# One component bounded functions

Carlo Bellavita and Artur Nicolau The second author is supported in part by
the Generalitat de Catalunya (grant 2017 SGR 395), the Spanish Ministerio de
Ciencia e Innovación (project MTM2017-85666-P) and the Spanish Research Agency
(María de Maeztu Program CEX2020-001084-M )

###### Abstract

Three different characterizations of one-component bounded analytic functions
are provided. The first one is related to the the inner-outer factorization,
the second one is in terms of the size of the reproducing kernels in the
corresponding de Branges-Rovnyak spaces and the last one concerns the
associated Clark measure.

##  1 Introduction

Let $\mathbb{D}$ be the open unit disc of the complex plane. Let
${H}^{\infty}(\mathbb{D})$ be the algebra of bounded analytic functions in
$\mathbb{D}$ and let $\|b\|_{\infty}=\sup\\{|b(z)|:z\in\mathbb{D}\\}$ for
$b\in{H}^{\infty}(\mathbb{D})$. An inner function is a bounded analytic
function in $\mathbb{D}$ whose radial limits have modulus one at almost every
point of the unit circle $\partial\mathbb{D}$. An inner function $b$ is called
one-component if there exists a number $0<C<1$ such that the sublevel set
$\\{z\in\mathbb{D}:|b(z)|<C\\}$ is connected. One-component inner functions
were introduced by B. Cohn in [C] in connection with Carleson measures for
model spaces and have been already widely studied. Indeed, as subsequently
proved by A. Volberg and S.Treil in [VT] and by A. Aleksandrov in [A2], it is
possible to provide an useful geometric characterization of Carleson measures
for model spaces corresponding to one-component inner functions. Other
descriptions of one-component inner functions have been given in [A3], [NR]
and [Be], algebraic properties of this family have been studied in [CM1], [R]
and [CM2], and further properties of the corresponding model spaces have been
given in [A1], [Ba], [Be] and [BBK].

This paper is devoted to the study of one-component bounded functions. We
start with the definition of this family. Given $b\in H^{\infty}(\mathbb{D})$
with $\|b\|_{\infty}=1$, its spectrum is defined as

$\text{spec}(b):=\left\\{\xi\in\partial\mathbb{D}:\
\liminf_{z\to\xi}|b(z)|<1\right\\}\ .$

######  Definition 1.

Let $b\in H^{\infty}(\mathbb{D})$ with $\|b\|_{\infty}=1$. Then, $b$ is called
one-component if there exists a constant $0<C<1$ such that the following two
conditions hold

  1. (a)

$\text{spec}(b)\subseteq\overline{\left\\{z\in\mathbb{D}:|b(z)|\leq
C\right\\}}$ ,

  2. (b)

The set $\\{z\in\mathbb{D}:\ |b(z)|<C\\}$ is connected.

One-component bounded functions were introduced by A. Baranov, E. Fricain and
J. Mashreghi in [BFM], where the authors studied Carleson measure for de
Branges-Rovnyak spaces. They provided an useful characterization for such
measures for de Branges-Rovnyak spaces corresponding to one-component bounded
functions. Note that one-component inner functions are one-component bounded
functions because if $b$ is inner, spec$(b)=\left\\{\xi\in\partial\mathbb{D}:\
\liminf_{z\to\xi}|b(z)|=0\right\\}$ (see [Ga]). Therefore, when $b$ is inner,
Condition _a)_ of Definition 1 is always satisfied.

In this paper, we provide several characterizations of one-component bounded
functions, which are analogous to the already known results holding for one-
component inner functions. We will extend to this bigger family the
description of [NR] in terms of the positions of the zeros, of [A2] in terms
of the growth of the reproducing kernels of the associated de Branges-Rovnyak
spaces and, finally, of [Be] in terms of the expression of the associated
Clark measures. It is worth mentioning that our proofs use some ideas from
these previous papers but some new arguments are also required to deal with
more general situation.

Given $b\in H^{\infty}(\mathbb{D})$, as already done in [Bi], we consider the
measure $\sigma_{b}$ with support in $\overline{\mathbb{D}}$ associated to
$|b(z)|$. Indeed, if $b=B_{\\{z_{n}\\}}S_{\mu}O_{b}$ is the classical
factorization of $b$ in a Blaschke product $B_{\\{z_{n}\\}}$ with zeros
$\\{z_{n}\\}$, a singular inner function $S_{\mu}$ and an outer factor
$O_{b}$, we define

$d\sigma_{b}(z):=\sum_{n}\left(1-|z_{n}|\right)\delta_{z_{n}}(z)+d\mu(z)+\log|b(z)|^{-1}dm(z),\quad
z\in\overline{\mathbb{D}},$

where $dm$ is the Lebesgue measure on $\partial\mathbb{D}$. We recall also
that if $z\in\mathbb{D}\setminus\\{0\\}$, the associated Carleson square
$Q(z)$ is defined as $Q(z):=\left\\{w\in\mathbb{D}:\ |z|<|w|<1,\
|w/|w|-z/|z||<(1-|z|)/2\right\\}\ .$ Given a Carleson square $Q(z)$,
$z\in\mathbb{D}\setminus\\{0\\}$, we denote its sidelenght by
$\ell(Q(z))=1-|z|$ and its top part by $T(Q(z)):=\left\\{w\in
Q(z):|z|<|w|<(1+|z|)/2\right\\}.$ Our first description of one-component
bounded functions uses the measure $\sigma_{b}$ and it is analogous to the
main result in [NR].

######  Theorem 1.1.

Let $b\in H^{\infty}(\mathbb{D})$ with $||b||_{\infty}=1$. Then, $b$ is one-
component if and only if there exists $0<C_{1}<1$ such that

(1) $\sigma_{b}\left(Q(z)\right)=0\quad\text{whenever}\quad|b(z)|\geq C_{1}\
.$

It will be shown that the spectrum of a one-component bounded function must be
a proper closed subset of the unit circle. As an application of Theorem 1.1,
we obtain the following converse result.

######  Theorem 1.2.

Let $E\subsetneq\mathbb{T}$ be a closed set. Then there exists a one-component
bounded function $b$ such that spec$(b)=E$.

We point out that, when $m(E)=0$, it is always possible to find a one-
component inner function whose spectrum is equal to $E$ ([NR]).

Our second characterization of one-component bounded functions uses the
reproducing kernels of the associated de Branges-Rovnyak spaces. In this paper
we will not work with the de Branges-Rovnyak spaces and we will not introduce
them. The interested reader may deepen this subject looking at [FM2]. The only
fact one needs to keep in mind, is that the de Branges-Rovnyak spaces are
reproducing kernel Hilbert spaces of holomorphic functions in the unit disk
$\mathbb{D}$. Let $b\in H^{\infty}(\mathbb{D})$ with $||b||_{\infty}=1$. The
reproducing kernel of $H(b)$ at $a\in\mathbb{D}$, is

$k^{b}_{a}(z):=\frac{1-\overline{b(a)}b(z)}{1-\overline{a}z}\ ,\qquad
z\in\mathbb{D},$

that is, $\left\langle f,k^{b}_{a}\right\rangle_{H(b)}=f(a)$, for every $f\in
H(b)$. If the function $b$ is inner, the associated de Branges-Rovnyak space
is the classical model space. We next describe one-component bounded functions
in terms of the size of $k^{b}_{a}(z)$.

######  Theorem 1.3.

  1. a)

Let $b$ be a one-component bounded function. Then there exists a constant
$C=C(b)>0$ such that for every $a\in\mathbb{D}$ we have

(2)
$\sup_{z\in\mathbb{D}}\bigg{|}\frac{1-\overline{b(a)}b(z)}{1-\overline{a}z}\bigg{|}\leq
C\frac{1-|b(a)|^{2}}{1-|a|^{2}}\ .$

  2. b)

Let $b\in H^{\infty}(\mathbb{D})$ with $\|b\|_{\infty}=1$. Assume there exists
a constant $0<C<1$ such that

(3) $\text{spec}(b)\subseteq\overline{\left\\{z\in\mathbb{D}:|b(z)|\leq
C\right\\}}\ .$

If

(4)
$\sup_{z\in\mathbb{D}}\bigg{|}\frac{1-\overline{b(a)}b(z)}{1-\overline{a}z}\bigg{|}=o\left(\frac{1}{1-|a|^{2}}\right)\
,\text{ as }|b(a)|\to 1,$

then $b$ is one-component.

In Theorem 1.3 we have considered the reproducing kernels $k^{b}_{a}(z)$ at a
point $a\in\mathbb{D}$. In the following results, we deal also with
$a\in\partial\mathbb{D}$. It is easy to see that any one-component bounded
function $b$ extends analytically across $\partial\mathbb{D}\setminus$
spec$(b)$, see [FM].

######  Theorem 1.4.

  1. a)

Let $b$ be a one-component bounded function. Then there exists a constant
$C=C(b)>0$ such that

(5) $\frac{|b(z)-b(\xi)|}{|z-\xi|}\leq C\big{|}b^{\prime}(\xi)\big{|}\ ,\qquad
z\in\mathbb{D},\,\xi\in\partial\mathbb{D}\setminus spec(b)\ .$

  2. b)

Let $b\in H^{\infty}(\mathbb{D})$ with $\|b\|_{\infty}=1$. Assume there exists
$0<C<1$ such that spec$(b)\subset\overline{\left\\{z\in\mathbb{D}:\ |b(z)|\leq
C\right\\}}$. If estimate (5) holds, the function $b$ is one-component.

Theorem 1.4 is applied to obtain the following result.

######  Theorem 1.5.

  1. a)

Let $b$ be a one-component bounded function. Then the function

(6) $H(\xi):=\begin{cases}1/b^{\prime}(\xi)&\ \text{ if
}\xi\notin\text{spec}(b)\\\ 0&\ \text{ if }\xi\in\text{spec}(b)\end{cases}$

is Lipschitz.

  2. b)

Let $b\in H^{\infty}(\mathbb{D})$ with $\|b\|_{\infty}=1$. Assume there exists
$0<C<1$ such that spec$(b)\subset\overline{\left\\{z\in\mathbb{D}:\ |b(z)|\leq
C\right\\}}$. If the function $H$ defined in (6) is Lipschitz, then $b$ is
one-component.

Theorem 1.5 is also equivalent to the following corollary.

######  Corollary 1.6.

Let $b\in H^{\infty}(\mathbb{D})$ with $\|b\|_{\infty}=1$. Then $b$ is one-
component if and only if the following three conditions hold:

  1. a)

There exists a constant $0<C<1$ such that
$\text{spec}(b)\subset\overline{\left\\{z\in\mathbb{D}:\ |b(z)|\leq
C\right\\}}\ .$

  2. b)

There exists a constant $C_{1}>0$ such that $|b^{\prime\prime}(\xi)|\leq
C_{1}|b^{\prime}(\xi)|^{2}$, for every $\xi\in\partial\mathbb{D}\setminus$
spec$(b)$

  3. c)

$|b^{\prime}(\xi)|\to\infty$, as $0<$dist$(\xi,$spec$(b))\to 0$.

