$, where

$G_{k}=\sum^{T-1}_{n=0}a_{kT+n}f^{kT+n},\quad k=0,1,2,\dotsc$

Next split $G_{2k}$ into successive blocks of length $T^{\prime}$ and pick the
subblock for which the sum of the modulus of the coefficients is the least. In
other words, pick $\mathcal{G}_{k}$ a subset of $T^{\prime}$ consecutive
integers in the interval $[2kT,(2k+1)T)$ such that

$\sum_{n\in\mathcal{G}_{k}}|a_{n}|\leq\sum_{\ell\in\mathcal{G}}|a_{\ell}|$

for any subset $\mathcal{G}$ of $T^{\prime}$ consecutive integers in
$[2kT,(2k+1)T)$. Since the number of disjoint subblocks of length $T^{\prime}$
in $[2kT,(2k+1)T)$ is $T/T^{\prime}$ we deduce

(3.27)
$\sum_{n\in\mathcal{G}_{k}}|a_{n}|\leq\frac{T^{\prime}}{T}\sum^{T-1}_{n=0}|a_{2kT+n}|.$

Each set of indices $\mathcal{G}_{k}$, $k\geq 1$, has $T^{\prime}$ elements
and the corresponding block

$S_{k}=\sum_{n\in\mathcal{G}_{k}}a_{n}f^{n}$

will be called a short block. The long blocks are defined as the blocks
between two short blocks. More concretely

$L_{1}=\sum^{N_{1}}_{n=0}a_{n}f^{n},$

where $N_{1}+1$ is the first index in $\mathcal{G}_{1}$ and

$L_{k}=\sum^{N_{k}}_{n=M_{k}}a_{n}f^{n},\quad k>1,$

where $M_{k}-1$ is the largest index in $\mathcal{G}_{k-1}$ and $N_{k}+1$ is
the smallest index in $\mathcal{G}_{k}$. Note that each short block has
$T^{\prime}$ terms while the number of terms in a long block is between $T$
and $3T$. Note also that if $\mathcal{L}_{k}=\\{n\in\mathbb{Z}:M_{k}\leq n\leq
N_{k}\\}$ denotes the set of indices appearing in the long block $L_{k}$, then
(3.27) implies

(3.28)
$\sum^{L}_{k=1}\sum_{n\in\mathcal{L}_{k}}|a_{n}|\geq\left(1-\frac{T^{\prime}}{T}\right)\sum^{N_{L}}_{n=1}|a_{n}|,\quad
L=1,2,\dotsc$

The idea is that (3.28) will imply that the short blocks are irrelevant and
the construction of the arcs $\\{I_{k}\\}$ and the sequence of integers
$\\{M_{k}\\}$ verifying (3.26) will depend on the long blocks. Moreover, the
fact that two long blocks are separated by a short block will provide a sort
of independence between the long blocks.

Let $s_{k}$ denote the $\ell^{2}$-norm of the coefficients in the long block
$L_{k}$, that is,

$s_{k}^{2}=\sum_{n\in\mathcal{L}_{k}}|a_{n}|^{2},\quad k=1,2,\dotsc$

Let $\varepsilon=\varepsilon(f)>0$ and $c=c(f)>0$ be the constants given by
Lemma 3.6. Let $0<\gamma=\gamma(f)<1$ be a constant to be fixed later
satisfying $1-\gamma<\varepsilon$. Let $D(0,R)$ denote the disc centered at
the origin of radius $R$. By the Denjoy–Wolff Theorem, $f^{n}$ tends to $0$
uniformly on compact sets of $\mathbb{D}$. The integer $T^{\prime}$ will be
taken large enough so that

(3.29) $f^{n}(D(0,1-c(1-\gamma)))\subset D(0,\varepsilon),\quad n\geq
T^{\prime}.$

Fix an arc $I^{*}\subset I$ such that $c^{-1}I^{*}\subset I$. The construction
starts with the first long block $L_{k_{0}}$ such that
$|f^{M_{k_{0}}}(z(I^{*}))|<\varepsilon$. Without loss of generality we can
assume that the constant $c$ is smaller than the constants appearing in Lemma
3.5. Apply Lemma 3.5 to find an arc $I_{k_{0}}$ with $c^{-1}I_{k_{0}}\subset
c^{-1}I^{*}\subset I$ such that

$\displaystyle c(1-\gamma)\leq 1-|f^{N_{k_{0}}}(z(I_{k_{0}}))|\leq 1-\gamma,$
$\displaystyle\frac{1}{m(I_{k_{0}})}\int_{I_{k_{0}}}\operatorname{Re}L_{k_{0}}\,dm\geq
cs_{k_{0}}.$

