#  $M_{z}$-invariant subspaces in the Bloch space

Adem Limani & Artur Nicolau

###### Abstract

We consider weak-star closed invariant subspaces of the shift operator in the
classical Bloch space. We prove that any bounded analytic function decomposes
into two factors, one which is cyclic and another one generating a proper
invariant subspace, satisfying a permanence property, which in a certain way
is opposite to cyclicity. Singular inner functions play the crucial role in
this decomposition. We show in several different ways that the description of
shift invariant subspaces generated by inner functions in the Bloch spaces
deviates substantially from the corresponding description in the Bergman
spaces, provided by the celebrated Korenblum-Roberts Theorem. Furthermore, the
relationship between invertibility and cyclicity is also investigated and we
provide an invertible function in the Bloch space which is not cyclic therein.
Our results answer several open questions stated in the early nineties.

##  1 Introduction

Let $\mathbb{D}$ denote the open unit disc in the complex plane $\mathbb{C}$
and let $\mathcal{B}$ be the classical Bloch space of analytic functions $f$
in $\mathbb{D}$ satisfying

$\left\lVert f\right\rVert_{\mathcal{B}}:=\left\lvert
f(0)\right\rvert+\sup_{z\in\mathbb{D}}(1-|z|)|f^{\prime}(z)|<\infty.$

Taking the closure of analytic polynomials in the norm above we obtain the so-
called little Bloch space $\mathcal{B}_{0}$, which is a separable Banach
subspace of $\mathcal{B}$ consisting of functions $f$ satisfying

$\lim_{|z|\to 1^{-}}(1-|z|)f^{\prime}(z)=0.$

Functions in the Bloch space are intimately related to conformal mappings, and
as such they are regarded as a crucial objects in geometric function theory,
see [21] for further details. Let $H^{\infty}$ designate the Banach space of
bounded analytic functions $f$ in $\mathbb{D}$ equipped with the usual norm
$\|f\|_{\infty}=\sup\\{|f(z)|:z\in\mathbb{D}\\}$, and recall that
$H^{\infty}\subset\mathcal{B}$ by Schwarz lemma. The Bloch space may be
regarded as the limit when $p\to\infty$ of the Bergman spaces $B_{p}$ of
analytic functions $f$ in $\mathbb{D}$ such that

$\int_{\mathbb{D}}|f(z)|^{p}dA(z)<\infty,$

where $dA$ denotes the area measure on $\mathbb{D}$. For $1\leq p<\infty$, we
denote by $W^{p}$ the Sobolev space of analytic functions $g$ on $\mathbb{D}$
satisfying

$\left\lVert
g\right\rVert^{p}_{W^{p}}:=|g(0)|^{p}+\int_{\mathbb{D}}\left\lvert
g^{\prime}(z)\right\rvert^{p}dA(z)<\infty.$

Let $dm$ denotes Lebesgue measure in $\partial\mathbb{D}$. It is well-known
that the dual space of $W^{1}$ can be identified with $\mathcal{B}$ considered
in the Cauchy-pairing. More precisely, for any $f\in\mathcal{B}$ and $g\in
W^{1}$, the limit

$\ell_{f}(g):=\lim_{r\to
1-}\int_{\partial\mathbb{D}}f(r\zeta)\overline{g(r\zeta)}dm(\zeta)$

exists and $f$ induces a unique bounded linear functional on $W^{1}$.
Similarly, the dual of $\mathcal{B}_{0}$ can be identified with $W^{1}$, see
for instance [4]. Let $M_{z}$ denote the multiplication operator by the
independent variable $z$, that is, the linear operator $M_{z}f(z)=zf(z)$,
$f\in\mathcal{B}$. Our main purpose is to investigate a certain class of
$M_{z}$-invariant subspaces in $\mathcal{B}$, and to adjust for the fact that
$\mathcal{B}$ is not separable, we shall instead consider weak-star closed
subspaces in $\mathcal{B}$ which are $M_{z}$-invariant. Given a function
$f\in\mathcal{B}$, we shall denote by $\left[f\right]_{\mathcal{B}}$ the
smallest weak-star closed $M_{z}$-invariant subspace containing $f$, that is,
$\left[f\right]_{\mathcal{B}}$ is the weak-star closure of polynomial
multiples of $f$. If $f\in\mathcal{B}_{0}$, then
$\left[f\right]_{\mathcal{B}}$ is actually the norm-closure of polynomial
multiples of $f$ by Mazur’s Theorem. A function $f\in\mathcal{B}$ is called
_cyclic_ in $\mathcal{B}$ if $\left[f\right]_{\mathcal{B}}=\mathcal{B}$. Since
the set of analytic polynomials is weak-star dense in $\mathcal{B}$, we have
that $f$ is cyclic in $\mathcal{B}$ if and only if
$1\in\left[f\right]_{\mathcal{B}}$. It is worth mentioning that
$\left[f\right]_{\mathcal{B}}$ may be much larger than the subspace of weak-
star limits of sequences of polynomial multiples of $f$, as we shall
demonstrate in THM:Seqcyc. A function $f\in H^{\infty}$ is declared to satisfy
the _permanence property_ (in $\mathcal{B}$) if the corresponding
$M_{z}$-invariant subspace satisfies $\left[f\right]_{\mathcal{B}}\cap
H^{\infty}=fH^{\infty}$. The permanence property should be understood as an
antithesis to cyclicity, in the sense that any bounded analytic function in
$\left[f\right]_{\mathcal{B}}$ is divisible by $f$.

In this note, we shall mainly be concerned with $M_{z}$-invariant subspaces in
$\mathcal{B}$ generated by bounded analytic functions. Recall that any $f\in
H^{\infty}$ can be factored as $f=FBS_{\mu}$, where $F$ denotes the outer
factor of $f$ defined by

$F(z)=\exp\left(\int_{\partial\mathbb{D}}\frac{\zeta+z}{\zeta-z}\log|f(\zeta)|dm(\zeta)\right),\qquad
z\in\mathbb{D},$

the function $S_{\mu}$ is the singular inner factor of $f$ given by

$S_{\mu}(z)=\exp\left(-\int_{\partial\mathbb{D}}\frac{\zeta+z}{\zeta-z}d\mu(\zeta)\right),\qquad
z\in\mathbb{D},$

where $\mu$ is a positive finite Borel measure on $\partial\mathbb{D}$ which
is singular with respect $dm$, while $B$ denotes the Blaschke product encoding
the zeros of $f$ on $\mathbb{D}$, that is,

$B(z)=\prod_{a:f(a)=0}\frac{\left\lvert
a\right\rvert}{a}\frac{a-z}{1-\overline{a}z},\qquad z\in\mathbb{D}.$

Functions in $H^{\infty}$ having radial limits of modulus one at almost every
point of the unit circle are called _inner functions_ , and they are always of
the form $\Theta=BS_{\mu}$. Given $f\in H^{\infty}$ with inner-outer
factorization $f=FBS_{\mu}$, we denote by $\nu_{f}$ its associated Herglotz-
Nevanlinna measure defined by

$d\nu_{f}=-\log|f(\zeta)|dm(\zeta)+d\mu(\zeta)+\sum_{z:f(z)=0}(1-|z|^{2})\delta_{z},$

where $\delta_{z}$ denotes the Dirac point mass measure at $z\in\mathbb{D}$.
Note that if $\left\lVert f\right\rVert_{H^{\infty}}\leq 1$, then $\nu_{f}$ is
positive. It was established in [12] that classical outer functions belonging
to $\mathcal{B}$ must be cyclic, and the existence of a cyclic singular inner
function in $\mathcal{B}_{0}$ was proved in [3]. Our first result provides a
structural theorem for $M_{z}$-invariant subspaces generated by bounded
analytic functions, where singular inner functions play the decisive role.