These last three results are similar to their analogue versions for one-
component inner functions, presented in [A1] and [A3]. However, we point out
that our proofs are different. Indeed, unlike [A3], we do not make use of the
explicit expression of the norms of the reproducing kernels in de Branges-
Rovnyak spaces.

We describe also one-component bounded functions in terms of their Clark
measures. Clark measures of one-component inner functions were described in
[Be] by R.V. Bessonov. In the following, we give the corresponding result for
one-component bounded functions. Before stating the result, we recall the
definition and main properties of Clark measures. For a more complete
description we refer to [PS], [S] and [CMR].

Let $b\in H^{\infty}(\mathbb{D})$ with $\|b\|_{\infty}=1$ and $|\alpha|=1$.
Since $(\alpha+b)/(\alpha-b)$ is an analytic function in $\mathbb{D}$ having
positive real part, there exists a unique positive measure $\mu_{\alpha}$ in
the unit circle such that

(7)
$\frac{\alpha+b(z)}{\alpha-b(z)}=\int_{\partial\mathbb{D}}\frac{\xi+z}{\xi-z}d\mu_{\alpha}(\xi)+2i\frac{\Im\left(\bar{\alpha}b(0)\right)}{|\alpha-b(0)|^{2}}\
,\quad z\in\mathbb{D}.$

The measure $\mu_{\alpha}$ is called the Clark measure of the function $b$ at
the value $\alpha$.

We note that if $b(0)=0$ the measure $\mu_{\alpha}$ is a probability measure.
It follows from (7) that the real part of $(\alpha+b(z))/(\alpha-b(z))$ is the
Poisson integral of the measure $\mu_{\alpha}$. The measure $\mu_{\alpha}$ is
singular if and only if the function $b$ is inner. In this case, the measure
$\mu_{\alpha}$ is carried by $\left\\{\xi\in\partial\mathbb{D}:\
b(\xi)=\alpha\right\\}$.

For every positive, Borel measure $\mu$ on the unit circle
$\partial\mathbb{D}$ we denote by $a(\mu)$ the set of all its isolated atoms.
The set $P(\mu):=$ spt$(\mu)\setminus a(\mu)$ consists of all the accumulation
points in the support of $\mu$, denoted by spt$(\mu)$. We will say that an
atom $\xi\in a(\mu)$ has two neighbours if there is an open arc
$(\xi_{-},\xi_{+})$ on the unit circle $\partial\mathbb{D}$ with endpoints
$\xi_{\pm}\in a(\mu)$ such that $(\xi_{-},\xi_{+})\cap$ spt$(\mu)=\\{\xi\\}$.

######  Theorem 1.7.

Let $\mu$ be a positive, Borel measure on $\partial\mathbb{D}$. The following
two condition are equivalent:

  1. a)

$\mu$ is the Clark measure of a one-component bounded function.

  2. b)

$\mu=\mu_{a}+\mu_{s}$ and there exists a constant $C=C(\mu)>0$ such that

    1. i.

The measure $\mu_{a}$ is absolutely continuous and
$\mu_{a}(\xi)=g(\xi)dm(\xi)$, with $1/C\leq g(\xi)\leq C$ for $\mu_{a}$-almost
all $\xi\in\partial\mathbb{D}$.

    2. ii.

The measure $\mu_{s}$ is purely atomic and every atom has two neighbours.
Moreover, there are infinitely many atoms in any connected component $I$ of
$\partial\mathbb{D}\setminus P(\mu)$, which accumulate to both boundary points
of $I$. Finally, for any $\xi\in a(\mu)$ we have

$C^{-1}|\xi-\xi_{\pm}|\leq\mu\\{\xi\\}\leq C|\xi-\xi_{\pm}|\ .$

    3. iii.

We have

$H^{*}(\mu)(\xi):=\limsup_{\epsilon\to
0}\bigg{|}\int_{\\{t\in\partial\mathbb{D}:|t-\xi|>\epsilon\\}}\frac{d\mu(t)}{1-\bar{t}\xi}\bigg{|}\leq
C$

for any $\xi\in$ spt$(\mu_{a})\cup a(\mu)$.

It is worth mentioning that the conditions $C^{-1}\leq g(\xi)\leq C$ for
$\mu_{a}$-almost every point $\xi\in\partial\mathbb{D}$ and
$C^{-1}|\xi-\xi_{\pm}|\leq\mu(\xi)\leq C|\xi-\xi_{\pm}|,\,\xi\in a(\mu),$ can
be rephrased in the following way: There exists a constant $C_{1}>0$ such that
$C_{1}^{-1}|I|\leq\mu(I)\leq C_{1}|I|$ for every arc
$I\subset\partial\mathbb{D}$ containing at least two different points of
spt$(\mu)$. It is also interesting to mention that in the statement of Theorem
1.7, condition _b.iii)_ can be replaced by the apparently stronger condition
that $H^{*}(\mu)(\xi)\leq C$ at any point $\xi\in\text{spt}(\mu)$.

This paper is divided in five short sections. Next section contains some
auxiliary results. Section 3 is devoted to the proofs of Theorems 1.1, 1.2 and
some consequences. Theorems 1.3, 1.4 and 1.5 are proved in Section 4. Last
Section contains the proof of Theorem 1.7.

##  2 Preliminary results

In this section, we collect some auxiliary results that will be used in the
rest of the paper.

We first give some comments about Definition 1. As already mentioned, the main
difference between one-component bounded functions and one-component inner
functions is Condition _a)_ in Definition 1. Our first auxiliary result says
that Condition _a)_ allows us to choose freely the level $C$ of the connected
sublevel set.

######  Lemma 2.1.

Let $b\in H^{\infty}(\mathbb{D})$ with $\|b\|_{\infty}=1$. Assume there exists
a constant $0<C<1$ such that $\Omega_{C}:=\\{z\in\mathbb{D}:|b(z)|<C\\}$ is
connected and spec$(b)\subset\overline{\Omega_{C}}$. Then
$\Omega_{C_{1}}:=\\{z\in\mathbb{D}:|b(z)|<{C_{1}}\\}$ is connected for every
$C_{1}>C$.

###### Proof.

We argue by contradiction and we assume that there exists $C<C_{1}<1$ such
that $\Omega_{C_{1}}$ is not connected. Note that
$\Omega_{C}\subset\Omega_{C_{1}}$. Let $\Omega_{1}$ be a connected component
of $\Omega_{C_{1}}$ such that $\Omega_{1}\cap\Omega_{C}=\emptyset$. Note that
$\Omega_{1}$ is simply connected. Let $\phi:\mathbb{D}\to\Omega_{1}$ be a
conformal map. Then the function $U=b\circ\phi/C_{1}$ is inner. Since
$\inf_{z\in\mathbb{D}}|U(z)|>\delta>0$, the function $U$ has to be constant,
which is impossible. This contradiction finishes the proof. ∎

There are some differences between the spectrum of an inner function and the
spectrum of a bounded analytic function. Let us consider a bounded function
$b$, which, according to its inner-outer factorization, can be written as
$b=BS_{\mu}O_{b}$. Therefore, as stated in [BFM], $\overline{\text{spec}(b)}$
is the smallest closed subset of $\partial\mathbb{D}$ containing the limit
points of the zeros of the Blaschke product $B$ and the supports of the
measures $\mu$ and $\log|b|^{-1}dm$. It is well known and easy to prove that
$b$ has an analytic extension and it is unimodular through any arc of the open
set $\partial\mathbb{D}\setminus\overline{\text{spec}(b)}$, see [FM]. Note
that there are $b\in H^{\infty}(\mathbb{D})$ whose spectrum is not closed.

######  Lemma 2.2.

Let $b\in H^{\infty}(\mathbb{D})$ with $\|b\|_{\infty}=1$. Assume there exists
a constant $0<C<1$ such that $\text{spec}(b)\subset\overline{\\{z\in
D:|b(z)|\leq C\\}}$. Then, spec$(b)$ is closed.

###### Proof.

Let us consider a sequence $\\{\xi_{n}\\}\subset\text{spec}(b)$ such that
$\xi_{n}\to\xi$. For every $n$ the exists a sequence
$\\{t_{n,k}\\}\subset\mathbb{D}$ such that $t_{n,k}\to\xi_{n}$ as $k\to\infty$
and $|b(t_{n,k})|\leq C$. If we apply a diagonal argument to the sequences
$\\{t_{n,k}\\}$, we find a sequence $\\{t_{m}\\}\subset\mathbb{D}$ such that
$t_{m}\to\xi$ and $|b(t_{m})|\leq C$, which implies also that
$\xi\in$spec$(b)$. ∎

So, the spectrum of a one-component bounded funcion is a closed subset of the
unit circle. Given a closed connected set $E\subset\overline{\mathbb{D}}$ let
$w(a,\Gamma,\mathbb{D}\setminus E)$ be the harmonic measure at the point
$a\in\mathbb{D}\setminus E$ of the set
$\Gamma\subset\partial(\mathbb{D}\setminus E)$ in the domain
$\mathbb{D}\setminus E$, that is, the value at the point $a$ of the harmonic
function in the domain $\mathbb{D}\setminus E$ whose boundary values are $1$
at $\Gamma$ and $0$ elsewhere. We will use the following consequence of the
classical Hall’s Lemma. We recall that $\rho(z,w)$ is the pseudo-hyperbolic
distance between $z$ and $w$.

######  Lemma 2.3.

Let $\Gamma$ be a curve contained in the unit disc joining the origin with a
point in a Carleson square $Q(z)$ with $|z|>1/2$. Assume that
$\rho(z,\Gamma)\geq 1/2$ and $\mathbb{D}\setminus\Gamma$ is connected. Then
there exits a constant $C(\Gamma)>0$ such that
$w(z,\Gamma,\mathbb{D}\setminus\Gamma)\geq C(\Gamma)$.

###### Proof.

Let $\tau$ be the automorphism of the unit disc with $\tau^{-1}=\tau$ and
$\tau(z)=0$. The conformal invariance of harmonic measure gives
$w(z,\Gamma,\mathbb{D}\setminus\Gamma)=w(0,\tau(\Gamma),\mathbb{D}\setminus\tau(\Gamma))$.
Note that $\tau(\Gamma)$ is a curve satisfying
$\inf\\{|w|:w\in\tau(\Gamma)\\}\geq 1/2$ and having diameter bounded below by
an absolute constant. Thus, either the radial projection of $\tau(\Gamma)$
onto the unit circle or its circular projection onto the unit interval, is a
connected set whose diameter is bounded below by an absolute constant. Then
the radial or circular version of Hall’s Lemma (see page 124 of [GM]) finish
the proof. ∎

Lemma 2.3 will be used to prove the following auxiliary result .

######  Lemma 2.4.

Let $b$ be a one-component bounded function. Then there exists a constant
$0<C<1$ such that $\limsup_{r\to 1}|b(r\xi)|\leq C<1$, for every $\xi\in$
spec$(b)$.