Assume by induction that we have constructed arcs
$I_{k_{0}},I_{k_{0}+1},\dotsc,I_{k}$, $k\geq k_{0}$, with $c^{-1}I_{j}\subset
c^{-1}I_{j-1}$, $j=k_{0}+1,\dotsc,k$, such that the following two conditions
hold

(3.30) $\displaystyle c(1-\gamma)\leq 1-|f^{N_{k}}(z(I_{k}))|\leq 1-\gamma,$
(3.31)
$\displaystyle\frac{1}{m(I_{k})}\int_{I_{k}}\operatorname{Re}\left(\sum^{k}_{j=k_{0}}L_{j}\right)\,dm\geq
c\sum^{k}_{j=k_{0}}s_{j}-c^{-1}(1-\gamma)^{1/2}\sum^{k-1}_{\ell=k_{0}}\left(\sum^{\ell}_{j=k_{0}}t^{N_{\ell}-N_{j}}s_{j}^{2}\right)^{1/2}.$

Here $0<t<1$ is the constant appearing in Lemma 3.6 and when $k=k_{0}$, we
replace the right hand side term of (3.31) by $cs_{k_{0}}$. Recall that two
different long blocks are separated by a short block which has $T^{\prime}$
terms. Hence the estimates (3.29) and (3.30) give that
$|f^{M_{k+1}}(z(I_{k}))|\leq\varepsilon$. Apply Lemma 3.6 to
$F_{1}=\sum\limits_{j=k_{0}}^{k}L_{j}$ and $F_{2}=L_{k+1}$, to find an arc
$I_{k+1}$ with $c^{-1}I_{k+1}\subset c^{-1}I_{k}$ such that

$\displaystyle c(1-\gamma)\leq 1-|f^{N_{k+1}}(z(I_{k+1}))|\leq 1-\gamma,$
(3.32)
$\displaystyle\frac{1}{m(I_{k+1})}\int_{I_{k+1}}\operatorname{Re}L_{k+1}\,dm\geq
cs_{k+1},$ (3.33)
$\displaystyle\frac{1}{m(I_{k+1})}\int_{I_{k+1}}\operatorname{Re}\sum^{k}_{j=k_{0}}L_{j}\,dm\geq\frac{1}{m(I_{k})}\int_{I_{k}}\operatorname{Re}\sum^{k}_{j=k_{0}}L_{j}\,dm-c^{-1}(1-\gamma)^{1/2}\left(\sum^{k}_{j=k_{0}}t^{N_{k}-N_{j}}s_{j}^{2}\right)^{1/2}.$

Then (3.33) and the induction assumption (3.31) give

$\frac{1}{m(I_{k+1})}\int_{I_{k+1}}\operatorname{Re}\left(\sum^{k}_{j=k_{0}}L_{j}\right)\,dm\geq
c\sum^{k}_{j=k_{0}}s_{j}-c^{-1}(1-\gamma)^{1/2}\sum^{k}_{\ell=k_{0}}\left(\sum^{\ell}_{j=k_{0}}t^{N_{\ell}-N_{j}}s_{j}^{2}\right)^{1/2}.$

Applying (3.32) we deduce

$\frac{1}{m(I_{k+1})}\int_{I_{k+1}}\operatorname{Re}\left(\sum^{k+1}_{j=k_{0}}L_{j}\right)\,dm\geq
c\sum^{k+1}_{j=k_{0}}s_{j}-c^{-1}(1-\gamma)^{1/2}\sum^{k}_{\ell=k_{0}}\left(\sum^{\ell}_{j=k_{0}}t^{N_{\ell}-N_{j}}s_{j}^{2}\right)^{1/2}.$

This concludes the inductive step.

Note that for any $k>k_{0}$, we have

$\sum^{k-1}_{\ell=k_{0}}\left(\sum^{\ell}_{j=k_{0}}t^{N_{\ell}-N_{j}}s_{j}^{2}\right)^{1/2}\leq\sum^{k-1}_{\ell=k_{0}}\sum^{\ell}_{j=k_{0}}t^{(N_{\ell}-N_{j})/2}s_{j}=\sum^{k-1}_{j=k_{0}}s_{j}\sum^{k-1}_{\ell=j}t^{(N_{\ell}-N_{j})/2}\leq(1-t^{1/2})^{-1}\sum^{k-1}_{j=k_{0}}s_{j}.$

Pick $0<\gamma<1$ sufficiently close to $1$ so that
$c^{-1}(1-\gamma)^{1/2}(1-t^{1/2})^{-1}\leq c/2$. The estimate (3.31) gives

(3.34)
$\frac{1}{m(I_{k})}\int_{I_{k}}\operatorname{Re}\left(\sum^{k}_{j=k_{0}}L_{j}\right)\,dm\geq\frac{c}{2}\sum^{k}_{j=k_{0}}s_{j}.$