######  Theorem 1.1.

THM:M_zLat Let $f=FS_{\mu}B$ be in $H^{\infty}$, where $F$ is outer, $S_{\mu}$
is singular inner and $B$ is a Blaschke product. Then there exists a unique
(up to sets of $\mu$-measure zero) decomposition $\mu=\mu_{P}+\mu_{C}$ of
$\mu$, where $\mu_{P},\mu_{C}$ are mutually singular positive measures giving
rise to the following dichotomy.

  1. (i)

The inner function $\Theta_{0}:=BS_{\mu_{P}}$ generates a proper
$M_{z}$-invariant subspace on $\mathcal{B}$ satisfying the permanence
property, that is, $\left[\Theta_{0}\right]_{\mathcal{B}}\cap
H^{\infty}=\Theta_{0}H^{\infty}$.

  2. (ii)

$FS_{\mu_{C}}$ is cyclic in $\mathcal{B}$.

THM:M_zLat should be viewed as a Bloch space version of Theorem 1 in [28] by
Roberts in the context of Bergman spaces. Our result shows that in order to
understand $M_{z}$-invariant subspaces in $\mathcal{B}$ generated by functions
in $H^{\infty}$, it suffices to study the permanence property and cyclicity of
singular inner functions. It turns out that these two notions have natural
interpretations in the context of model spaces. For $0<p<\infty$ let $H^{p}$
denote the classical Hardy space of analytic functions $f$ in $\mathbb{D}$
such that

$\|f\|_{p}^{p}=\sup_{0<r<1}\int_{\partial\mathbb{D}}|f(r\zeta)|^{p}dm(\zeta)<\infty.$

Given an inner function $\Theta$, the associated model space $K_{\Theta}$ is
defined as $K_{\Theta}=H^{2}\ominus\Theta H^{2},$ that is, the orthogonal
complement of $\Theta H^{2}$ in the classical Hardy space $H^{2}$. It follows
from the celebrated Beurling Theorem that the model spaces are the only
invariant subspaces for the adjoint $(M_{z})^{*}$ viewed as an operator on
$H^{2}$. A deep result of Aleksandrov says that functions with continuous
extensions to $\partial\mathbb{D}$ in any model space are always dense
therein, see [1] and also [2] and [25] for recent generalizations. Recall that
a closed set $E\subset\partial\mathbb{D}$ of Lebesgue measure zero is said to
be a Beurling-Carleson set if

$\int_{\partial\mathbb{D}}\log\text{dist}\left(\zeta,E\right)dm(\zeta)>-\infty.$

It was recently established that $\bigcap_{p>1}W^{p}\cap K_{\Theta}$ is dense
in $K_{\Theta}$ if and only if $\Theta=BS_{\mu}$ and $\mu$ is carried on a
countable union of Beurling-Carleson sets [26], while $\cup_{p>1}W^{p}\cap
K_{S_{\mu}}=\\{0\\}$ if and only if $\mu$ does not charge any Beurling-
Carleson set [17]. We shall now consider what happens when $p=1$ in our next
result. Recall that for an inner function $\Theta$, one defines the associated
model space $K^{1}_{\Theta}:=H^{1}\cap\Theta\overline{zH^{1}}$, interpreted in
the sense of boundary values on $\partial\mathbb{D}$. In other words, a
function $f\in H^{1}$ belongs to $K^{1}_{\Theta}$ if there exists a function
$g\in H^{1}$ with $g(0)=0$ such that
$\Theta(\zeta)\overline{f(\zeta)}=g(\zeta)$ for $m$-a.e.
$\zeta\in\partial\mathbb{D}$.

######  Theorem 1.2.

THM:Modelspaces Let $\mu$ be a positive, finite, Borel, singular measure in
$\partial\mathbb{D}$ and let $S_{\mu}$ be the corresponding singular inner
function. Then

  1. (i)

$S_{\mu}$ is cyclic in $\mathcal{B}$ if and only if $K^{1}_{S_{\mu}}\cap
W^{1}=\\{0\\}$.

  2. (ii)

The permanence property $\left[S_{\mu}\right]_{\mathcal{B}}\cap
H^{\infty}=S_{\mu}H^{\infty}$ holds if and only if $K^{1}_{S_{\mu}}\cap W^{1}$
is dense in $K^{1}_{S_{\mu}}$.

Viewing $\mathcal{B}$ as a limiting case of the Bergman spaces, we shall now
make a few comparisons. In the context of Bergman spaces, the celebrated
Theorem of Korenblum and Roberts asserts that any singular measure $\mu$
uniquely decomposes as $\mu=\mu_{\mathcal{C}}+\mu_{\mathcal{K}}$, where
$\mu_{\mathcal{C}}$ is concentrated on a countable union of Beurling-Carleson
sets and gives rise to a proper $M_{z}$-invariant subspace generated by
$S_{\mu_{\mathcal{C}}}$ satisfying the permanence property on the Bergman
space, while $\mu_{\mathcal{K}}$ charges no Beurling-Carleson set and induces
a cyclic vector $S_{\mu_{\mathcal{K}}}$ therein, see [23], [28]. We shall see
that the situation in the Bloch space is very different and the results of
Korenblum and Roberts do not carry over to this setting. Although we have not
been able to find complete geometric descriptions of singular measures $\mu$
for which the corresponding singular inner functions $S_{\mu}$ are cyclic or
satisfy the permanence property in $\mathcal{B}$, we are still able to provide
several related conditions which allows us to answer various open questions
and problems posed in [12], [3], [4] and [17].

We declare a continuous non-decreasing and sub-additive function $w$ on
$[0,1]$ with $w(0)=0$ to be a _majorant_. A closed set
$E\subset\partial\mathbb{D}$ of Lebesgue measure zero is said to have finite
$w$-entropy if

$\int_{\partial\mathbb{D}}\log
w\left(\text{dist}\left(\zeta,E\right)\right)dm(\zeta)>-\infty.$

Of course, when $w(t)=t^{\alpha}$ for some $0<\alpha<1$, one retains the
classical Beurling-Carleson sets. Various descriptions of sets having finite
$w$-entropy have recently been treated in [22] and in [24]. Sets of finite
$w$-entropy are precisely the boundary zero sets of analytic functions in
$\mathbb{D}$ which extend continuously to $\partial\mathbb{D}$ whose modulus
of continuity do not exceed $w$ on $\overline{\mathbb{D}}$, see [13] for
classical Beurling-Carleson sets and [30] for general majorants. As mentioned
above, a singular inner function is cyclic in the Bergman space if and only if
its associated singular measure does not charge any Beurling-Carleson set. It
turns out that if a singular inner function is cyclic in the Bloch space, its
associated singular measure can not charge a far wider range of sets.