###### Proof.

Let $\xi\in$ spec$(b)$. Since $b$ is one-component, there exists a constant
$0<C_{1}<1$ and a curve $\Gamma\subset\mathbb{D}$ connecting the origin with
$\xi$ such that $|b(w)|<C_{1}$ for any $w\in\Gamma$. The Maximum Principle
gives that

$\log|b(z)|\leq(\log C_{1})w(z,\Gamma,\mathbb{D}\setminus\Gamma),\quad
z\in\mathbb{D}\setminus\Gamma.$

Hence Lemma 2.3 gives that $|b(r\xi)|\leq C_{1}^{C(\Gamma)}$ if
$\rho(r\xi,\Gamma)\geq 1/2$. Note that if $\rho(r\xi,\Gamma)<1/2$, Schwarz’s
Lemma gives $|b(r\xi)|\leq(C_{1}+1/2)/(1+C_{1}/2)$. This finishes the proof. ∎

Since one-component bounded functions $b$ satisfy $\|b\|_{\infty}=1$, Lemma
2.4 gives that the spectrum of a one-component bounded functions is a proper
subset of the unit circle.

Before concluding this preliminary section and moving to the proofs of the
main results, we recall some properties of the derivatives of bounded analytic
functions at points in the complement of their spectrum. For a more complete
discussion on this subject, we refer to [FM].

######  Lemma 2.5.

Let $b\in H^{\infty}(\mathbb{D})$ with $||b||_{\infty}=1$. Then,
$|b^{\prime}(r\xi)|\leq 4|b^{\prime}(\xi)|$ for every
$\xi\in\partial\mathbb{D}\setminus\overline{\text{spec}(b)}$ and $1>r>1/2$.

###### Proof.

Without loss of generality we can assume $\xi=1$. Let
$b=B_{\\{z_{n}\\}}S_{\mu}O_{b}$ be the inner-outer factorization of $b$. It is
sufficient to prove the lemma separately for the three factors. Firstly

$|B^{\prime}(r)|\leq\sum_{n}\frac{1-|z_{n}|^{2}}{|1-\overline{z_{n}}r|^{2}}\leq\frac{1}{r^{2}}\sum_{n}\frac{1-|z_{n}|^{2}}{|1/r-\overline{z_{n}}|^{2}}\leq
4\sum_{n}\frac{1-|z_{n}|^{2}}{|1-\overline{z_{n}}|^{2}}=4|B^{\prime}(1)|\ .$

For the singular inner factor we use the estimate
$|\xi-r|^{2}\geq|\xi-1|^{2}/2$, $\xi\in\partial\mathbb{D}$, $1/2<r<1$. Then

$|S_{\mu}^{\prime}(r)|\leq\int_{\partial\mathbb{D}}\frac{2}{|\xi-r|^{2}}d\mu(\xi)\leq\int_{\partial\mathbb{D}}\frac{4}{|\xi-1|^{2}}d\mu(\xi)=2|S^{\prime}_{\mu}(1)|.$

The same argument also gives

$|O_{b}^{\prime}(r)|\leq\bigg{|}\int_{\partial\mathbb{D}}\frac{2\xi}{(\xi-r)^{2}}\log|b^{-1}(\xi)|dm(\xi)\bigg{|}\leq\int_{\partial\mathbb{D}}\frac{4}{|\xi-1|^{2}}\log|b^{-1}(\xi)|dm(\xi)=2|O_{b}^{\prime}(1)|\
.$

This finishes the proof. ∎

######  Lemma 2.6.

Let $b\in H^{\infty}(\mathbb{D})$ with $||b||_{\infty}=1$. Let
$\xi_{0}\in\partial\mathbb{D}$ such that $b(z)$ extends analytically at a
neighbourhood $\mathcal{U}$ of $\xi_{0}$, having values of modulus one on
$\partial\mathbb{D}\cap\mathcal{U}$. Then

$\frac{\xi b^{\prime}(\xi)}{b(\xi)}>0$

for every $\xi\in\mathcal{U}\cap\partial\mathbb{D}$.

###### Proof.

Let $b=B_{\\{z_{n}\\}}S_{\mu}O_{b}$ be the inner-outer factorization of $b$.
With a straight computation we obtain that

$\frac{\xi b^{\prime}(\xi)}{b(\xi)}=\sum_{n}\frac{1-|z_{n}|^{2}}{|\xi-
z_{n}|^{2}}+\int_{\partial\mathbb{D}}\frac{2}{|\xi-t|^{2}}d\mu(t)+\int_{\partial\mathbb{D}}\frac{2}{|\xi-t|^{2}}\log|b^{\prime}(t)|^{-1}dm(t)\
,\quad\xi\in\partial\mathbb{D}\cap\mathcal{U},$

from which the lemma follows. ∎

##  3 Inner-Outer Factorization

In this section we will prove Theorem 1.1 and Theorem 1.2, providing also some
easy corollaries. A Carleson square $Q$ is called dyadic if its closure
intersected with the unit circle is a dyadic arc of the unit circle. Note that
each dyadic Carleson square contains two dyadic Carleson squares of half
sidelength.

###### Proof of Theorem 1.1.

We define $\Omega_{K}:=\\{z\in\mathbb{D}:\ |b(z)|\leq K\\}$, $K>0$. We first
prove that condition (1) is necessary. Pick a constant $0<C_{1}<1$ such that
$|b(0)|<C_{1}<1$, $\Omega_{C_{1}}$ is connected and

(8) $\left\\{\xi\in\partial\mathbb{D}\ :\
\liminf_{z\to\xi}|b(z)|<1\right\\}\subseteq\overline{\Omega_{C_{1}}}\ .$

We argue by contradiction. Assume that there exists a sequence of points
$\\{z_{n}\\}\subset\mathbb{D}$ such that $\lim_{n\to\infty}|b(z_{n})|=1$ and
$\sigma_{b}(Q(z_{n}))>0$ for every $n$. Note that $\Omega_{C_{1}}\cap
Q(z_{n})=\emptyset$ if $n$ is sufficiently large, since, otherwise, there
would exists a connected path $\Gamma$ joining the origin and a point in
$Q(z_{n})$ with $|b(w)|\leq{C_{1}}$ for any point $w\in\Gamma$. Schwarz’s
Lemma gives that $\rho(z_{n},\Gamma)\to 1$ as $n\to\infty$. The Maximum
Principle gives that

$\log|b(z)|\leq(\log C_{1})w(z,\Gamma,\mathbb{D}\setminus\Gamma),\quad
z\in\mathbb{D}\setminus\Gamma.$

Now Lemma 2.3 gives that $|b(z_{n})|<C_{1}^{C(\Gamma)}<1$ if $n$ is
sufficiently large, which contradicts the assumption. Therefore, if $n$ is
sufficiently large the set $\Omega_{C_{1}}\cap Q(z_{n})$ has to be empty and,
in particular, $b(z)\neq 0$ for every $z\in Q(z_{n})$. On the other hand,
because of (8), we deduce also that $\lim_{z\to\xi}|b(z)|=1$, for every
$\xi\in\overline{Q(z_{n})}\cap\partial\mathbb{D}$, which implies that
$\sigma_{b}(Q(z_{n}))=0$.

We now prove that condition (1) is sufficient. Let $C_{1}<C_{2}<1$ be a
constant which will be fixed later. Let $\mathcal{A}=\\{Q_{j}\\}$ be the
collection of maximal dyadic Carleson squares such that

(9) $\sup_{z\in T(Q_{j})}|b(z)|\geq C_{2}\ .$

By maximality, $|b(w)|\leq C_{2}$ for any
$w\in\mathbb{D}\setminus\cup_{j}Q_{j}$. We note that if
$(1-C_{2})(1-C_{1})^{-1}$ is sufficiently small, Schwarz’s Lemma shows that
Condition (9) implies that $|b(z)|\geq C_{1}$ for any $z\in T(2Q_{j})$. We fix
$0<C_{2}<1$ with this property. Therefore, the assumption (1) implies that
$\sigma_{b}(2Q_{j})=0$. In particular, $b$ extends analytically through
$\overline{2Q_{j}}\cap\partial\mathbb{D}$ and $|b(\xi)|=1$ for every
$\xi\in\overline{2Q_{j}}\cap\partial\mathbb{D}$. Hence, we deduce that

$\left\\{\xi\in\partial\mathbb{D}\ :\
\liminf_{z\to\xi}|b(z)|<1\right\\}\subseteq\overline{\mathbb{D}\setminus\cup
Q_{j}}\cap\partial\mathbb{D}\ .$

We fix $C_{3}>C_{2}$. We note that
$\Omega_{C_{3}}\supseteq\mathbb{D}\setminus\cup Q_{j}$. Hence

$\left\\{\xi\in\partial\mathbb{D}\ :\
\liminf_{z\to\xi}|b(z)|<1\right\\}\subseteq\overline{\Omega_{C_{3}}}\ .$

It only remains to show that $\Omega_{C_{3}}$ is connected. Let $\Omega_{1}$
be the connected component of $\Omega_{C_{3}}$ containing
$\mathbb{D}\setminus\cup Q_{j}$. If, by contradiction, we assume that there
exists another connected component $\Omega_{2}\neq\Omega_{1}$ of
$\Omega_{C_{3}}$, then $\Omega_{2}\subset\cup Q_{j}$. Note that $\Omega_{2}$
is simply connected and $\partial\Omega_{2}\cap\partial\mathbb{D}$ can have at
most two points. If $\phi:\ \mathbb{D}\to\Omega_{2}$ is a conformal map, then
$(C_{3})^{-1}(b\circ\phi)$ would be an inner function. Since it cannot be
constant, we deduce that $\inf_{\Omega_{2}}|b(z)|=0$. However, this would
contradict the following claim.

######  Claim 3.1.

There exists a constant $C>0$ such that for any $j=1,2,\ldots$ and any $w\in
Q_{j}=Q(z_{j})$ we have

$C^{-1}\frac{\log|b(z_{j})|^{-1}}{1-|z_{j}|}\leq\frac{\log|b(w)|^{-1}}{1-|w|}\leq
C\frac{\log|b(z_{j})|^{-1}}{1-|z_{j}|}.$

###### Proof.

We note that

$\log|b(w)|^{-1}\cong\int_{\overline{\mathbb{D}}}P_{w}(z)d\sigma_{b}(z)\ ,$

when $w\in Q(z_{j})$. Since $\sigma_{b}(2Q(z_{j}))=0$, we deduce

$\frac{\log|b(w)|^{-1}}{1-|w|}\cong\int_{\overline{\mathbb{D}}\setminus
2Q(z_{j})}\frac{1}{|1-\bar{z}w|^{2}}d\sigma_{b}(z)\cong\int_{\overline{\mathbb{D}}\setminus
2Q(z_{j})}\frac{1}{|1-\bar{z}z_{j}|^{2}}d\sigma_{b}(z)\cong\frac{\log|b(z_{j})|^{-1}}{1-|z_{j}|}\
.$

∎

This claim finishes the proof. ∎

Using the Theorem 1.1, it is easy to verify that the product of two one-
component bounded functions is still one-component.