Recall that each long block has at most $3T$ terms. Hence Cauchy-Schwarz’s
inequality gives

$\sum_{n\in\mathcal{L}_{j}}|a_{n}|\leq s_{j}(3T)^{1/2},\quad j=1,2,\dotsc$

Then apply (3.34) to deduce

(3.35)
$\frac{1}{m(I_{k})}\int_{I_{k}}\operatorname{Re}\sum^{k}_{j=k_{0}}(L_{j}+S_{j})\,dm\geq\frac{c}{2}(3T)^{-1/2}\sum^{k}_{j=k_{0}}\sum_{n\in\mathcal{L}_{j}}|a_{n}|-\sum^{k}_{j=k_{0}}\sum_{n\in\mathcal{G}_{j}}|a_{n}|,\quad
k\geq k_{0}.$

Let $\mathcal{A}_{k}$ be the set of indices appearing in
$\sum\limits_{j=k_{0}}^{k}(L_{j}+S_{j})$, that is,
$\mathcal{A}_{k}=\bigcup\limits_{j=k_{0}}^{k}(\mathcal{L}_{j}\cup\mathcal{G}_{j})$,
$k\geq k_{0}$. Applying (3.35), (3.27) and (3.28) we obtain

$\frac{1}{m(I_{k})}\int_{I_{k}}\operatorname{Re}\sum^{k}_{j=k_{0}}(L_{j}+S_{j})\,dm\geq\frac{c}{2}(3T)^{-1/2}\left(1-\frac{2T^{\prime}}{T}\right)\sum_{n\in\mathcal{A}_{k}}|a_{n}|-\frac{2T^{\prime}}{T}\sum_{n\in\mathcal{A}_{k}}|a_{n}|.$

We now choose $T$ so that $T^{\prime}T^{-1/2}$ is sufficiently small so that

$\frac{c}{2}(3T)^{-1/2}\left(1-\frac{2T^{\prime}}{T}\right)-\frac{2T^{\prime}}{T}\geq\frac{c}{4}T^{-1/2}.$

Then

(3.36)
$\frac{1}{m(I_{k})}\int_{I_{k}}\operatorname{Re}\sum^{k}_{j=k_{0}}(L_{j}+S_{j})\,dm\geq\frac{c}{4}T^{-1/2}\sum_{n\in\mathcal{A}_{k}}|a_{n}|.$

As observed previously, $|f^{M_{k+1}}(z(I_{k}))|\leq\varepsilon$. By Corollary
2.2 there exists a constant $0<c_{1}<1$ such that $|f^{n}(z(I_{k}))|\leq
c_{1}^{n-M_{k+1}}$, $n\geq M_{k+1}$. Then

$\sum_{n\geq
M_{k+1}}|a_{n}||f^{n}(z(I_{k}))|\leq(1-c_{1}^{2})^{-1/2}\left(\sum_{n\geq
M_{k+1}}|a_{n}|^{2}\right)^{1/2}.$

Since there exists a constant $c_{2}=c_{2}(f)>0$ such that

$\left|\frac{1}{m(I_{k})}\int_{I_{k}}\sum_{n\geq
M_{k+1}}a_{n}f^{n}\,dm-\sum_{n\geq M_{k+1}}a_{n}f^{n}(z(I_{k}))\right|\leq
c_{2}\left(\sum_{n\geq M_{k+1}}|a_{n}|^{2}\right)^{1/2},$

the estimate (3.36) gives

$\frac{1}{m(I_{k})}\int_{I_{k}}\operatorname{Re}\sum^{\infty}_{n=1}a_{n}f^{n}\,dm\geq\frac{c}{4}T^{-1/2}\sum_{n\in\mathcal{A}_{k}}|a_{n}|-(c_{2}+(1-c_{1}^{2})^{-1/2})\left(\sum_{n\geq
1}|a_{n}|^{2}\right)^{1/2}.$

This gives (3.26) and finishes the proof. ∎

##  4 Other function spaces

Let $A(\mathbb{D})$ denote the disc algebra, that is, the space of analytic
functions in $\mathbb{D}$ which extend continuously to
$\overline{\mathbb{D}}=\\{z\in\mathbb{C}:|z|\leq 1\\}$. The only inner
functions which belong to $A(\mathbb{D})$ are the finite Blaschke products.
See [Ga, p. 72]. Our next result is an easy consequence of Theorem 1.5.