######  Theorem 1.3.

THM:wsets Let $w$ be a majorant. Assume that there exists $0<\gamma<1$ such
that $w(t)/t^{\gamma}$ is non-increasing and that $w$ satisfies the Dini-
condition

$\int_{0}^{1}\frac{w(t)}{t}dt<\infty.$

Assume $\mu$ is carried on a countable union of sets having finite
$w$-entropy. Then the associated singular inner function $S_{\mu}$ satisfies
the permanence property: $\left[S_{\mu}\right]_{\mathcal{B}}\cap
H^{\infty}=S_{\mu}H^{\infty}$.

We shall in fact deduce THM:wsets from a slightly stronger statement in
Section 3 (see THM:PPBMOAw therein), which essentially asserts that inner
factors of certain weighted BMOA spaces gives rise to the permanence property.
In order for $S_{\mu}$ to be cyclic in $\mathcal{B}$, THM:wsets says that
$\mu$ cannot charge any set of finite $w$-entropy. Note that the class of
eligible majorants $w$ for which our Theorem holds, goes far beyond classical
Beurling-Carleson sets, as it also applies to slowly increasing majorants of
type $w(t)=\log^{-\alpha}(e/t)$, with $\alpha>1$. Since any function cyclic in
$\mathcal{B}$ must necessarily be cyclic in the Bergman spaces, we have that
$S_{\mu}$ is cyclic in $\mathcal{B}$ implies that $\mu$ cannot charge
Beurling-Carleson sets. The question whether the converse is true was raised
in 1991 by Brown and Shields [12]. THM:wsets above shows that this is not the
case, but we shall below illustrate that this actually fails in a very strong
sense. Before stating our next result, note that if $S_{\mu}\in W^{1}$, then
so are the reproducing kernels of $K_{S_{\mu}}$, and since they span a dense
subspace of $K^{1}_{S_{\mu}}$, part (ii) of THM:Modelspaces implies that the
permanence property holds whenever $S_{\mu}\in W^{1}$.

######  Theorem 1.4.

THM:NoMOC Let $w$ be a majorant with $w(t)/t\to\infty$ as $t\to 0^{+}$. Then
there exists a singular probability measure $\mu=\mu(w)$ on
$\partial\mathbb{D}$ satisfying the following properties:

  1. (i)

For any arc $I\subset\partial\mathbb{D}$, we have $\mu(I)\leq w(|I|).$

  2. (ii)

$S_{\mu}\in W^{1}$, hence it satisfies the permanence property, that is,
$\left[S_{\mu}\right]_{\mathcal{B}}\cap H^{\infty}=S_{\mu}H^{\infty}.$

If we take $w(t)=t^{\alpha}$, $0<\alpha<1$, condition (i) implies that $\mu$
charges no Beurling-Carleson set [29] while (ii) shows that $S_{\mu}$ is not
cyclic in $\mathcal{B}$. Hence not every cyclic singular inner function in the
Bergman space is cyclic in $\mathcal{B}$, answering the problem left open in
[12]. This brings to light a new remarkable discrepancy on $M_{z}$-invariant
subspaces between the Hardy spaces and Bergman spaces. Although Beurling’s
Theorem on the Hardy spaces naturally carries over their corresponding
limiting case $BMOA$ (see [1]), the Korenblum-Roberts Theorem on the Bergman
spaces breaks down in $\mathcal{B}$. It turns out that the negative answer to
the above mentioned question has an immediate consequence to another related
problem in the context of model spaces, mentioned in [17]. As a consequence of
THM:Modelspaces and THM:NoMOC with $w(t)=t\log(e/t)$, we obtain the following
conclusion.

######  Corollary 1.5.

Modelneg There exists a singular inner function $\Theta$ for which the
following distinctive phenomenons occur:

  1. (i)

$K_{\Theta}\cap\bigcup_{p>1}W^{p}=\\{0\\}$.

  2. (ii)

The reproducing kernels of $K_{\Theta}$ belong to $W^{1}$ and hence bounded
analytic functions in $K_{\Theta}\cap W^{1}$ form a dense subset in
$K_{\Theta}$.

It was also asked in [3] whether the condition
$|f(z)|\gtrsim{\log}^{-1}(e/(1-|z|))$ for any $z\in\mathbb{D}$ of a Bloch
function $f$, ensures that $f$ is cyclic in $\mathcal{B}$. Our THM:NoMOC shows
that this is not the case, answering Problem 1 in [3] in the negative. In
fact, given any majorant $w$ with $w(t)/t\to\infty$ as $t\to 0^{+}$, THM:NoMOC
provides a positive singular measure $\mu$ such that $S_{\mu}$ is not cyclic
in $\mathcal{B}$ while condition (i) readily translates to the estimate

$\left\lvert
S_{\mu}(z)\right\rvert\gtrsim\exp\left(-c\frac{w(1-|z|)}{1-|z|}\right),\qquad
z\in\mathbb{D},$

for some numerical constant $c>0$. We conclude that there cannot be any bound
from below of a Bloch function which ensures it to be cyclic in $\mathcal{B}$.
This pathological behavior is very different from the context of Bergman-type
spaces, see [7]. As previously mentioned, $S_{\mu}$ satisfies the permanence
property in Bergman spaces if and only if $\mu$ is concentrated on a countable
union of Beurling-Carleson sets. Our next result says that the situation in
the Bloch space is completely different and no condition on the support of
$\mu$ alone can describe singular inner functions satisfying the permanence
property in $\mathcal{B}$.

######  Theorem 1.6.

THM:Nosupp Let $E\subset\partial\mathbb{D}$ be a closed set of Lebesgue
measure zero. Then there exists a singular probability measure $\mu$ supported
on $E$, such that

  * (i)

$\mu(I)\geq|I|$, for any dyadic arc $I$ with $I\cap E\neq\emptyset$,

  * (ii)

$S_{\mu}\in W^{1}$, hence it satisfies the permanence property, that is,
$\left[S_{\mu}\right]_{\mathcal{B}}\cap H^{\infty}=S_{\mu}H^{\infty}.$

In order to construct a cyclic singular inner function in $\mathcal{B}$, the
authors in [3] provided a sufficient condition for functions in $\mathcal{B}$
to be cyclic. Our next result gives a different sufficient condition which in
a certain sense generalizes the above mentioned result.

######  Theorem 1.7.

THM:SuffcycB Let $f\in H^{\infty}$ without zeros in $\mathbb{D}$, $\left\lVert
f\right\rVert_{H^{\infty}}\leq 1$. Let $\nu=\nu_{f}$ denote its associated
Herglotz-Nevanlinna measure. Assume there exists a constant $C=C(f)>0$, such
that for any pair of contiguous arcs $I,I^{\prime}\subset\partial\mathbb{D}$
of same length $|I|=|I^{\prime}|$, we have

$\left\lvert\frac{\nu(I)}{|I|}-\frac{\nu(I^{\prime})}{|I^{\prime}|}\right\rvert\leq
C\inf\exp\left(-\frac{\nu(J)}{|J|}\right),$ (1)

where the infimum is taken over all arcs $J\subset\partial\mathbb{D}$ with
$J\supset I\cup I^{\prime}$. Then $f$ is cyclic in $\mathcal{B}$.