######  Corollary 3.1.

Let $b_{1},b_{2}$ be two one-component bounded functions. Then $b_{1}b_{2}$ is
a one-component bounded function.

###### Proof.

We need to verify the sufficient condition (1) of Theorem 1.1. We know that
there exists a constant $0<C<1$ such that

$\sigma_{b_{i}}(Q(z))=0,\ \text{ if }|b_{i}(z)|\geq C,\quad i=1,2.$

Pick a constant $C<C_{1}<1$. Assume $|b_{1}(z)b_{2}(z)|>C_{1}$. Then
$|b_{i}(z)|>C_{1}$, $i=1,2$ and hence

$\sigma_{b_{1}b_{2}}(Q(z))=\sigma_{b_{1}}(Q(z))+\sigma_{b_{2}}(Q(z))=0\ .$

This finishes the proof. ∎

We note that the same result for one-component inner functions has been
already proved in [CM1] with a different argument. Theorem 1.1 can be also
used to prove that any proper closed subset $E\subset\partial\mathbb{D}$ is
the spectrum of one-component bounded function.

###### Proof of Theorem 1.2.

Write $\partial\mathbb{D}\setminus E=\cup I_{k}$, where $I_{k}$ are open arcs.
For any $I_{k}$ consider its endpoints $e^{it_{k}(1)}$, $e^{it_{k}(2)}$ and
the isosceles triangle $T_{k}$ defined as

$T_{k}=\left\\{re^{it}:t_{k}(1)<t<t_{k}(2),0\leq
1-r<\min\\{|t-t_{k}(1)|,|t-t_{k}(2)|\\}\right\\}.$

We locate $\\{z_{k,j}\\}_{j}$ on $\partial T_{k}\cap\mathbb{D}$ such that the
pseudo-hyperbolic distance $\rho(z_{k,j+1},z_{k,j})=\delta$ is independent of
$k$ and $j$ and $\delta<1/2$. The sequence $\\{z_{k,j}\\}_{j,k}$ is a Blaschke
sequence since

$\sum_{k}\sum_{j}1-|z_{k,j}|\lesssim\sum_{k}|I_{k}|<\infty\ .$

Let $B$ be the Blaschke product with zeros $\\{z_{k,j}:k,j\\}$. Consider the
bounded analytic function $b$ defined as

$b(z):=\exp\left(\int_{E}\frac{\xi+z}{\xi-z}\log(1/2)dm(\xi)\right)B(z),\quad
z\in\mathbb{D}.$

Since $\partial\mathbb{D}\setminus E$ has positive measure we have
$\|b\|_{\infty}=1$. Because of the construction, it is clear that spec$(b)=E$.
The only thing left to prove is that the function $b$ is one-component. We
first note that if $z\in\cup_{k}\partial T_{k}\cap\mathbb{D}$, Schwarz’s lemma
gives $|B(z)|\leq 1/2$. Hence $|b(z)|<1/2$ for any $z\in\cup_{k}\partial
T_{k}\cap\mathbb{D}\cup E$. The Maximum Principle gives that $|b(z)|\leq 1/2$
for any $z\in\mathbb{D}\setminus\cup_{k}T_{k}$. Fix $C>1/2$. If $|b(z))|\geq
C$ then $z\in\cup_{k}T_{k}$ and consequently $\sigma_{b}(Q(z))=0$. We can now
apply Theorem 1.1 and we deduce that $b$ is one-component.

∎

We close this section with another consequence of Theorem 1.1.

######  Corollary 3.2.

Let $b\in H^{\infty}(\mathbb{D})$ with $\|b\|_{\infty}=1$. Assume there exists
a constant $0<C<1$ such that spec$(b)\subset\overline{\left\\{z\in\mathbb{D}:\
|b(z)|\leq C\right\\}}$. Then there exists a Blaschke product $B$ such that
the function $bB$ is one-component and spec$(b)=$spec$(bB)$.

###### Proof.

Lemma 2.2 gives that spec$(b)$ is closed. We write
$\partial\mathbb{D}\setminus\text{spec}(b)=\cup_{k}I_{k}$ where each $I_{k}$
is a closed arc of $\partial\mathbb{D}$ satisfying
$|I_{k}|=\text{dist}(I_{k},\overline{\Omega_{C}})$. We choose $r_{k}\in(0,1)$
such that, if $|z|\geq r_{k}$ and $e^{i\text{arg}(z)}\in I_{k}$, then
$|b(z)|\geq(1+C)/2$. Let $\Gamma$ be the region given by
$\Gamma=\cup_{k=1}^{\infty}\\{r\xi:r\geq r_{k}\text{ and }\xi\in I_{k}\\}$. We
fix $0<\varepsilon<C$ and we pick points
$\\{z_{j}\\}\subset\partial\Gamma\cap\mathbb{D}$ so that the pseudo-hyperbolic
distance $\rho(z_{j},z_{j+1})=\varepsilon$. We consider the Blaschke product
$B$ whose zeros are $\\{z_{j}\\}$. Due to Schwarz’s lemma, $|B(z)|\leq C$ when
$z\in\partial\Gamma\cap\mathbb{D}$ and, consequently, $|bB|<C$ on
$\left(\partial\Gamma\cap\mathbb{D}\right)\cup\text{spec}(b)$. The Maximum
Principle gives $|b(z)B(z)|\leq C$ for any $z\in\mathbb{D}\setminus\Gamma$. We
fix $C<C_{1}<1$. If $|b(z)B(z)|\geq C_{1}$ then $z\in\Gamma$ and consequently
$\sigma_{bB}(Q(z))=0$. We apply now Theorem 1.1 and we obtain that $bB$ is
one-component. ∎

Before concluding this section, we highlight that if $b_{1},b_{2}$ are one-
component bounded functions, even if $b_{1}/b_{2}\in H^{\infty}$, it may
happen that $b_{1}/b_{2}$ is not one-component. For sake of completeness, we
recall that the regularity of the quotient of two one-component inner
functions has been studied in [CM2].

##  4 Reproducing kernels for one-component bounded function

In this section we will prove Theorems 1.3, 1.4 and 1.5.

###### Proof of Theorem 1.3.

We start by proving _a)_. Let $0<C_{1}<1$ be the constant given by Theorem 1.1
and let $C_{1}<C_{2}=C_{2}(C_{1})<1$ be a constant to be fixed later. We
consider the family of maximal dyadic Carleson squares
$\left\\{Q(z_{j})\right\\}$ such that

$\sup_{z\in T(Q(z_{j}))}|b(z)|\geq C_{2}.$

We note that $|b(z)|\leq C_{2}$ when
$z\in\mathbb{D}\setminus\cup_{j}Q(z_{j})$. Taking $C_{2}>C_{1}$ sufficiently
close to $1$, Schwarz’s lemma gives that $\sup_{z\in
T(\widetilde{Q}(z_{j}))}|b(z)|\geq C_{1}$, where $\widetilde{Q}(z_{j})$ is the
dyadic Carleson square containing $Q(z_{j})$ with double sidelenght. We fix
$0<C_{2}<1$ with this property. Applying Theorem 1.1, we obtain that
$\sigma_{b}(\widetilde{Q}(z_{j}))=0$. Therefore, the function $b$ extends
analytically on $\overline{\widetilde{Q}(z_{j})}\cap\partial\mathbb{D}$ and
$|b(\xi)|=1$ for every
$\xi\in\overline{\widetilde{Q}(z_{j})}\cap\partial\mathbb{D}$. Claim 3.1 gives

(10)
$\frac{\log|b(w)|^{-1}}{1-|w|}\cong\frac{\log|b(z_{j})|^{-1}}{1-|z_{j}|}\cong\frac{1}{|\ell(Q(z_{j}))|}\
,$

for every $w\in Q(z_{j})$, uniformly on $j$.

With no loss of generality, we assume that $|b(a)|$ is close to $1$. Hence
$a\in\cup_{j}Q(z_{j})$. We fix $j$ so that $a\in Q(z_{j})$. Moreover, we can
also assume that the pseudo-hyperbolic distance
$\rho(a,\mathbb{D}\setminus\cup_{j}Q(z_{j}))$ is close to 1, since, otherwise,
by Schwarz’s lemma $|b(a)|$ would not be close to $1$. We want to show that
there exists a constant $C>0$ such that

$\bigg{|}\frac{1-\overline{b(z)}b(a)}{1-\overline{z}a}\bigg{|}\leq
C\frac{1-|b(a)|^{2}}{1-|a|^{2}}\ ,\quad z\in\mathbb{D}.$

If $z\notin 2Q(z_{j})$, then

$\bigg{|}\frac{1-\overline{b(z)}b(a)}{1-a\bar{z}}\bigg{|}\leq\frac{2}{|\ell(Q(z_{j}))|}\cong\frac{\log|b(a)|^{-1}}{1-|a|}\cong\frac{1-|b(a)|^{2}}{1-|a|^{2}}\
.$

If $z\in 2Q_{j}$, since

$\frac{1-\overline{b(z)}b(a)}{1-a\bar{z}}=\frac{1-|b(a)|^{2}}{1-a\bar{z}}-b(a)\frac{\overline{b(z)-b(a)}}{1-\overline{z}a}\
$

and

$\frac{1-|b(a)|^{2}}{|1-\overline{z}a|}\lesssim\frac{1-|b(a)|^{2}}{1-|a|^{2}}\
,$

it is sufficient to prove that

$\bigg{|}\frac{b(z)-b(a)}{1-\overline{z}a}\bigg{|}\lesssim\frac{1-|b(a)|^{2}}{1-|a|^{2}}\
.$

For $w\in\mathbb{D}\setminus\\{0\\}$, we denote $w^{*}=w/|w|$. We note that

$\frac{|b(z)-b(a)|}{|1-\overline{z}a|}\leq\frac{|b(z)-b(z^{*})|}{|1-\overline{z}a|}+\frac{|b(z^{*})-b(a^{*})|}{|1-\overline{z}a|}+\frac{|b(a^{*})-b(a)|}{|1-\overline{z}a|}\
,$

and we estimate the three terms separately. Firstly, we note that (10) gives
that

(11)
$|b^{\prime}(\xi)|\lesssim\frac{1}{\ell(Q(z_{j}))},\quad\xi\in\partial\mathbb{D}\cap\overline{Q(z_{j})}$

and consequently

$\frac{|b(z^{*})-b(a^{*})|}{|1-\overline{z}a|}\leq\frac{1}{|1-\overline{z}a|}\int_{z^{*}}^{a^{*}}|b^{\prime}(t)|dt\lesssim\frac{C}{|\ell(Q(z_{j}))|}\frac{|z-a|}{|1-\overline{z}a|}\lesssim\frac{1-|b(a)|^{2}}{1-|a|^{2}}\
,$

where in the last inequality we have used (10). Moreover, by using Lemma 2.5,
(11) and (10), we obtain that

$\frac{|b(a)-b(a^{*})|}{|1-\overline{z}a|}\lesssim\frac{1-|a|}{|\ell(Q(z_{j}))|}\frac{1}{|1-\overline{z}a|}\cong\frac{1-|b(a)|^{2}}{1-|a|^{2}}$

which proves the statement _a)_.