######  Theorem 4.1.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let
$\\{a_{n}\\}$ be a sequence of complex numbers not identically zero. Then
$\sum\limits_{n=1}^{\infty}a_{n}f^{n}\in A(\mathbb{D})$ if and only if $f$ is
a finite Blaschke product and $\sum\limits_{n=1}^{\infty}|a_{n}|<\infty$.

###### Proof.

Write $F=\sum\limits_{n=1}^{\infty}a_{n}f^{n}$. The sufficiency follows from
the uniform convergence of the partial sums $\sum\limits_{n=1}^{N}a_{n}f^{n}$.
We now prove the necessity. Assume $F\in A(\mathbb{D})$. Since
$\sup\\{|F(\xi)|:\xi\in\partial\mathbb{D}\\}<\infty$, Theorem 1.5 gives that
$\sum\limits_{n=1}^{\infty}|a_{n}|<\infty$. We now show that $f$ is a finite
Blaschke product arguing by contradiction. Assume that $f$ does not extend
analytically at any neighbourhood of the point $\xi\in\partial\mathbb{D}$. By
Frostman’s Theorem (see [Ga, p. 77]) there exists a set $E\subset\mathbb{D}$
of logarithmic capacity zero such that for any $\alpha\in\mathbb{D}\backslash
E$ one can find a sequence $\\{z_{k}\\}=\\{z_{k}(\alpha)\\}$, of points in
$\mathbb{D}$ converging to $\xi$, such that $f(z_{k})=\alpha$ for any $k$.
Then

$F(z_{k})=\sum^{\infty}_{n=1}a_{n}f^{n-1}(\alpha),\quad k=1,2,\dotsc$

Here $f^{0}(\alpha)=\alpha,\alpha\in\mathbb{D}$. Since $\\{z_{k}\\}$ converges
to $\xi$ and $F\in A(\mathbb{D})$, for any $\alpha\in\mathbb{D}\backslash E$
we have

$\sum_{n=1}^{\infty}a_{n}f^{n-1}(\alpha)=F(\xi).$

Hence the function $\sum\limits^{\infty}_{n=1}a_{n}f^{n-1}$ is constant in
$\mathbb{D}$. Since it vanishes at the origin, we
deduce$\sum\limits_{n=1}^{\infty}a_{n}f^{n-1}\equiv 0$. Hence
$\sum\limits_{n=1}^{\infty}a_{n}f^{n}\equiv 0$. By Lemma 2.3, $a_{n}=0$ for
any $n\geq 1$, which gives the contradiction. ∎

Let $\mathcal{D}$ denote the Dirichlet space of analytic functions $f$ in
$\mathbb{D}$ such that

$\|f\|^{2}_{\mathcal{D}}=\int_{\mathbb{D}}|f^{\prime}(z)|^{2}\,dA(z)<\infty,$

where $dA(z)$ denotes the area measure. Note that $\|f\|^{2}_{\mathcal{D}}$
can be understood as the area of the image domain $f(\mathbb{D})$, counting
multiplicities. Hence the only inner functions which belong to the Dirichlet
space are finite Blaschke products. Our next result is a description of linear
combinations of iterates of an inner function which belong to the Dirichlet
space.

######  Theorem 4.2.

Let $f$ be a finite Blaschke product with $N>1$ zeros and with $f(0)=0$. Let
$\\{a_{n}\\}$ be a sequence of complex numbers with
$\sum\limits_{n=1}^{\infty}|a_{n}|^{2}<\infty$ and
$F=\sum\limits_{n=1}^{\infty}a_{n}f^{n}$. Then $F\in\mathcal{D}$ if and only
if $\sum\limits^{\infty}_{n=1}|a_{n}|^{2}N^{n}<\infty$. Moreover there exists
a universal constant $C=C(f)>0$ such that

$C^{-1}\sum^{\infty}_{n=1}|a_{n}|^{2}N^{n}\leq\int_{\mathbb{D}}|F^{\prime}(z)|^{2}\,dA(z)\leq
C\sum^{\infty}_{n=1}|a_{n}|^{2}N^{n}.$

###### Proof.

Note that

$\int_{\mathbb{D}}|F^{\prime}(z)|^{2}\,dA(z)=\sum^{\infty}_{n=1}|a_{n}|^{2}b_{n,n}+2\operatorname{Re}\sum^{\infty}_{n=1}\overline{a}_{n}\sum_{k>n}a_{k}b_{k,n},$

where

$b_{k,n}=\int_{\mathbb{D}}\overline{(f^{n})^{\prime}(z)}(f^{k})^{\prime}(z)\,dA(z),\quad
k,n=1,2,\dotsc$

Since $f^{n}$ is a Blaschke product with $N^{n}$ zeros, the area counting
multiplicities, of $f^{n}(\mathbb{D})$ is $\pi N^{n}$. Then