Note that condition (1) clearly implies that $\nu$ is a Zygmund measure on
$\partial\mathbb{D}$, that is, the expression in the left hand side of (1) is
uniformly bounded over the collection of pairs of contiguous arcs of the same
length. However, condition (1) is considerably stronger as it asserts that if
the density of the measure over a an arc is large, then the oscillation of the
densities of the measure on subarcs must be substantially smaller. It is worth
mentioning that the construction in [3] of a singular measure $\mu$ for which
$S_{\mu}$ is cyclic in $\mathcal{B}$ actually implies that $\mu$ satisfies the
stronger condition

$\sup_{|I|\leq\delta}\left\lvert\frac{\mu(I)}{|I|}-\frac{\mu(I^{\prime})}{|I^{\prime}|}\right\rvert\leq
C\inf_{|J|\geq\delta}\exp\left(-\frac{\mu(J)}{|J|}\right),$

which readily implies (1). It turns out that if one removes the infimum in
condition (1), then one retains a description of bounded analytic functions
which are invertible in $\mathcal{B}$.

######  Theorem 1.8.

THM:Invertibility B Let $f\in H^{\infty}$ without zeros in $\mathbb{D}$,
$\left\lVert f\right\rVert_{H^{\infty}}\leq 1$. Let $\nu=\nu_{f}$ denote its
associated Herglotz-Nevanlinna measure. Then $1/f\in\mathcal{B}$ if and only
if there exists a constant $C=C(f)>0$ such that for any pair of contiguous
arcs $I,I^{\prime}\subset\partial\mathbb{D}$ of the same length, we have

$\left\lvert\frac{\nu(I)}{|I|}-\frac{\nu(I^{\prime})}{|I^{\prime}|}\right\rvert\leq
C\exp\left(-\frac{\nu(I)}{|I|}\right).$ (2)

For a description of elements in $H^{\infty}\cap\mathcal{B}_{0}$ of similar
flavour, we refer the reader to the work of Bishop in [6]. We now turn our
attention to the problem of determining the relationship between invertibility
and cyclicity, which is a vastly investigated topic in various spaces of
analytic functions. It it is well-known that in the setting of commutative
Banach algebras of analytic functions with units, the notions of cyclicity and
invertibility are equivalent, while moving towards classes of analytic
function for which the algebra property fails, such as the classical Dirichlet
spaces and the Hardy spaces, invertibility is strictly stronger than
cyclicity, see [11]. However, the situation for Bergman spaces attracted
considerable amount of attention and remained open for quite some time until
it was resolved by Borichev and Hedenmalm in [8]. In their deep work, they
constructed invertible functions in the Bergman Spaces, which are not cyclic
therein, showing that the two notions are quite different in that setting. Our
next result asserts that a similar phenomenon occurs in the Bloch space. This
answers Problem 2 from [3] in the negative.

######  Theorem 1.9.

THM:CycvsInv There exists $f\in\mathcal{B}$ with $1/f\in\mathcal{B}$ such that
$f$ is not cyclic in $\mathcal{B}$.

The crucial property of a function $f$ as above is that $\left\lvert
f^{\prime}(z)\right\rvert$ is as huge as possible for a considerably large set
of points $z\in\mathbb{D}$. This aligns well with the principal philosophy
surrounding the work in [8], that an impediment to cyclicity of a Bergman
space function is that the function enjoys maximal growth at a ”massive” set
in $\mathbb{D}$.

Our final result should be viewed as a pre-cautionary warning when considering
weak-star closures of $M_{z}$-invariant subspaces generated by Bloch
functions. In fact it may be viewed as a Bloch space version of Theorem 1.2 by
Borichev and Hedenmalm in [9]. It asserts that sequential limits are not
enough in order to capture the behavior of $M_{z}$-invariant subspaces in
$\mathcal{B}$. Recall that classical outer functions in $\mathcal{B}$ are
always cyclic therein.

######  Theorem 1.10.

THM:Seqcyc There exists an outer function $f\in\mathcal{B}$ with $1/f\in
H^{\infty}$, such that the set of all weak-star sequential limits in
$\mathcal{B}$ of functions of the type $f(z)Q(z)$, where $Q$ is an analytic
polynomial, is a proper subspace of $\mathcal{B}$.

The paper is organized as follows. Section 2 is devoted to establish
THM:M_zLat and THM:Modelspaces in the more general framework of so-called
regular spaces. In Section 3, we deduce THM:wsets on the the permanence
property induced by certain boundary zero sets, while Section 4 contains the
proofs of THM:NoMOC and THM:Nosupp, which illustrate that the results of
Korenblum and Roberts in Bergman spaces do not extend to the Bloch space.
Finally, Section 5 is devoted to establishing our results on the theme of
invertibility versus cyclicity, containing THM:SuffcycB, THM:Invertibility B,
THM:CycvsInv and THM:Seqcyc.

##  2 Beurling-type theorems and model spaces

In this section, we shall keep a broad point of view and illustrate a close
relationship between Beurling-type theorems on $M_{z}$-invariant subspaces
generated by inner functions $\Theta$ and the ample of functions with
sufficiently regular boundary values belonging the model space $K_{\Theta}$.

###  2.1 Regular spaces

A Banach space $X$ of analytic functions in $\mathbb{D}$ will be referred to
as a _regular space_ if the following three properties hold:

  * (i)

$X\subseteq H^{1}$.

  * (ii)

The set of analytic polynomials and the set of functions analytic in a
neighborhood of $\overline{\mathbb{D}}$ are dense in $X$.

  * (iii)

For any $\ell$ in the dual space $X^{*}$, we have
$\limsup_{n\to\infty}\,\left\lvert\ell(z^{n})\right\rvert^{1/n}\leq 1.$

Assumption $(i)$ ensures that the boundary values of functions in $X$ are
integrable on $\partial\mathbb{D}$ and by the closed graph theorem, there
exists a constant $C>0$ such that $\|f\|_{1}\leq C\|f\|_{X}$ for any $f\in X$.
Assumption $(ii)$ implies that $X$ is a separable Banach space, thus Helly’s
selection theorem ensures that any closed and bounded set in the norm of its
Banach space dual $X^{*}$ is sequentially compact wrt to the weak-star
topology. In conjunction with $(iii)$, we may substitute the abstract Banach
space dual-pairing between $X,X^{*}$ with the more practical _Cauchy dual_
$X^{\prime}$, considered via the $H^{2}$-pairing

$\lim_{r\to
1-}\int_{\partial\mathbb{D}}f(r\zeta)\overline{g(r\zeta)}dm(\zeta),\qquad f\in
X,g\in X^{\prime}.$

Note that condition $(i)$ also gives the containment $H^{\infty}\subset
X^{\prime}$. For our purposes, we shall equip $X^{\prime}$ with the weak-star
topology. Let us briefly verify that $X=W^{1}$ is indeed a regular space. Note
that if $g\in W^{1}$, then

$\int_{\partial\mathbb{D}}|g(r\zeta)-g(0)|dm(\zeta)\leq\int_{\partial\mathbb{D}}\int_{0}^{r}|g^{\prime}(s\zeta)|dsdm(\zeta)\leq\left\lVert
g\right\rVert_{W^{1}},\quad 0<r<1.$

Hence $W^{1}\subset H^{1}$ as required. It is clear that analytic polynomials
are dense in $W^{1}$. Finally since the Cauchy-dual relation
$(W^{1})^{\prime}\cong\mathcal{B}$ holds, condition (iii) is also fulfilled as
the Taylor coefficients of Bloch functions are well-known to be bounded, see
[4]. We thus conclude that $X=W^{1}$ indeed is a regular space. More
generally, classical examples of regular spaces are provided by the analytic
Sobolev spaces $W^{p}$ with $p\geq 1$ and their counter-parts involving
multiple derivatives.