We prove statement _b)_ by contradiction. We assume there exist points
$a_{n}\in\mathbb{D}$ with $|b(a_{n})|\to 1$, but $\sigma_{b}(Q(a_{n}))>0$. Let
$0<C<1$ be a constant to be fixed later and let
$\Omega_{C}=\\{z\in\mathbb{D}:\ |b(z)|<C\\}$. The assumption (4) gives that
$Q(a_{n})\cap\Omega_{C}=\emptyset$ for $n$ large enough. Since
spec$(b)\subseteq\overline{\Omega_{C}}$, we deduce also that
spec$(b)\cap\overline{Q(a_{n})}=\emptyset$. Hence $\sigma_{b}(Q(a_{n}))=0$
which is a contradiction. ∎

######  Remark.

We note that condition (3) of Theorem 1.3 is necessary. Indeed, $b(z)=(1+z)/2$
does not satisfy (3), and consequently it is not one-component, while (4)
holds.

We move now to the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

We start by proving _a)_. We apply Theorem 1.3 and obtain a constant $C>0$
such that

$\bigg{|}\frac{1-\overline{b(a)}b(z)}{1-\overline{a}z}\bigg{|}\leq
C\frac{1-|b(a)|^{2}}{1-|a|^{2}}\ ,$

for every $a,z\in\mathbb{D}$. For any
$\xi\in\partial\mathbb{D}\setminus$spec$(b)$, we pick a sequence
$\\{a_{n}\\}\subset\mathbb{D}$, which tends to $\xi$, and we obtain

$\frac{|b(z)-b(\xi)|}{|z-\xi|}\leq
C\big{|}b^{\prime}(\xi)\big{|},\,z\in\mathbb{D}\ .$

We move now to the proof of statement _b)_. For sake of clarity, we split the
argument in four claims. Since by Lemma 2.2 the set spec$(b)$ is closed, we
write $\partial\mathbb{D}\setminus$ spec$(b)=\cup_{j}I_{j}$, where $I_{j}$ are
open arcs. Let $0<C_{1}<1$ be a constant to be fixed later. For every $\xi\in
I_{j}$, we choose $r(\xi)$ such that $1-r(\xi)=C_{1}/|b^{\prime}(\xi)|$.

######  Claim 4.1.

There exists a constant $C_{2}>0$ such that

$\frac{1}{|b^{\prime}(\xi)|}\leq C_{2}\text{dist}(\xi,\text{spec}(b))\
,\quad\xi\in\partial\mathbb{D}\setminus\text{spec}(b).$

###### Proof.

For every $\eta\in$ spec$(b)$, we take $\\{z_{n}\\}\subset\Omega_{C}$
approaching $\eta$. Since
$1-C\leq|b(z_{n})-b(\xi)|\lesssim|b^{\prime}(\xi)||z_{n}-\xi|$, we deduce that

$\frac{1}{|b^{\prime}(\xi)|}\lesssim|\eta-\xi|\ .$

∎

######  Claim 4.2.

There exist constants $C_{3},C_{4}>0$ such that for every
$\xi\in\partial\mathbb{D}\setminus$spec$(b)$ and for every $|z-\xi|\leq
C_{3}/|b^{\prime}(\xi)|$, we have

(12) $|b(z)-b(\xi)-b^{\prime}(\xi)(z-\xi)|\leq
C_{4}|z-\xi|^{2}|b^{\prime}(\xi)|^{2}\ .$

###### Proof.

Fix $\xi\in\partial\mathbb{D}\setminus$spec$(b)$. Consider the disc
$D(\xi)=\\{z\in\mathbb{C}:|z-\xi|\leq C_{3}/{|b^{\prime}(\xi)|}\\}$. Due to
the previous claim, the two functions in (12) are analytic in $D(\xi)$. Note
that by assumption, estimate (12) holds when $z\in\partial D(\xi)$ and by the
Maximum Principle the estimate (12) holds for any $z\in D(\xi)$. ∎

######  Claim 4.3.

There exists a constant $0<C_{5}<1$ such that $|b(r(\xi)\xi)|\leq C_{5}$ for
every $\xi\in\partial\mathbb{D}\setminus$spec$(b)$.

###### Proof.

Since $|b(\xi)|=1$, we rewrite (12) as

(13)
$\bigg{|}\frac{b(z)}{b(\xi)}-1-\frac{b^{\prime}(\xi)}{b(\xi)}\xi\left(\frac{z}{\xi}-1\right)\bigg{|}\leq
C_{4}|z-\xi|^{2}|b^{\prime}(\xi)|^{2}\ .$

We consider $z=r(\xi)\xi$. Then

$\bigg{|}\frac{b(r(\xi)\xi)}{b(\xi)}-1-\frac{b^{\prime}(\xi)}{b(\xi)}\xi\left(r(\xi)-1\right)\bigg{|}\leq
C_{4}(1-r(\xi))^{2}|b^{\prime}(\xi)|^{2}\ ,$

which implies that ${b(r(\xi)\xi)}/{b(\xi)}-1$ belongs to the disk $B$ with
centre at the negative point $b^{\prime}(\xi)\xi\left(r(\xi)-1\right))/b(\xi)$
with radius comparable to $(1-r(\xi))^{2}|b^{\prime}(\xi)|^{2}$. Since
$C_{1}<1$, $B$ is entirely contained in the left half plane. Therefore, the
point ${b(r(\xi)\xi)}/{b(\xi)}$ is contained in $1+B\subset\mathbb{D}$, which
is separated from $\partial\mathbb{D}$. We deduce that there exists a constant
$0<C_{5}<1$ such that $|b(r(\xi)\xi)|\leq C_{5}$ for every
$\xi\in\partial\mathbb{D}\setminus$ spec$(b)$. ∎

The Claim 4.3 proves that

$|b|\leq C_{5}\text{ on }\cup_{j}\left\\{r(\xi)\xi:\ \xi\in I_{j}\right\\}\ .$

Consider the domain $\Omega:=\left\\{r\xi:\ 0<r<1\text{ if
}\xi\in\text{spec}(b)\ ,\ 0<r<r(\xi)\text{ if }\xi\in\cup_{j}I_{j}\right\\}$.
By the Maximum Principle

$|b|\leq\max\\{C,C_{5}\\}<1\text{ on }\Omega.$

######  Claim 4.4.

There exists a constant $C_{6}>0$ such that $|b(z)|>C_{6}$ for every
$z\in\mathbb{D}\setminus\Omega$.

###### Proof.

By assumption, we know that

$|b(z)-b(\xi)|\leq C|b^{\prime}(\xi)||z-\xi|\ .\quad
z\in\mathbb{D},\xi\in\partial\mathbb{D}\setminus\text{spec}(b).$

Taking $0<C_{1}<1$ sufficiently small, we deduce that

$|b(z)|\geq 1-CC_{1}>0\ ,\quad z\in\mathbb{D}\setminus\Omega.$

∎

We are now ready to prove that the function $b$ is one-component. We fix a
constant $1>C_{7}>C_{i}$, $i=1,\ldots,6$ and consider the sublevel set
$\mathcal{W}:=\left\\{z\in\mathbb{D}:\ |b(z)|<C_{7}\right\\}$. Therefore
$\Omega\subset\mathcal{W}$. We assume there exists a non empty connected
component $\Omega^{*}$ of $\mathcal{W}$ such that
$\Omega^{*}\cap\Omega=\emptyset$. Then $\Omega^{*}$ would be simply connected
and contained in $\mathbb{D}\setminus\Omega$. Consider a conformal map
$\phi:\mathbb{D}\to\Omega^{*}$. Since
$\overline{\Omega^{*}}\cap\partial\mathbb{D}$ contains at most two points, the
function $b\circ\phi/C_{7}$ is inner. Claim 4.4 gives that $|b(z)|>C_{6}$ and
hence the inner function $b\circ\phi/C_{7}$ is bounded from below. Therefore
it has to be constant which is impossible. Consequently, $\mathcal{W}$ is
connected, which concludes the proof. ∎

We note that if $b$ is one-component, Theorem 1.4 tells us that

(14)
$\sup_{z\in\mathbb{D}}\frac{|1-\overline{b(\xi)}b(z)|}{|1-\bar{\xi}z|}\leq
C|b^{\prime}(\xi)|,\quad\xi\in\partial\mathbb{D}\setminus\text{spec}(b).$

We now deduce the following useful estimate.

######  Corollary 4.1.

Let $b\in H^{\infty}$ with $||b||_{\infty}=1$. Fix $0<C<1$ and consider the
sublevel set $\Omega_{C}:=\left\\{z\in\mathbb{D}:|b(z)|\leq C\right\\}$.

  1. a)

There exist a constant $C_{1}>0$ such that dist$(\xi,\Omega_{C}\ \cup$
spec$(b))\leq C_{1}/|b^{\prime}(\xi)|,$ for any $\
\xi\in\partial\mathbb{D}\setminus$ spec$(b)$.

  2. b)

If $b$ is one-component and spec$(b)\subset\overline{\Omega_{C}}$, then there
exits a constant $C_{2}>0$ such that dist$(\xi,\Omega_{C})\geq
C_{2}/|b^{\prime}(\xi)|,$ for any $\ \xi\in\partial\mathbb{D}\setminus$
spec$(b)$.

###### Proof.

The proof of statement _a)_ is contained in [BFM]. On the other hand,
statement _b)_ is an immediate consequence of (14). ∎

We note that when $b$ is one-component, $|b^{\prime}(\xi)|$ is (uniformly)
comparable to dist$(\xi,\Omega_{C})$, $\xi\in\partial\mathbb{D}\setminus
spec(b)$. Here $\Omega_{C}={\left\\{z\in\mathbb{D}:\ |b(z)|\leq C\right\\}}$
is the connected sublevel set of $b$.

We are now ready to prove Theorem 1.5.

###### Proof of Theorem 1.5.

We start by proving statement _a)_. Claim 4.2 in the proof of Theorem 1.4
provide constants $C_{1},C_{2}>0$ such that

$|b(z)-b(\xi)-b^{\prime}(\xi)(z-\xi)|\leq
C_{1}|z-\xi|^{2}|b^{\prime}(\xi)|^{2}$

when $\xi\in\partial\mathbb{D}\setminus$ spec$(b)$ and $|z-\xi|\leq
C_{2}/|b^{\prime}(\xi)|$. Dividing by $|z-\xi|^{2}$ and taking $z$ tending to
$\xi$, we obtain

$|b^{\prime\prime}(\xi)|\leq 2C_{1}|b^{\prime}(\xi)|^{2}\ $

for every $\xi\in\partial\mathbb{D}\setminus$ spec$(b)$. We note that
Corollary 4.1 tells us that for any
$\eta\in\partial\left(\text{spec}(b)\right)$ we have

$\lim_{\xi\to\eta,\xi\in\partial\mathbb{D}\setminus
spec(b)}|b^{\prime}(\xi)|=\infty\ .$

Therefore the function $H(\xi)$ defined in (6) is Lipschitz.