$\int_{\mathbb{D}}|(f^{n})^{\prime}(z)|^{2}\,dA(z)=\pi N^{n}.$

On the other hand, the change of variables formula with multiplicities (see
[EG, p. 122]) gives

$b_{k,n}=\int_{\mathbb{D}}|(f^{n})^{\prime}(z)|^{2}(f^{k-n})^{\prime}(f^{n}(z))\,dA(z)=\int_{\mathbb{D}}N^{n}(f^{k-n})^{\prime}(w)\,dA(w),\quad
k>n.$

The mean value property gives $b_{k,n}=\pi N^{n}f^{\prime}(0)^{k-n}$, $k>n$.
Hence

$\frac{1}{\pi}\int_{\mathbb{D}}|F^{\prime}(z)|^{2}\,dA(z)=\sum^{\infty}_{n=1}|a_{n}|^{2}N^{n}+2\operatorname{Re}\sum^{\infty}_{n=1}\overline{a}_{n}N^{n}\sum_{k>n}a_{k}f^{\prime}(0)^{k-n}=\vec{a}^{\,t}T\vec{a},$

where $\vec{a}$ denotes the vector $(a_{n}N^{n/2})_{n\geq 1}$ and $T$ is the
Toeplitz matrix whose entries are

$t_{n,k}=(f^{\prime}(0)N^{-1/2})^{k-n},\quad k\geq n;\qquad
t_{n,k}=(\overline{f^{\prime}(0)}N^{-1/2})^{n-k},\quad n\geq k.$

Consider the symbol

$t(\xi)=\sum^{\infty}_{n=-\infty}t_{n,0}\xi^{n}=\frac{1-|f^{\prime}(0)|^{2}N^{-1}}{|1-f^{\prime}(0)N^{-1/2}\xi|^{2}},\quad\xi\in\partial\mathbb{D}.$

It is well known that $T$ diagonalizes and its eigenvalues are between the
minimum and the maximum of $t$. See [BG]. Since $C(f,N)^{-1}\leq t(\xi)\leq
C(f,N)$, a.e. $\xi\in\partial\mathbb{D}$, where

$C(f,N)=\frac{1+|f^{\prime}(0)|N^{-1/2}}{1-|f^{\prime}(0)|N^{-1/2}},\quad\text{a.e.\
}\xi\in\partial\mathbb{D},$

we deduce that

$C(f,N)^{-1}\sum^{\infty}_{n=1}|a_{n}|^{2}N^{n}\leq\frac{1}{\pi}\int_{\mathbb{D}}|F^{\prime}(z)|^{2}\,dA(z)\leq
C(f,N)\sum^{\infty}_{n=1}|a_{n}|^{2}N^{n}.$

∎

Let $\mathcal{B}$ denote the Bloch space of analytic functions $f$ in
$\mathbb{D}$ such that

$\|f\|_{\mathcal{B}}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|f^{\prime}(z)|<\infty.$

It is well known that a lacunary series is in the Bloch space if and only if
its coefficients are uniformly bounded. See [ACP]. In our setting this
condition is still sufficient but we will see that it is not necessary.

######  Theorem 4.3.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let
$\\{a_{n}\\}$ be a bounded sequence of complex numbers. Then
$F=\sum\limits_{n=1}^{\infty}a_{n}f^{n}\in\mathcal{B}$ and there exists a
constant $C=C(f)>0$ such that $\|F\|_{\mathcal{B}}\leq C\sup\limits_{n\geq
1}|a_{n}|$.

###### Proof.

Let $0<r_{0}=r_{0}(f)<1$ and $0<c_{0}=c_{0}(f)<1$ be the constants given by
Corollary 2.2. Since $f^{n}$ tends to $0$ uniformly on compacts of
$\mathbb{D}$, for any $z\in\mathbb{D}$ we can pick $N(z)$ to be the minimum
positive integer $n$ such that $|f^{n}(w)|\leq r_{0}$ for any $w\in\mathbb{D}$
with $\rho(w,z)\leq 1/2$. Corollary 2.2 gives

(4.1) $\sup\\{|f^{n}(w)|:\rho(w,z)\leq r_{0}\\}\leq c_{0}^{n-N(z)},\quad n\geq
N(z),\quad z\in\mathbb{D}.$

By Cauchy’s estimate, there exist a universal constant $c_{1}>0$ such that

$(1-|z|)|(f^{n})^{\prime}(z)|\leq c_{1}c_{0}^{n-N(z)},\quad n\geq N(z),\quad
z\in\mathbb{D}.$

Hence

(4.2) $(1-|z|)\sum_{n\geq N(z)}|(f^{n})^{\prime}(z)|\leq
c_{1}(1-c_{0})^{-1},\quad z\in\mathbb{D}.$