###  2.2 $M_{z}$-invariant subspaces in duals of Regular Spaces

Let $X$ be a regular space and let $X^{\prime}$ denote its Cauchy dual. In
this subsection we collect several general properties of $M_{z}$-invariant
subspaces in $X^{\prime}$. Given a function $f\in X^{\prime}$, we denote by
$\left[f\right]_{X^{\prime}}$ the weak-star closure in $X^{\prime}$ of the
smallest $M_{z}$-invariant subspace containing $f$. We declare $f\in
X^{\prime}$ to be cyclic in $X^{\prime}$ if
$\left[f\right]_{X^{\prime}}=X^{\prime}$. Similarly, given an inner function
$\Theta$, we say that $\Theta$ satisfies the _permanence property_ in
$X^{\prime}$ if

$\left[\Theta\right]_{X^{\prime}}\cap H^{\infty}\subseteq\Theta H^{\infty}.$

The permanence property was recently introduced in the papers [25] and [27],
in more specific context than ours. We also mention that it implicitly
appeared in the earlier work of J. Roberts [28], in the context of Bergman
spaces. Let $X$ be a regular space, let $\Theta$ be an inner function and let
$\mathcal{P}$ be the set of analytic polynomials. Observe that $X\cap
K_{\Theta}^{1}$ regarded as a subset of $X$ is precisely the pre-annihilator
of $\Theta\mathcal{P}\subset X^{\prime}$, denoted by
$\Theta\mathcal{P}_{\perp}$. A standard argument involving the Hahn-Banach
separation theorem gives the following equality of sets

$(X\cap
K_{\Theta}^{1})^{\perp}=(\Theta\mathcal{P}_{\perp})^{\perp}=\left[\Theta\right]_{X^{\prime}}.$
(3)

Since the Cauchy reproducing kernels are in $X$, weak-star convergence in
$X^{\prime}$ implies convergence on compact subsets of $\mathbb{D}$. Hence the
permanence property always holds for Blaschke products. In what follows, we
shall thus restrict our attention to the permanence property for singular
inner functions. Recall that $H^{\infty}$ is the Cauchy dual of the Banach
space $L^{1}(dm)/H^{1}_{0}$, where $H^{1}_{0}$ denotes the subspace of
functions in $H^{1}$ vanishing at the origin. Our first simple observation is
that for regular spaces $X$, the weak-star topology in $X^{\prime}$ is coarser
than the weak-star topology in $H^{\infty}$.

######  Lemma 2.1.

HooX’

Let $X$ be a regular space. Then $H^{\infty}\subset X^{\prime}$ holds, and
whenever $\\{f_{n}\\}_{n}$ converges to $f$ in weak-star of $H^{\infty}$, then
$f_{n}$ converges to $f$ in weak-star of $X^{\prime}$.

###### Proof.

Both these statements follow immediately from the assumption $X\subseteq
H^{1}$ of regular spaces. ∎

Our next observation allows us to substitute multiples of analytic polynomials
by multiples of $H^{\infty}$-functions when considering (weak-star) cyclicity
of a bounded analytic function in $X^{\prime}$.

######  Proposition 2.2.

bddcyc Let $f\in H^{\infty}$. Then the weak-star closure in $X^{\prime}$ of
$fH^{\infty}:=\\{fh:h\in H^{\infty}\\}$ equals $\left[f\right]_{X^{\prime}}$.

###### Proof.

It suffices to verify that $fh\in\left[f\right]_{X^{\prime}}$ for any $h\in
H^{\infty}$. Let $\\{Q_{n}\\}_{n}$ be a sequence of polynomials which converge
weak-star in $H^{\infty}$ to $h$. Then since $f\in H^{\infty}$, $Q_{n}f$
converges weak-star in $H^{\infty}$ to $fh$. The conclusion now follows from
HooX’. ∎

Our next results illustrates a certain ordering for $M_{z}$-invariant
subspaces generated by bounded analytic functions.

######  Proposition 2.3 (Division principle).

cycdivprinc Let $f,g\in H^{\infty}$ with the property that $f/g\in
H^{\infty}$. Then
$\left[f\right]_{X^{\prime}}\subseteq\left[g\right]_{X^{\prime}}$. In
particular, whenever $f$ is weak-star cyclic in $X^{\prime}$, then $g$ is
weak-star cyclic in $X^{\prime}$.

###### Proof.

Set $h=f/g\in H^{\infty}$ and note that it suffices to show that
$f=hg\in\left[g\right]_{X^{\prime}}$. This however follows from the same
argument as in bddcyc. ∎

For singular inner functions $S_{\mu}$, the division principle implies that
whenever $S_{\mu}$ is cyclic in $X^{\prime}$, then so is any divisor
$S_{\mu_{0}}$ of $S_{\mu}$, where $0\leq\mu_{0}\leq\mu$. In the opposite
direction, a monotonicity principle for the permanence property of singular
inner functions in $X^{\prime}$ holds.

######  Proposition 2.4 (Monotonicity principle).

Ppropmonprinc Let $\\{\mu_{n}\\}_{n}$ be a sequence of positive, finite,
Borel, singular measures with $\mu_{n}\leq\mu_{n+1}$, $n\geq 1$, such that
$\mu_{n}\to\mu$ weak-star in the space of complex finite Borel measures on
$\partial\mathbb{D}$. Then if for any $n\geq 1$, the associated singular inner
functions $S_{\mu_{n}}$ satisfies the permanence property in $X^{\prime}$,
then so does $S_{\mu}$.