For the proof of statement _b)_ , we use some ideas from the proof of Theorem
1.4. Due to Lemma 2.2, spec$(b)$ is a closed subset of $\partial\mathbb{D}$.
We denote $\partial\mathbb{D}\setminus$ spec$(b)=\cup_{j}I_{j}$ and for every
$\xi\in I_{j}$ consider $0<r(\xi)<1$ defined by

$1-r(\xi)=\frac{1}{16}\frac{1}{|b^{\prime}(\xi)|}\ .$

Since the function $H$ is Lipschitz, there exists a constant $C>0$ such that

$\bigg{|}\frac{1}{b^{\prime}(\xi)}\bigg{|}\leq C|\xi-\eta|\
,\quad\xi\in\partial\mathbb{D}\setminus\text{spec}(b),\ \eta\in\text{spec}(b)\
.$

Therefore

$\bigg{|}\frac{1}{b^{\prime}(\xi)}\bigg{|}\leq C\text{
dist}(\xi,\text{spec}(b))\ .$

Moreover, for every $t,\xi\in I_{j}$ we have

$\bigg{|}\frac{1}{b^{\prime}(t)}-\frac{1}{b^{\prime}(\xi)}\bigg{|}\leq
C|t-\xi|$

which implies that

(15)
$\frac{1}{2}\bigg{|}\frac{1}{b^{\prime}(\xi)}\bigg{|}\leq\bigg{|}\frac{1}{b^{\prime}(t)}\bigg{|}\leq\frac{3}{2}\bigg{|}\frac{1}{b^{\prime}(\xi)}\bigg{|}$

when $|t-\xi|<(2C|b^{\prime}(\xi)|)^{-1}$. For $z,\xi\in I_{j}$,

$b(z)-b(\xi)-b^{\prime}(\xi)(z-\xi)=\int_{\xi}^{z}b^{\prime\prime}(t)(z-t)dt\
.$

Since $|b^{\prime\prime}(t)|\leq C|b^{\prime}(t)|^{2}$ for $t\notin$
spec$(b)$, we deduce that

$\big{|}b(z)-b(\xi)-b^{\prime}(\xi)(z-\xi)\big{|}\leq
C\int_{\xi}^{z}|b^{\prime}(t)|^{2}|z-t|dt\ .$

Using (15), we obtain a constant $C_{1}>0$ such that

(16) $\big{|}b(z)-b(\xi)-b^{\prime}(\xi)(z-\xi)\big{|}\leq
C_{1}|b^{\prime}(\xi)|^{2}|z-\xi|^{2}\ ,\quad z,\xi\in I_{j}$

For sake of clarity, we split the rest of the proof in two claims.

######  Claim 4.5.

There exists $C_{2}<1$ such that $|b(r(\xi)\xi)|\leq C_{2},\ \xi\in I_{j}$ .

###### Proof.

We argue by contradiction. Assume there exist $\xi_{n}\in I_{j}$ such that
$|b(r(\xi_{n})\xi_{n})|\to 1$. Since $||b||_{\infty}=1$ and

$b(r(\xi_{n})\xi_{n})=\int_{\partial\mathbb{D}}P_{r(\xi_{n})\xi_{n}}(t)b(t)dm(t)\
,$

for any $\delta>0$, we have

$\frac{1}{1-r(\xi_{n})}m\left\\{t:\ |t-\xi_{n}|<1-r(\xi_{n})\text{ with
}|b(t)-b(r(\xi_{n})\xi_{n})|>\delta\right\\}\to 0\ ,\quad\text{ as
}n\to\infty.$

This contradicts (16) and finishes the proof of the Claim. ∎

We consider

$\Omega:=\left\\{r\xi:\xi\in\partial\mathbb{D},0<r<1\text{ if
}\xi\in\text{spec}(b),1\geq 1-r>C_{2}/|b^{\prime}(\xi)|\text{ if
}\xi\in\partial\mathbb{D}\setminus\text{spec}(b)\right\\}.$

We note that $|b(\xi)|\leq C$ for almost every $\xi\in$ spec$(b)$. On the
other hand, Claim 4.5 tells us that $|b(r(\xi)\xi)|\leq C_{2}$ for every
$\xi\in\cup_{j}I_{j}$. Then, the Maximum Principle gives

(17) $\sup_{\Omega}|b(z)|\leq\max\left(C,C_{2}\right)<1\ .$

######  Claim 4.6.

We have $|b(z)|\geq 1/2$ for any $z\in\mathbb{D}\setminus\Omega$.

###### Proof.

We fix $z\in\mathbb{D}\setminus\Omega$ and we consider $\xi=z/|z|$. There
exists $0<r<1$ such that

$|b(z)-b(\xi)|\leq|b^{\prime}(r\xi)|(1-|z|)\ .$

We apply Lemma 2.5 to deduce that

$|b(z)-b(\xi)|\leq 4|b^{\prime}(\xi)|(1-|z|)\ .$

Since $z\in\mathbb{D}\setminus\Omega$, we have $1-|z|<1-r(\xi)=1/16\
|b^{\prime}(\xi)|$ and we obtain that

$|b(z)-b(\xi)|<1/2\ .$

We deduce that $|b(z)|\geq 1/2$, which proves the claim. ∎

We can now conclude the proof of statement _b)_. We fix
$1>C_{3}>\max\\{C,C_{1},C_{2},1/2\\}$ and we prove that
$\Omega_{C_{3}}=\\{z\in\mathbb{D}:|b(z)|<C_{3}\\}$ is connected. We note that
(17) implies that $\Omega$ is contained in $\Omega_{C_{3}}$. Let
$\Omega_{1}\subset\mathbb{D}\setminus\Omega$ be another connected component of
$\Omega_{C_{3}}$. Then $\Omega_{1}$ is simply connected and there exists $j$
such that

$\Omega_{1}\subset\left\\{r\xi:\ \xi\in I_{j},1>r>r(\xi)\right\\}\ .$

Let $\phi:\mathbb{D}\to\Omega_{1}$ be a conformal mapping. Since
$\partial\Omega_{1}\cap\partial\mathbb{D}$ contains at most two points, the
function $U:=(C_{3})^{-1}b\circ\phi$ is inner. However, Claim 4.6 says that
$U$ is bounded from below and this gives that $\Omega_{1}$ is empty, that is,
$\Omega_{C_{3}}$ is connected. ∎

We finally prove Corollary 1.6.

###### Proof of Corollary 1.6.

We have just to prove that (6) is equivalent to the conditions _b)_ and _c)_
of Corollary 1.6. Indeed, if _b)_ and _c)_ hold, the function $H$ defined in
(6) has to be Lipschitz because it has bounded derivative at almost every
point of the unit circle.

On the other hand, if $H$ is Lipschitz, then, due to Rademacher’s theorem and
the regularity of $b^{\prime}(\xi)$ when $\xi\notin\text{spec}(b)$, we obtain
condition _b)_. Analogously, condition _c)_ follows directly from the
definition of $H$. ∎

##  5 Clark measures of one-component bounded function

In this section, we prove Theorem 1.7, characterizing Clark measures of one-
component bounded functions. We need two preliminary results.

######  Lemma 5.1.

Let $b\in H^{\infty}(\mathbb{D})$ with $||b||_{\infty}=1$. Let $\mu$ be the
Clark measure of $b$ at the value $\alpha=1$. Then
$P(\mu)=\overline{\text{spec}(b)}$, where $P(\mu):=$spt$(\mu)\setminus
a(\mu)$.

###### Proof.

We need to verify the two inclusions. If $\xi\notin\overline{\text{spec(b)}}$,
the function $b$ extends analytically across an arc containing $\xi$. Then
either $(1+b(z))/(1-b(z))$ is analytic through an arc
$I(\xi)\subset\partial\mathbb{D}$ containing $\xi$ or $b(\xi)=1$. In the first
case, $\mu_{a}(I(\xi))=0$ and consequently, since $\mu_{s}$ is carried on
$\\{t:b(t)=1\\}\subset\partial\mathbb{D}\setminus I(\xi)$, we deduce
$\mu(I(\xi))=0$, which implies that $\xi\notin$spt$(\mu)$. If, on the other
hand, $b(\xi)=1$, then $\xi\in a(\mu)$. Consequently, in both the cases,
$\xi\notin P(\mu)$.

Let us now assume $\xi\notin P(\mu)$. If $\xi\in a(\mu)$, then $b(z)$ has an
analytic extension at a neighbourhood of $\xi$ and $b(\xi)=1$, which implies
that $\xi\notin\overline{\text{spec}(b)}$. If on the other hand,
$\xi\notin$spt$(\mu)$, then $(1+b(z))/(1-b(z))$ extends analytically at a
neighbourhood $I(\xi)$ of $\xi$ and its real part is equal to zero at
$I(\xi)$. For this reason, $|b(\xi)|=1$ at $I(\xi)$, which means that
$I(\xi)\subset\partial\mathbb{D}\setminus\overline{\text{spec}(b)}$, which
implies that, in both the cases, $\xi\notin\overline{\text{spec}(b)}$. ∎

######  Lemma 5.2.

Let $\mu$ be a positive, finite Borel measure on the unit circle satisfying
the assumptions b i) and b ii) in Theorem 1.7. Then there exists a constant
$C>0$ such that

(18)
$\bigg{|}\int_{\partial\mathbb{D}}\frac{d\mu(t)}{1-\bar{t}r\xi}-\int_{\\{|t-\xi|>1-r\\}}\frac{d\mu(t)}{1-\bar{t}\xi}\bigg{|}\leq
C,\quad 0<r<1,\ \xi\in P(\mu).$

###### Proof.

In order to obtain (18), it is sufficient to prove that that there exists a
constant $C_{1}>0$ such that

(19)
$\bigg{|}\int_{\\{|t-\xi|>1-r\\}}\left(\frac{1}{1-\bar{t}r\xi}-\frac{1}{1-\bar{t}\xi}\right)d\mu(t)\bigg{|}+\bigg{|}\int_{\\{|t-\xi|<1-r\\}}\frac{d\mu(t)}{1-\bar{t}r\xi}\bigg{|}\leq
C_{1}\ .$

For $n\geq 0$ let $I(n)$ be the arc centered at $\xi$ of measure
$2^{n+1}(1-r)$. We note that there exists a constant $C_{2}>0$ such that

(20)
$\displaystyle\int_{\\{|t-\xi|>1-r\\}}\frac{1-r}{|1-\bar{t}r\xi||1-\bar{t}\xi|}d\mu(t)\leq$
$\displaystyle\sum_{n=0}^{\infty}\int_{I(n+1)\setminus
I(n)}\frac{1-r}{2^{2n}(1-r)^{2}}d\mu(t)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{\mu(I(n+1)\setminus
I(n))}{2^{2n}(1-r)}\leq C_{2}\ ,$

because by _b.i)_ and _b.ii)_ , $\mu(I)\lesssim|I|$ if $I$ contains two
different points of spt$(\mu)$. Similarly there exists a constant $C_{3}>0$
such that

(21)
$\int_{\\{|t-\xi|<1-r\\}}\frac{d\mu(t)}{|1-\bar{t}r\xi|}\leq\frac{1}{1-r}\mu\left(\\{t:\
|t-\xi|<1-r\\}\right)\leq C_{3}\ .$

Consequently, (20) and (21) imply (19), from which estimate (18) follows. ∎

We are now ready to prove Theorem 1.7. Even if some parts of this proof follow
the ideas of the analogue version for inner functions in [Be], for the sake of
completeness we have decided to present them here again.