Let $M=N(z)-1$. Note that there exists a point $w$ with $\rho(w,z)\leq 1/2$
such that $|f^{M}(w)|\geq r_{0}$. According to part (a) of Lemma 2.1, there
exists a constant $0<c_{2}<1$ such that $1-|f^{n}(w)|\leq c_{2}^{M-n}$ for any
$n<N(z)$. By Schwarz’s Lemma there exists a constant $c_{3}>0$ such that
$1-|f^{n}(z)|\leq c_{3}c_{2}^{M-n}$ for any $n<N(z)$. Since
$(1-|z|^{2})|(f^{n})^{\prime}(z)|\leq 1-|f^{n}(z)|^{2}$, $z\in\mathbb{D}$, we
deduce

(4.3) $(1-|z|^{2})\sum_{n=1}^{M}|(f^{n})^{\prime}(z)|\leq
c_{3}(1-c_{2})^{-1},\quad z\in\mathbb{D}.$

The estimates (4.2) and (4.3) give that $\|F\|_{\mathcal{B}}\leq
c\sup\limits_{n}|a_{n}|$ with $c=c_{1}(1-c_{0})^{-1}+c_{3}(1-c_{2})^{-1}$. ∎

Next we will show that the converse estimate in Theorem 4.3 does not hold,
that is, there exist an inner function $f$ with $f(0)=0$ and unbounded
sequence $\\{a_{n}\\}$ of complex numbers such that $\sum
a_{n}f^{n}\in\mathcal{B}$. We start with an auxiliary result on the hyperbolic
derivative $D_{h}f$ defined in (2.1). Denote $f^{0}(z)=z,z\in\mathbb{D}$.

######  Lemma 4.4.

Let $f\in H^{\infty}(\mathbb{D})$, $\|f\|_{\infty}\leq 1$. Then for any
$n=1,2,\dotsc$, we have

$D_{h}(f^{n})(z)=\prod^{n-1}_{k=0}D_{h}(f)(f^{k}(z)),\quad z\in\mathbb{D}.$

###### Proof.

Note that $(f^{n})^{\prime}(z)=\prod\limits_{k=0}^{n-1}f^{\prime}(f^{k}(z))$,
$n\geq 1$. Then

$(1-|z|^{2})\frac{(f^{n})^{\prime}(z)}{1-|f^{n}(z)|^{2}}=\prod^{n-1}_{k=0}\frac{(1-|f^{k}(z)|^{2}f^{\prime}(f^{k}(z))}{1-|f^{k+1}(z)|^{2}}.$

∎

Let $f\in H^{\infty}$ with $\|f\|_{\infty}\leq 1$. Schwarz’s Lemma gives that
$D_{h}f(z)\leq 1$, $z\in\mathbb{D}$. Given any positive gauge function
$w\colon[0,1]\to(0,+\infty)$ satisfying a mild regularity condition, such that

$\int^{1}_{0}\frac{w^{2}(t)}{t}\,dt=\infty,$

there exists an inner function $f$ with $f(0)=0$ such that $D_{h}f(z)\leq
w(1-|z|)$, $z\in\mathbb{D}$. See [AAN] or [Sm]. In particular, given any
$0<\tau<1$, there exists an inner function $f$ with $f(0)=0$ such that
$D_{h}f(z)\leq\tau$. Let $\\{a_{n}\\}$ be any sequence of complex numbers such
that

(4.4) $\sum^{\infty}_{n=1}|a_{n}|\tau^{n}<\infty.$

We next show that the function $F=\sum\limits^{\infty}_{n=1}a_{n}f^{n}$ is in
the Bloch space. Note that

$(1-|z|^{2})|F^{\prime}(z)|\leq\sum^{\infty}_{n=1}|a_{n}||(f^{n})^{\prime}(z)|(1-|z|^{2})=\sum^{\infty}_{n=1}|a_{n}|(1-|f^{n}(z)|^{2})D_{h}(f^{n})(z),\quad
z\in\mathbb{D}.$

Apply Lemma 4.4 to deduce $D_{h}(f^{n})(z)\leq\tau^{n}$, $n\geq 1$,
$z\in\mathbb{D}$. Then

$(1-|z|^{2})|F^{\prime}(z)|\leq\sum^{\infty}_{n=1}|a_{n}|\tau^{n},\quad
z\in\mathbb{D}.$

Hence if condition (4.4) holds, we have $F\in\mathcal{B}$.

We finish this section showing the following converse of part (b) of Corollary
1.3.