###### Proof.

Note that by monotonicity each $S_{\mu_{n}}$ is a divisor of $S_{\mu}$ hence
$K^{1}_{S_{\mu_{n}}}\subseteq K^{1}_{S_{\mu}}$ for each $n$. Since
$S_{\mu_{n}}$ satisfies the permanence property for any $n\geq 1$, then
according to Proposition LABEL:cycdivprinc, we obtain

$\left[S_{\mu}\right]_{X^{\prime}}\cap
H^{\infty}\subseteq\left[S_{\mu_{n}}\right]_{X^{\prime}}\cap
H^{\infty}\subseteq S_{\mu_{n}}H^{\infty},\qquad n\geq 1.$

Let $f\in\left[S_{\mu}\right]_{X^{\prime}}\cap H^{\infty}$ be arbitrary and
note that by the above containment, we can for any $n$ find a bounded analytic
function $h_{n}$ such that $f=S_{\mu_{n}}h_{n}$. Now since $\left\lVert
h_{n}\right\rVert_{\infty}=\left\lVert f\right\rVert_{\infty},$ for $n\geq 1$,
we may extract a subsequence $\\{h_{n_{k}}\\}_{k}$ which converges pointwise
in $\mathbb{D}$ to some $h\in H^{\infty}$. Since $S_{\mu_{n}}$ converges
pointwise in $\mathbb{D}$ to $S_{\mu}$, we conclude that $f=S_{\mu}h$. Thus
$S_{\mu}$ satisfies the permanence property in $X^{\prime}$ as desired. ∎

Our final lemma asserts that $M_{z}$-invariant subspaces generated by bounded
analytic functions are invariant under the multiplication of outer functions
in $H^{\infty}$. In particular, outer functions in $H^{\infty}$ always cyclic
in $X^{\prime}$.

######  Lemma 2.5.

Fcycstab Let $F\in H^{\infty}$ be an outer function and $g\in H^{\infty}$.
Then $\left[Fg\right]_{X^{\prime}}=\left[g\right]_{X^{\prime}}$.

###### Proof.

The containment $Fg\in\left[g\right]_{X^{\prime}}$ follows from the proof of
bddcyc and gives the inclusion $\subseteq$. For the converse inclusion, we
apply Theorem 7.4 in [20] which yields a sequence of bounded analytic
functions $\\{F_{n}\\}_{n}$ satisfying the following properties

  1. (i)

$\left\lvert F_{n}(z)F(z)\right\rvert\leq 1$, $z\in\mathbb{D}$,

  2. (ii)

$F_{n}(\zeta)F(\zeta)$ converges pointwise to $1$ for $m$-a.e
$\zeta\in\partial\mathbb{D}$.

Then $FgF_{n}$ converges weak star in $H^{\infty}$ to $g$. The result now
follows from HooX’ and bddcyc. ∎

###  2.3 Proof of THM:Modelspaces in the context of Regular Spaces

Our main purpose is to establish the connection between the containment of a
regular space $X$ in the model spaces $K_{\Theta}$ to $M_{z}$-invariant
subspaces generated by $\Theta$ in $X^{\prime}$ in the weak-star topology.
However, the assumption $X\subseteq H^{1}$ actually makes the model spaces
$K^{1}_{\Theta}$ more appropriate in this regard. We recall that the
reproducing kernels of $K^{1}_{\Theta}$ are explicitly given by

$\kappa_{\Theta}(z,\lambda):=\frac{1-\overline{\Theta(\lambda)}\Theta(z)}{1-\overline{\lambda}z},\qquad\lambda,z\in\mathbb{D}$

and their linear span forms a dense subset in $K^{1}_{\Theta}$. Our first
observation is related to cyclicity of inner functions $\Theta$ in
$X^{\prime}$ and the existence of non-trivial functions in $X\cap
K^{1}_{\Theta}$. We remark that similar results were known for a wide range of
analytic Sobolev spaces $X$ in [17] and more recently, it has appeared
implicitly in a more general setting similar to ours in [25].

######  Proposition 2.6.

prop2.1 Let $X$ be a regular space. An inner function $\Theta$ is weak-star
cyclic in $X^{\prime}$ if and only if $X\cap K_{\Theta}^{1}$ contains no non-
trivial function.

###### Proof.

Any $g\in X\cap K^{1}_{\Theta}$ is annihilated by polynomial multiples of
$\Theta$, hence if we assume that $\Theta$ is weak-star cyclic in
$X^{\prime}$, then $g\equiv 0$. Conversely, if $\Theta$ is not weak-star
cyclic in $X^{\prime}$, then there exists a non-trivial $g\in X$ such that

$\int_{\partial\mathbb{D}}\overline{g(\zeta)}\Theta(\zeta){\zeta}^{n}dm(\zeta)=0,\qquad
n\geq 0.$

The F. and M. Riesz Theorem implies that there exists $g_{0}\in H^{1}$ with
$g_{0}(0)=0$ such that $g=\Theta\overline{g_{0}}$ at almost every point of
$\partial\mathbb{D}$. Hence $g\in
H^{1}\cap\Theta\overline{zH^{1}}=K_{\Theta}^{1}$. ∎

Next we shall relate the density of regular spaces in the model spaces
$K^{1}_{\Theta}$ to the _permanence property_ of $\Theta$ in $X^{\prime}$. The
following result is very similar to Theorem 1.3 in [25], but we shall need a
more general version for our setting.

######  Proposition 2.7.

Pprinc Let $X$ be a regular space and let $\Theta$ be an inner function. Then
$X\cap K_{\Theta}^{1}$ is dense in $K_{\Theta}^{1}$ if and only if $\Theta$
satisfies the permanence property in $X^{\prime}$.

###### Proof.

Suppose that $X\cap K_{\Theta}^{1}$ is dense in $K_{\Theta}^{1}$ and let
$f\in\left[\Theta\right]_{X^{\prime}}\cap H^{\infty}$. According to (3), $f$
must necessarily annihilate the family of reproducing kernels
$\\{k_{\Theta}(\cdot,\lambda)\\}_{\lambda\in\mathbb{D}}$ of $K_{\Theta}^{1}$,
that is

$f(\lambda)=\Theta(\lambda)\int_{\partial\mathbb{D}}\frac{\overline{\Theta(\zeta)}f(\zeta)}{1-\overline{\zeta}\lambda}dm(\zeta),\qquad\lambda\in\mathbb{D}.$

Hence $f\in\Theta H^{2}$, but since $f\in H^{\infty}$, we get that $f\in\Theta
H^{\infty}$ as desired. Conversely, if the permanence property holds, then in
fact we verify that the following stronger condition holds:

$\left[\Theta\right]_{X^{\prime}}\cap H^{2}\subseteq\Theta H^{2}.$ (4)

To see this, note that if $f\in\left[\Theta\right]_{X^{\prime}}\cap H^{2}$,
then $\left[f\right]_{X^{\prime}}\subseteq\left[\Theta\right]_{X^{\prime}}$.
Let $f=F\Phi$, where $F$ is outer and $\Phi$ inner. We want to show that
$\Theta$ divides $\Phi$ in $H^{2}$. For $n\geq 1$, consider the outer function
$F_{n}$ defined as $\left\lvert F_{n}\right\rvert=\min\left(n,\left\lvert
F\right\rvert\right)$ on $\partial\mathbb{D}$ and note that if we set
$f_{n}:=F_{n}\Phi$, then $f_{n}/f\in H^{\infty}$ and hence by the division
principle cycdivprinc, we get

$\left[f_{n}\right]_{X^{\prime}}\subseteq\left[f\right]_{X^{\prime}}\subseteq\left[\Theta\right]_{X^{\prime}},\qquad
n\geq 1.$

However since $f_{n}=F_{n}\Phi$ is bounded for each $n\geq 1$, Fcycstab
implies that $\left[f_{n}\right]_{X^{\prime}}=\left[\Phi\right]_{X^{\prime}}$
and thus we conclude that $\Phi\in\left[\Theta\right]_{X^{\prime}}\cap
H^{\infty}\subseteq\Theta H^{\infty}$, proving the claim. With this at hand,
note that (4) in conjunction with (3) may be rephrased as