###### Proof of Theorem 1.7.

We start by proving that _a)_ implies _b)_. We assume that there exists a one-
component bounded function $b$ such that

$\frac{1+b(z)}{1-b(z)}=\int_{\partial\mathbb{D}}\frac{\xi+z}{\xi-z}d\mu(\xi)+2i\frac{\Im\left(b(0)\right)}{|1-b(0)|^{2}}\
,\quad z\in\mathbb{D}.$

Since spec$(b)$ is closed due to Lemma 2.2, Lemma 5.1 gives that
$P(\mu)=$spec$(b)$. First we note that spt$(\mu_{a})\subset\text{spec}(b)$.
Indeed, if $I$ is an arc contained in $\partial\mathbb{D}\setminus$ spec$(b)$,
then $b$ extends analytically through $I$ and $|b|=1$ on $I$. Therefore the
real part of $(1+b(r\xi))/(1-b(r\xi))$ tends to $0$ as $r$ tends to $1$, for
any $\xi\in I$. Hence $\mu_{a}(I)=0$.

Due to Lemma 2.4, there exists a constant $0<C<1$ such that

$\text{spec}(b)=\left\\{\xi\in\partial\mathbb{D}:\ \limsup_{r\to
1}|b(r\xi)|\leq C\right\\}.$

Therefore there exists a constant $C_{1}>0$, such that

$1/C_{1}\leq\Re\left(\frac{1+b(r\xi)}{1-b(r\xi)}\right)=\int_{\partial\mathbb{D}}\frac{1-r^{2}}{|t-r\xi|}d\mu(t)\leq
C_{1}\ ,$

when $1>r>r_{0}(\xi)$ and $\xi\in$ spec$(b)$. Hence

$1/C_{1}\leq g(\xi)\leq C_{1}$

for $m$-almost every $\xi\in$ spec$(b)$. Consequently, $\mu$ satisfies _b.i)_.

Since $\limsup_{r\to 1}|b(r\xi)|\leq C<1$ for any $\xi\in$ spec$(b)=P(\mu)$,
we deduce

$P(\mu)\cap\left\\{\xi\in\partial\mathbb{D}:\ \limsup_{r\to 1}\Re
b(r\xi)=1\right\\}=\emptyset.$

Since the singular measure $\mu_{s}$ is carried by the set
$\left\\{\xi\in\partial\mathbb{D}:\ \lim_{r\to 1}b(r\xi)=1\right\\}$, we
deduce that $\mu_{s}(P(\mu))=0$, which implies that $\mu_{s}$ is purely
atomic.

Let $I=(\alpha,\beta)$ be a connected component of
$\partial\mathbb{D}\setminus P(\mu)$. Since spec$(b)=P(\mu)$, the function $b$
extends analytically through $I$. Hence, the set $a(\mu):=\left\\{\xi\in I:\
b(\xi)=1\right\\}$ is discrete. Corollary 4.1 says that

$|b^{\prime}(\xi)|^{-1}\cong\text{dist}\left(\xi,\\{z\in\mathbb{D}:\
|b(z)|\leq C\\}\right)$

when $\xi\in\partial\mathbb{D}\setminus$spec$(b)$. Hence

$\lim_{I\ni\xi\to\alpha}|b^{\prime}(\xi)|=\infty.$

Since, due to Lemma 2.6, $\xi b^{\prime}(\xi)b(\xi)^{-1}\geq 0$, when $\xi\in
I$, we deduce that

$\lim_{I\ni\xi\to\alpha}\frac{\xi b^{\prime}(\xi)}{b(\xi)}=+\infty\ .$

Therefore, there exist infinitely many different points $\xi_{k}\in I$, such
that $\xi_{k}\to\alpha$ and $b(\xi_{k})=1$. A similar argument replacing
$\alpha$ by $\beta$ shows that the set $a(\mu)\cap I$ is infinite and it
accumulates to both $\alpha$ and $\beta$. We deduce that every $\xi\in a(\mu)$
has two neighbours.

In order to prove that $C^{-1}|\xi-\xi_{\pm}|\leq\mu\\{\xi\\}\leq
C|\xi-\xi_{\pm}|$ when $\xi\in a(\mu)$ and $\xi_{\pm}$ are its neighbours, we
apply an argument due to A. Baranov and K. Dyakonov [BD]. Fix $\xi\in a(\mu)$.
We pick $t\in(\xi,\xi_{+})$ such that $b(t)=-1$. Then, using Theorem 1.4, we
get

$|\xi_{+}-\xi|>|t-\xi|\gtrsim|b^{\prime}(\xi)|^{-1}|b(t)-b(\xi)|=2|b^{\prime}(\xi)|^{-1}\
.$

Hence $\mu\\{\xi\\}\lesssim|\xi-\xi_{+}|$. Let $N$ be a positive integer to be
fixed later. We split the arc $(\xi,\xi_{+})$ into $N$ pairwise disjoints sub-
arcs $J_{k}=(\alpha_{k},\beta_{k})$, with $k=1,\dots,N,$ such that

$\text{Arg}(b(\beta_{k}))-\text{Arg}(b(\alpha_{k}))=\frac{2\pi}{N}\ ,\quad
k=1,\ldots,N.$

Let $s_{k}\in(\alpha_{k},\beta_{k})$, such that
$|b^{\prime}(s_{k})|=\min\left\\{|b^{\prime}(s)|:\
s\in[\alpha_{k},\beta_{k}]\right\\}$. Then

(22)
$\frac{2\pi}{N}=\int_{\alpha_{k}}^{\beta_{k}}\frac{sb^{\prime}(s)}{b(s)}dm(s)\geq|b^{\prime}(s_{k})||\beta_{k}-\alpha_{k}|\
,\quad k=1,\ldots,N.$

when $k=1,\dots,N$. Due to Corollary 4.1,

(23) $|\beta_{k}-\alpha_{k}|\leq
C_{1}\frac{2\pi}{N}\text{dist}\left(s_{k},\\{z\in\mathbb{D}:|b(z)|\leq
C\\}\right)\ ,\quad k=1,\ldots,N.$

We now take $t\in(\alpha_{k},\beta_{k})$ and observe that

$|b^{\prime}(t)|=\int_{\overline{\mathbb{D}}}\frac{1}{|1-\bar{t}z|^{2}}d\sigma_{b}(z)\
.$

We note that
$|t-z|\geq|s_{k}-z|-|s_{k}-t|\geq|s_{k}-z|-|\beta_{k}-\alpha_{k}|\ .$
Consequently, if we choose $N>4C_{1}\pi$, (23) gives us that

$|t-z|\geq\frac{1}{2}|s_{k}-z|\,\quad\text{ if }|b(z)|\leq C.$

Since spt$(\sigma_{b})\subset\overline{\left\\{z\in\mathbb{D}:\ |b(z)|\leq
C\right\\}}$, we deduce

$|b^{\prime}(t)|\leq 4|b^{\prime}(s_{k})|$

and, by using (22), we deduce that

$|\xi_{+}-\xi|\lesssim\sum_{k=1}^{N}|\beta_{k}-\alpha_{k}|\leq
C(N)\frac{1}{|b^{\prime}(\xi)|}\ .$

This finishes the proof of _b.ii)_.

Finally we prove _b.iii_). An easy calculation (see [CMR]), shows that

(24)
$\frac{1}{1-b(z)}=\int_{\partial\mathbb{D}}\frac{d\mu(t)}{1-\bar{t}z}-\frac{||\mu||-1}{2}-i\frac{\Im\left(b(0)\right)}{|1-b(0)|^{2}},\quad
z\in\mathbb{D}.$

Using (24), (18) and the fact that $\limsup_{r\to 1}|b(r\xi)|\leq C<1$ for
$\xi\in$ spec$(b)$, we obtain that $H^{*}(\mu)(\xi)$ is bounded when $\xi\in
P(\mu)$.

On the other hand, if $\xi\in a(\mu)$, we have

$\displaystyle H^{*}(\mu)(\xi)=$
$\displaystyle\lim_{z\to\xi}\left(\frac{1}{1-b(z)}-\frac{1}{b^{\prime}(\xi)(1-\bar{\xi}z)}\right)+C$
$\displaystyle=$
$\displaystyle\lim_{z\to\xi}\frac{1}{(1-\bar{\xi}z)}\left(\frac{1-\bar{\xi}z}{1-b(z)}-\frac{1}{b^{\prime}(z)}\right)+C$
$\displaystyle=$
$\displaystyle-\frac{b^{\prime\prime}(\xi)}{b^{\prime}(\xi)^{2}}+C\ ,$

which is uniformly bounded due Corollary 1.6. This estimate finishes the proof
of _b.iii)_ and, consequently, of _b)_.

Let us now show that _b)_ implies _a)_. Given the measure $\mu$ satisfying the
assumption in _b)_ , we need to show that the function $b$ defined as

(25)
$\frac{1+b(z)}{1-b(z)}=\int_{\partial\mathbb{D}}\frac{t+z}{t-z}d\mu(t)\,\quad
z\in\mathbb{D}$

is one-component. The main ingredient will be Corollary 1.6. We start by
showing the first condition in Corollary 1.6. Let $\xi\in$ spec$(b)$. If $\xi$
lies in the spectrum of the inner factor of $b$ then
$\xi\in\overline{\\{z\in\mathbb{D}:|b(z)|<\varepsilon\\}}$, for any
$\varepsilon>0$. If $\xi\in\text{spt}(\mu_{a})$ we will show that there exists
a constant $C<1$ such that

$\limsup_{r\to 1}|b(r\xi)|\leq C.$

We note the (25) gives

(26)
$\frac{1-|b(r\xi)|^{2}}{|1-b(r\xi)|^{2}}=\int_{\partial\mathbb{D}}\frac{1-r^{2}}{|t-r\xi|^{2}}d\mu(t)\geq\frac{\mu\left(\\{t:\
|t-\xi|<1-r\\}\right)}{4(1-r)}\geq C_{1}\ ,$

according to assumptions _b.i)_ and _b.ii)_. On the other hand, _b.iii)_ and
(18) give that there exists a constant $C_{2}>0$ such that

(27) $\limsup_{r\to
1}\bigg{|}\int_{\partial\mathbb{D}}\frac{d\mu(t)}{1-\bar{t}r\xi}\bigg{|}\leq
C_{2}\ .$

Hence (24), (26) and (27) provide a constant $0<C_{3}<1$ such that

$\limsup_{r\to 1}|b(r\xi)|\leq C_{3}\ .$

From this it follows that the function $b$ satisfies Condition $a)$ of
Corollary 1.6. Next we will show that condition (b) in Corollary 1.6 holds. We
need the following preliminary result.