######  Corollary 4.5.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let
$\\{a_{n}\\}$ be a non-identically zero sequence of complex numbers with
$\sum\limits^{\infty}_{n=1}|a_{n}|^{2}<\infty$. Assume
$F=\sum\limits_{n=1}^{\infty}a_{n}f^{n}\in\operatorname{VMOA}(\mathbb{D})$.
Then $f$ is a finite Blaschke product.

###### Proof.

Assume $f$ is not a finite Blaschke product. then there exists a sequence of
points $\\{z_{k}\\}$ in $\mathbb{D}$ with $|z_{k}|\to 1$ such that
$f(z_{k})\to 0$, as $k\to\infty$. Lemma 3.2 gives that

$\int_{\partial\mathbb{D}}\left|F(\xi)-F(z_{k})\right|^{2}P(z_{k},\xi)\,dm(\xi)\geq\frac{1}{2}\,\frac{1-|f^{\prime}(0)|}{1+|f^{\prime}(0)|}\sum^{\infty}_{n=1}|a_{n}|^{2},$

if $k$ is sufficiently large. Since the integral above tends to $0$ as $k$
tends to $\infty$, we deduce that $a_{n}=0$ for any $n$. This finishes the
proof. ∎

##  5 Open problems

We close the paper mentioning some open problems we have not explored.

###  Analytic continuation and smoothness.

Hadamard’s Theorem says that if a lacunary power series can be extended
analytically across an arc of the unit circle, it can actually be extended
analytically to a neighbourhood of the closed unit disc (see [Zy, p. 208]). It
is natural to ask if the following analogous result holds.

### Problem 1.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Let
$\\{a_{n}\\}$ be a sequence of complex numbers whose modulus are square
summable and consider $F=\sum\limits^{\infty}_{n=1}a_{n}f^{n}$. Assume that
$F$ extends analytically across an arc $I\subset\partial\mathbb{D}$. Is it
true that there exist $n_{0}>0$ such that $a_{n}=0$ for any $n>n_{0}$ and that
$f^{n}$ extends analytically across $I$ for any $n\leq n_{0}$?

For $0<\alpha<1$, let $\operatorname{Lip}_{\alpha}(\overline{\mathbb{D}})$
denote the space of continuous functions $g$ in $\overline{\mathbb{D}}$ for
which there exists a constant $C=C(g)>0$ such that $|g(z)-g(w)|\leq
C|z-w|^{\alpha}$, for any pair of points $z,w\in\overline{\mathbb{D}}$.
Lacunary series which belong to
$\operatorname{Lip}_{\alpha}(\overline{\mathbb{D}})$ can be described in terms
of their coefficients. It is then natural to ask for the corresponding result
in our context.

### Problem 2.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Describe the sequences of complex numbers $\\{a_{n}\\}$ with square summable
modulus, such that
$F=\sum\limits^{\infty}_{n=1}a_{n}f^{n}\in\operatorname{Lip}_{\alpha}(\overline{\mathbb{D}})$.

###  Peano curves.

Let $\sum a_{k}z^{n_{k}}$ be a lacunary series with $\sum|a_{k}|=\infty$ and
$\lim a_{k}=0$. Then for any $w\in\mathbb{C}$, there exists
$\xi\in\partial\mathbb{D}$ such that $\sum a_{k}\xi^{n_{k}}$ converges to $w$
([We]). It is then natural to ask for the following analogous result.

### Problem 3.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Let
$\\{a_{n}\\}$ be a sequence of complex numbers with
$\sum\limits_{n=1}^{\infty}|a_{n}|=\infty$, but $\lim_{n\to\infty}a_{n}=0$. Is
it true that for any $w\in\mathbb{C}$, there exists $\xi\in\partial\mathbb{D}$
such that $\sum\limits_{n=1}^{\infty}a_{n}f^{n}(\xi)$ converges to $w$?

There is also a version of the question above in the interior of the unit
disc. Murai proved that any lacunary series $\sum a_{k}z^{n_{k}}$ convergent
in $\mathbb{D}$ with $\sum|a_{k}|=\infty$, takes any complex value infinitely
often. See [Mu]. It is natural to ask for the following analogous result.

### Problem 4.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Let
$\\{a_{n}\\}$ be any sequence of complex numbers such that
$F(z)=\sum\limits_{n=1}^{\infty}a_{n}f^{n}(z)$ is analytic in $\mathbb{D}$.
Assume $\sum\limits_{n=1}^{\infty}|a_{n}|=\infty$. Is true that for any
$w\in\mathbb{C}$ there exists $z\in\mathbb{D}$ such that $F(z)=w$?