$(X\cap K_{\Theta}^{1})^{\perp}\cap H^{2}\subseteq\Theta H^{2}.$

Now since the Cauchy dual $(K_{\Theta}^{1})^{\prime}$ is contained in
$K^{2}_{\Theta}$ (see [19]), we see that any annihilator of $X\cap
K_{\Theta}^{1}$ is contained in $K^{2}_{\Theta}\cap\Theta H^{2}=\\{0\\}$. This
is enough to conclude that $X\cap K_{\Theta}^{1}$ is dense in
$K_{\Theta}^{1}$. ∎

###### Proof of THM:Modelspaces.

Taking $X=W^{1}$, part (a) follows from prop2.1, while $(b)$ follows from
Pprinc. ∎

###  2.4 An abstract decomposition for singular measures

Inspired from the Korenblum-Roberts Theorem on $M_{z}$-invariant subspaces
generated by inner functions, this section is devoted to extending their
result to the context of Regular Spaces. Here we shall follow ideas developed
by Roberts in [28], with certain required adaptations in order to deal with
our general framework.

######  Theorem 2.8 (Abstract decomposition).

Absdec Let $X$ be a regular space and let $S_{\mu}$ be a singular inner
function. Then there exists a unique decomposition of $\mu=\mu_{P}+\mu_{C}$
(up to sets of $\mu$-measure zero) into mutually singular parts
$\mu_{P},\mu_{C}$ satisfying the following properties:

  * (i)

$S_{\mu_{P}}$ has the permanence property in $X^{\prime}$, that is,
$\left[S_{\mu_{P}}\right]_{X^{\prime}}\cap H^{\infty}\subseteq
S_{\mu_{P}}H^{\infty}$,

  * (ii)

$S_{\mu_{C}}$ is weak-star cyclic in $X^{\prime}$, that is,
$\left[S_{\mu_{C}}\right]_{X^{\prime}}=X^{\prime}$.

###### Proof.

Let $\mu\lvert_{E}$ denote the restriction of $\mu$ to the Borel set $E$.
Consider the the following collection of Borel subsets of $\partial\mathbb{D}$

$\mathcal{M}(X^{\prime},\mu):=\left\\{E\subset\partial\mathbb{D}:\,\text{E
Borel set},\,\,\,\mu(E)>0,\left[S_{\mu\lvert_{E}}\right]_{X^{\prime}}\cap
H^{\infty}\subseteq S_{\mu\lvert_{E}}H^{\infty}\right\\}.$

If the set $\mathcal{M}(X^{\prime},\mu)$ is empty, then we simply take
$\mu_{C}=\mu$. For the being moment, we shall assume that the collection
$\mathcal{M}(X^{\prime},\mu)$ is non-empty, and our goal is to primarily
establish that $\mathcal{M}(X^{\prime},\mu)$ is closed under the formation of
unions. To this end, pick two sets $E,F\in\mathcal{M}(X^{\prime},\mu)$ which
we may assume are not contained in one or another. Set $\nu:=\mu\lvert_{E\cup
F}$ and note that the containment of $K_{S_{\mu\lvert_{E}}}^{1}\subset
K_{S_{\nu}}^{1}$ in conjunction with (3) implies

$\left[S_{\nu}\right]_{X^{\prime}}\cap H^{\infty}=\left(X\cap
K_{S_{\nu}}^{1}\right)^{\perp}\cap H^{\infty}\subseteq(X\cap
K_{S_{\mu\lvert_{E}}}^{1})^{\perp}\cap
H^{\infty}=\left[S_{\mu\lvert_{E}}\right]_{X^{\prime}}\cap H^{\infty}\subseteq
S_{\mu\lvert_{E}}H^{\infty}.$

Therefore, any $f\in\left[S_{\nu}\right]_{X^{\prime}}\cap H^{\infty}$ is
divisible by $S_{\mu\lvert_{E}}$, and by switching the roles of $E$ and $F$,
we deduce in a similar way that $f$ is also divisible by $S_{\mu\lvert_{F}}$.
Altogether, we obtain $f/S_{\nu}\in H^{\infty}$ and therefore $E\cup
F\in\mathcal{M}(X^{\prime},\mu)$. Now consider the quantity

$\gamma(X^{\prime},\mu):=\sup\left\\{\mu(E):\,E\in\mathcal{M}(X^{\prime},\mu)\right\\},$

and observe that since $\mathcal{M}(X^{\prime},\mu)$ is closed under finite
unions, we can find a sequence of Borel sets $\\{E_{n}\\}_{n}$ with
$E_{n}\in\mathcal{M}(X^{\prime},\mu)$ and $E_{n}\subset E_{n+1}$ for any $n$,
such that $\mu(E_{n})\to\gamma(X^{\prime},\mu)$ as $n\to\infty$. Set
$E:=\cup_{n}E_{n}$ and we claim that our candidates are given by
$\mu_{P}=\mu\lvert_{E}$ and $\mu_{C}=\mu\lvert_{\partial\mathbb{D}\setminus
E}$. We first check that $E\in\mathcal{M}(X^{\prime},\mu)$. To this end, let
$f\in\left[S_{\mu\lvert_{E}}\right]_{X^{\prime}}\cap H^{\infty}$ and note that
since each $E_{n}\in\mathcal{M}(X^{\prime},\mu)$, an application of (3)
implies

$\left[S_{\mu\lvert_{E}}\right]_{X^{\prime}}\cap H^{\infty}=(X\cap
K_{S_{\mu\lvert_{E}}}^{1})^{\perp}\cap H^{\infty}\subseteq(X\cap
K_{S_{\mu\lvert_{E_{n}}}}^{1})^{\perp}\cap
H^{\infty}=\left[S_{\mu\lvert_{E_{n}}}\right]_{X^{\prime}}\cap
H^{\infty}\subseteq S_{\mu\lvert_{E_{n}}}H^{\infty},$

for all $n$. Hence for each $n$, there exists $\\{h_{n}\\}_{n}\subset
H^{\infty}$ such that $f=S_{\mu\lvert_{E_{n}}}h_{n}$. But since $\left\lVert
h_{n}\right\rVert_{\infty}=\left\lVert f\right\rVert_{\infty},$ we may extract
a subsequence $\\{h_{n_{k}}\\}_{k}$ that converges pointwise in $\mathbb{D}$
to a certain function $h\in H^{\infty}$. Now since the sequence of sets
$\\{E_{n}\\}_{n}$ is increasing, we actually obtain $f=S_{\mu_{E}}h$. This
proves that $E\in\mathcal{M}(X^{\prime},\mu)$ and consequently we have
established claim $(i)$ with $\mu_{P}=\mu\lvert_{E}$.
We now proceed to establish claim $(ii)$ by showing that if
$\mu_{C}:=\mu\lvert_{\partial\mathbb{D}\setminus E}$ then $S_{\mu_{C}}$ is
weak-star cyclic in $X^{\prime}$. To this end, suppose that $S_{\mu_{C}}$ is
not weak-star cyclic in $X^{\prime}$, then there exists a non-trivial $g\in X$
such that