######  Lemma 5.3.

Let $\mu$ be a Borel measure which satisfies condition _b)_ of Theorem 1.7 and
let $\xi_{0}\in a(\mu)$. Let $k>0$ be a constant sufficiently small compared
with the constant $C=C(\mu)$ appearing in condition _b)_ of Theorem 1.7. We
denote

(28) $D_{\xi_{0}}(k):=\left\\{z\in\mathbb{C}:\
|z-\xi_{0}|<k\mu(\xi_{0})\right\\}\ .$

Then, the function $b$ defined in (25) has an analytic extension at
$D_{\xi_{0}}(k)$ and

(29) $\bigg{|}\frac{1-b(z)}{\xi_{0}-z}\bigg{|}\cong\frac{1}{\mu(\xi_{0})}\
,\quad z\in D_{\xi_{0}}(k).$

###### Proof of Lemma 5.3.

We know that there exists a constant $C_{1}$ such that

(30)
$\frac{1}{1-b(z)}=\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{d\mu(t)}{1-\bar{t}z}+\frac{\mu(\xi_{0})}{1-\bar{\xi_{0}}z}+C_{1}\
,\quad z\in\mathbb{D}.$

We note that

$\displaystyle\bigg{|}\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{d\mu(t)}{1-\bar{t}z}\bigg{|}$
$\displaystyle=\bigg{|}\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{1-\bar{t}\xi_{0}}{1-\bar{t}z}\frac{d\mu(t)}{1-\bar{t}\xi_{0}}\bigg{|}$
$\displaystyle\leq\bigg{|}\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{d\mu(t)}{1-\bar{t}\xi_{0}}\bigg{|}+\bigg{|}\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{\bar{t}(z-\xi_{0})}{(1-\bar{t}z)(1-\bar{t}\xi_{0})}d\mu(t)\bigg{|}\
.$

Due to _b.iii)_ , the first integral is uniformly bounded. The second integral
is estimated as follows,

$\bigg{|}\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{\bar{t}(z-\xi_{0})}{(1-\bar{t}z)(1-\bar{t}\xi_{0})}d\mu(t)\bigg{|}\leq|z-\xi_{0}|\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{d\mu(t)}{|1-\bar{t}z||1-\bar{t}\xi_{0}|}.$

Since $|z-\xi_{0}|\leq k\mu(\xi_{0})$ and
$|t-\xi_{0}|\geq|\xi_{\pm}-\xi_{0}|\cong{\mu(\xi_{0})}$, we deduce that
$|t-z|\geq\frac{1}{2}|t-\xi_{0}|$ if $k$ is sufficiently small. Hence

$|z-\xi_{0}|\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{d\mu(t)}{|1-\bar{t}z||1-\bar{t}\xi_{0}|}\leq
2|z-\xi_{0}|\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{d\mu(t)}{|1-\bar{t}\xi_{0}|^{2}}$

Using the fact that $\mu(I)$ is comparable to $|I|$ when the arc $I$ contains
two different points of spt$(\mu)$, we obtain that

$\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{d\mu(t)}{|t-\xi_{0}|^{2}}\cong\frac{1}{|\xi_{+}-\xi_{0}|}\
.$

Consequently there exists a constant $C_{2}>0$ such that

$\bigg{|}\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{\bar{t}(z-\xi_{0})}{(1-\bar{t}z)(1-\bar{t}\xi_{0})}d\mu(t)\bigg{|}\leq
C_{2}\ .$

At this point, identity (30) implies that there exits a constant $C_{3}>0$
such that

$\bigg{|}\frac{\xi_{0}-z}{1-b(z)}-\mu(\xi_{0})\bar{\xi_{0}}\bigg{|}\leq
C_{3}|\xi_{0}-z|\ .$

By taking $k$ sufficiently small so that $C_{3}|\xi_{0}-z|\leq
C_{3}k\mu(\xi_{0})\leq 1/2\ \mu(\xi_{0})$, we deduce that

$\bigg{|}\frac{\xi_{0}-z}{1-b(z)}-\mu(\xi_{0})\bar{\xi_{0}}\bigg{|}\leq\frac{1}{2}\mu(\xi_{0})\
,$

for every $z\in D_{\xi_{0}}(k)$, which proves the Lemma. ∎

We can now check condition (b) of Corollary 1.6. The identity (25) gives

(31)
$\frac{b^{\prime}(z)}{(1-b(z))^{2}}=\int_{\partial\mathbb{D}}\frac{\bar{\xi}}{(1-\bar{\xi}z)^{2}}d\mu(\xi),\quad
z\in\mathbb{D}$

and

(32)
$\frac{b^{\prime\prime}(z)}{(1-b(z))^{2}}+\frac{{b^{\prime}(z)}^{2}}{(1-b(z))^{3}}=2\int_{\partial\mathbb{D}}\frac{\bar{\xi}^{2}}{(1-\bar{\xi}z)^{3}}d\mu(\xi)\
,\quad z\in\mathbb{D}.$

Let $\xi_{0}\in a(\mu)$. By Lemma 5.3,

$|1-b(z)|\gtrsim 1$

when $z\in\partial D_{\xi_{0}}(k)$. Hence, (31) gives that

$|b^{\prime}(z)|\lesssim\frac{\mu(\xi_{0})}{|1-\bar{\xi_{0}}z|^{2}}+\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{d\mu(\xi)}{|1-\bar{\xi}z|^{2}}$

when $z\in\partial D_{\xi_{0}}(k)$. Since $\mu(I)\cong|I|$ if $I$ is an arc
containg two points of spt$(\mu)$, we deduce that the last integral can be
estimated by $|\xi_{0}-\xi_{+}|^{-1}\cong 1/\mu(\xi_{0})$. Therefore

$|b^{\prime}(z)|\lesssim\frac{1}{\mu(\xi_{0})}$

when $z\in\partial D_{\xi_{0}}(k)$. Now (32) implies that

$|b^{\prime\prime}(z)|\lesssim|b^{\prime}(z)|^{2}+\frac{2\mu(\xi_{0})}{|1-\bar{\xi_{0}}z|^{3}}+2\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{d\mu(\xi)}{|1-\bar{\xi}z|^{3}}\
,\quad z\in\partial D_{\xi_{0}}(k).$

Acting as before, we obtain that

$|b^{\prime\prime}(z)|\lesssim\frac{1}{\mu(\xi_{0})^{2}}$

when $z\in\partial D_{\xi_{0}}(k)$, and hence, by Maximum Principle

$|b^{\prime\prime}(z)|\lesssim|b^{\prime}(\xi_{0})|^{2}$

for any $z\in D_{\xi_{0}}(k)$. Now identity (31) gives

$|b^{\prime}(z)|=|1-b(z)|^{2}\bigg{|}\frac{\bar{\xi_{0}}}{(1-z\bar{\xi_{0}})^{2}}\mu(\xi_{0})+\int_{\partial\mathbb{D}\setminus\\{\xi_{0}\\}}\frac{\bar{\xi}}{(1-\bar{\xi}z)^{2}}d\mu(\xi)\bigg{|}\
,$

for any $z\in\partial\mathbb{D}\setminus\text{spec}(b)$. Applying Lemma 5.3,
we deduce

(33)
$|b^{\prime}(z)|\gtrsim\frac{\mu(\xi_{0})}{|1-\bar{\xi_{0}}z|^{2}}|1-b(z)|^{2}\cong\frac{1}{\mu(\xi_{0})}$

when $z\in D_{\xi_{0}}(k)$. Consequently

(34) $|b^{\prime\prime}(z)|\leq|b^{\prime}(z)|^{2}$

for any $z\in D_{\xi_{0}}(k)$.

We need to obtain the same estimate also for the other points
$z\in\partial\mathbb{D}\setminus\left(\text{spec}(b)\cup_{a(\mu)}D_{\xi}(k)\right)$.
We start by showing that

(35) $|b(z)-1|>k$

for every
$z\in\partial\mathbb{D}\setminus\left(\text{spec}(b)\cup_{a(\mu)}D_{\xi}(k)\right)$.
Indeed, let $z\in(\xi_{-},\xi_{+})$ where $\xi_{\pm}$ are neighbour atoms of
$\mu$. We note that, due to Lemma 5.3, $|b(w)-1|\cong k$ if $w\in\partial
D_{\xi_{\pm}}(k)\cap\partial\mathbb{D}$. Since Arg$(b)$ is increasing for
$t\in(\xi_{-},\xi_{+})$, estimate (35) follows. Therefore if
$z\notin\cup_{a(\mu)}D_{\xi}(k)$, we have $|z-\xi_{\pm}|\geq k\mu(\xi_{\pm})$.
On the other hand, $|z-\xi_{\pm}|\leq|\xi_{+}-\xi_{-}|\cong\mu(\xi_{\pm})$.
Therefore

(36) $k\mu(\xi_{\pm})\leq|z-\xi_{\pm}|\leq C\mu(\xi_{\pm})\ .$

Applying the previous argument with $\xi_{0}$ replaced by $\xi_{\pm}$, we
deduce that $|b^{\prime\prime}(z)|\lesssim|b^{\prime}(z)|^{2}$. This finishes
the proof of property _b)_ of Corollary 1.6. Finally we need to verify that
$|b^{\prime}(z)|\to\infty$ as $0<$dist$(z,\text{spec}(b))\to 0$. We note that
(33) and assumption _b.ii)_ gives it for $z\in\cup_{a(\mu)}D_{\xi}(k)$. For
$z\notin\cup_{a(\mu)}D_{\xi}(k)$, let, as before, $z\in(\xi_{-},\xi_{+})$
where $\xi_{\pm}$ are neighbour atoms of $\mu$. Then (35), (36) and (33) give
that $|b^{\prime}(z)|\gtrsim|b^{\prime}(\xi_{\pm})|$ which, since
dist$(z,\text{spec}(b))\to 0$, finishes the proof. ∎

######  Remark.

We note that if $b$ is an inner function, Theorem 1.7 reduces to the result of
[Be] on Clark measures of one-component inner functions. Indeed, in this case,
the absolute continuous part of the measure $\mu$ would be identically zero
and the Hilbert transform $H^{*}(\mu)$ would be equal to

$H(\mu)(\xi):=\int_{\partial\mathbb{D}\setminus\\{\xi\\}}\frac{d\mu(t)}{1-\bar{t}\xi}\
,$

for any $\xi\in a(\mu)$.

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C. Bellavita, Department of Mathematics, Università degli Studi di Milano,
Milan, Italy

E-mail address, C. Bellavita<EMAIL_ADDRESS>
A. Nicolau , Departament de Matemàtiques, Universitat Autònoma de Barcelona
and Centre de Recerca Matemàtica, 08193 Barcelona

E-mail address, A. Nicolau<EMAIL_ADDRESS>