###  Abel’s Theorem.

Let $F(z)=\sum a_{n}z^{n}$ be a power series with radius of convergence $1$.
Let $\xi\in\partial\mathbb{D}$ such that $\sum a_{n}{\xi}^{n}$ converges. The
classical Abel’s Theorem says that $F(z)$ tends to $\sum a_{n}{\xi}^{n}$, as
$z$ approaches non-tangentially to $\xi$. The classical Hardy-Littlewood High
Indices Theorem asserts that for lacunary series, the converse holds. In other
words, if $F(z)=\sum a_{k}z^{n_{k}}$ is a lacunary series which has limit $L$
when $z$ approaches non-tangentially a point $\xi\in\partial\mathbb{D}$, then
$\sum a_{k}{\xi}^{n_{k}}$ converges to $L$. It is then natural to ask for the
following analogous results.

### Problem 5.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Let
$\\{a_{n}\\}$ be any sequence of complex numbers such that
$F(z)=\sum\limits_{n=1}^{\infty}a_{n}f^{n}(z)$ is analytic in $\mathbb{D}$.
Let $\xi\in\partial\mathbb{D}$ such that $f^{n}(\xi)=\lim\limits_{r\to
1}f^{n}(r\xi)$ exists for any positive integer $n$. Assume
$\sum\limits_{n=1}^{\infty}a_{n}f^{n}(\xi)$ converges. Is it true that
$\lim\limits_{r\to 1}F(r\xi)$ exists?

### Problem 6.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Let
$\\{a_{n}\\}$ be any sequence of complex numbers such that
$F(z)=\sum\limits_{n=1}^{\infty}a_{n}f^{n}(z)$ is analytic in $\mathbb{D}$.
Assume $F(z)$ has limit when $z$ approaches non-tangentially a point
$\xi\in\partial\mathbb{D}$. Is it true that
$\sum\limits_{n=1}^{\infty}a_{n}f^{n}(\xi)$ converges?

## References

  * [AAN] Aleksandrov, A. B.; Anderson, J. M.; Nicolau, A. Inner functions, Bloch spaces and symmetric measures. _Proc. London Math. Soc. (3)_ 79 (1999), no. 2, 318–352.
  * [ACP] Anderson, J. M.; Clunie, J.; Pommerenke, Ch. On Bloch functions and normal functions. _J. Reine Angew. Math._ 270 (1974), 12–37.
  * [Ba] Baernstein, A. Analytic functions of bounded mean oscillation _Aspects of contemporary complex analysis_. Proc. NATO Adv. Study Inst., Univ. Durham, 1979, pp. 3–36. Academic Press, London-New York, 1980.
  * [BG] Böttcher, A.; Grudsky, S. M. _Toeplitz matrices, asymptotic linear algebra and functional analysis_. Birkäuser Verlag, Basel, 2000.
  * [EG] Evans, L. C.; Gariepy, R. F. _Measure theory and fine properties of functions_. Revised edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.
  * [Ga] Garnett, J. B. _Bounded analytic functions_. Revised first edition. Graduate Texts in Mathematics 236. Springer, New York, 2007.
  * [GJ] Garnett, J. B.; Jones, P. W. The distance in BMO to $L^{\infty}$ _Ann. of Math._ 108 (1978), no. 2, 373–393.
  * [Ka] Kahane, J.-P. _Some random series of functions_. Second edition. Cambridge Studies in Advanced Mathematics 5. Cambridge University Press, Cambridge, 1985.
  * [Mu] Murai, T. The value-distribution of lacunary series and a conjecture of Paley. _Ann. Inst. Fourier_ 31 (1981), no. 1. vii, 135–156.
  * [Ne] Neuwirth, J. H. Ergodicity of some mappings of the circle and the line. _Israel J. Math._ 31 (1978), no. 3.–4, 359–367.
  * [NS1] Nicolau, A.; Soler i Gibert, O. Approximation in the Zygmund class. _J. London Math. Soc. (2)_ 101 (2020), no. 1, 226–246.
  * [NS2] Nicolau, A.; Soler i Gibert, O. A central limit theorem for inner functions. Preprint, 2020. arXiv:2006.12105
  * [Po] Pommerenke, Ch. On ergodic properties of inner functions. _Math. Ann._ 256 (1981), no. 1, 43–50.
  * [SS] Saksman, E.; Soler i Gibert, O. Approximation in the Zygmund and Hölder classes on $\mathbb{R}^{n}$. arXiv:2009.09752.
  * [Sh] Shapiro, J. H. _Composition operators and classical function theory_. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.
  * [Sm] Smith, W. Inner functions in the hyperbolic little Bloch class. _Michigan Math. J._ 45 (1998), no. 1, 103–114.
  * [We] Weiss, M. Concerning a theorem of Paley on lacunary power series. _Acta Math._ 102 (1959), 225–238.
  * [Zy] Zygmund, A. _Trigonometric series_. Vol. I, II. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002.