$\int_{\partial\mathbb{D}}\overline{g(\zeta)}S_{\mu_{C}}(\zeta)\zeta^{n}dm(\zeta)=0,\qquad
n\geq 0.$

By the F. and M. Riesz Theorem, there exists a function $h_{1}\in H^{1}$ with
$h_{1}(0)=0$ such that

$\overline{g(\zeta)}=\frac{h_{1}(\zeta)}{S_{\mu_{C}}(\zeta)}=\frac{h_{2}(\zeta)}{S_{\nu}(\zeta)},\quad
m-\text{a.e.}\,\,\,\zeta\in\partial\mathbb{D}.$

Here $h_{2}\in H^{1}$, $h_{2}(0)=0$, $0\leq\nu\leq\mu_{C}$ and the quotient
$h_{1}/h_{2}$ is a singular inner function dividing $S_{\mu_{C}}$ (possibly a
unimodular constant). Now if $\nu\equiv 0$, then $g\in
H^{1}\cap\overline{H^{1}}$ on $\partial\mathbb{D}$ with $g(0)=0$, which
implies $g\equiv 0$, thus contradicting the assumption that $g$ is non-
trivial. Hence we may assume that $\nu\neq 0$, and our aim is to establish
that $S_{\nu}$ satisfies the permanence property, which will contradict the
maximality of $S_{\mu_{P}}$. To this end, note that

$\int_{\partial\mathbb{D}}\overline{g(\zeta)}S_{\nu}(\zeta)\zeta^{n}dm(\zeta)=\int_{\partial\mathbb{D}}h_{2}(\zeta)\zeta^{n}dm(\zeta)=0,\qquad
n\geq 0,$

hence by the F. and M. Riesz Theorem, we have $g\in K^{1}_{S_{\nu}}:=H^{1}\cap
S_{\nu}\overline{H^{1}_{0}}$. Now if
$u\in\left[S_{\nu}\right]_{X^{\prime}}\cap H^{\infty}$, then according to (3)
we actually get

$\int_{\partial\mathbb{D}}\overline{g(\zeta)}u(\zeta)\zeta^{n}dm(\zeta)=0,\qquad
n\geq 0.$

Applying the F. and M. Riesz Theorem once again, we can find $h_{3}\in H^{1}$
with $h_{3}(0)=0$ such that

$\overline{g(\zeta)}=\frac{h_{3}(\zeta)}{u(\zeta)}=\frac{h_{2}(\zeta)}{S_{\nu}(\zeta)},\quad
m-\text{a.e.}\,\,\,\zeta\in\partial\mathbb{D}$

which implies that $u\in S_{\nu}H^{\infty}$. Hence $S_{\nu}$ satisfies the
permanence property in $X^{\prime}$. We have thus obtained the desired
contradiction and it follows that $S_{\mu_{C}}$ is weak-star cyclic in
$X^{\prime}$. Obviously the same argument works in case the collection
$\mathcal{M}(X^{\prime},\mu)$ is void. The claim regarding uniqueness of the
above decomposition of $\mu$ follows from maximality of $\mu_{P}$ with respect
to the permanence property in $X^{\prime}$.

∎

###### Proof of THM:M_zLat.

The following proof of THM:M_zLat actually holds for any regular space $X$.
Note that statement $(i)$ in Absdec regarding the permanence property is
easily shown to remain true if $S_{\mu_{P}}$ is multiplied by a Blaschke
product. The claim on cyclicity in $(ii)$ of THM:M_zLat follows from the
cyclicity of $S_{\mu_{C}}$ in conjunction with Fcycstab. This completes the
proof.

∎

As a consequence of Absdec in conjunction with the results in Section 2.3, we
conclude this section with the following description regular spaces in model
spaces.

######  Corollary 2.9.

strmodthm Let $X$ be a regular space, let $\Theta=BS_{\mu}$ where $B$ is a
Blaschke product and $S_{\mu}$ is a singular inner function. Consider the
decomposition $\mu=\mu_{P}+\mu_{C}$ given in Absdec. Then the following holds:

Abundance of regular functions:

$X\cap K^{1}_{BS_{\mu_{P}}}$ is a dense subspace of $K^{1}_{BS_{\mu_{P}}}$,

Absence of regular functions:

$K^{1}_{S_{\mu_{C}}}\cap X=\\{0\\}$.

##  3 The permanence property and boundary zero sets

This section is devoted to showing that inner factors of certain weighted BMOA
spaces give rise to the permanence property in $\mathcal{B}$, for which
THM:wsets will be derived as a corollary.

###  3.1 Inner factors in weighted BMOA

Let $w$ be a majorant and denote by $BMO_{w}(\partial\mathbb{D})$ the space of
Lebesgue integrable functions $h$ on $\partial\mathbb{D}$ equipped with the
semi-norm

$\left\lVert
h\right\rVert_{BMO_{w}}:=\sup_{I}\frac{1}{w(|I|)}\frac{1}{|I|}\int_{I}|h-h_{I}|dm<\infty.$

We denote by $C_{w}(\partial\mathbb{D})$ the space of continuous functions $h$
on $\partial\mathbb{D}$ equipped with the semi-norm

$\left\lVert h\right\rVert_{C_{w}}:=\sup_{\zeta\neq\xi}\frac{\left\lvert
h(\zeta)-h(\xi)\right\rvert}{w(\left\lvert\zeta-\xi\right\rvert)}<\infty.$

It is straightforward to verify the containment
$C_{w}(\partial\mathbb{D})\subseteq BMO_{w}(\partial\mathbb{D})$ and it turns
out that these spaces share very intimate features. If the majorant $w$
satisfies the Dini-condition

$\int_{0}^{1}\frac{w(t)}{t}dt<\infty,$ (5)

then functions in $BMO_{w}(\partial\mathbb{D})$ extend continuously to
$\partial\mathbb{D}$ with modulus of continuity not exceeding a constant
multiple of $\int_{0}^{\delta}\frac{w(t)}{t}dt$. Conversely, if the Dini-
condition (5) on $w$ fails then $BMO_{w}(\partial\mathbb{D})$ contains
discontinuous and unbounded functions. See [31] for details on these matters.
Analytic counter-parts of the above defined spaces will be important, namely
we consider the spaces $BMOA_{w}:=H^{2}\cap BMO_{w}(\partial\mathbb{D})$ and
$A_{w}:=H^{\infty}\cap C_{w}(\partial\mathbb{D})$, hence the containment
$A_{w}\subseteq BMOA_{w}$ holds. Moving forward, we shall now restrict our
attention to majorants $w$ satisfying the following condition. There exists a
constant $0<\gamma<1$ such that

$w(t)/t^{\gamma}\text{ is non-increasing on }[0,1].$ (6)

We now state our main result in this section.

######  Theorem 3.1.

THM:PPBMOAw Let $w$ be a majorant satisfying condition (6) and the Dini
condition (5). Then any non-trivial singular inner factor $S_{\mu}$ of a
function in $BMOA_{w}$ satisfies the permanence property in $\mathcal{B}$,
that is, $\left[S_{\mu}\right]_{\mathcal{B}}\cap
H^{\infty}=S_{\mu}H^{\infty}.$ In particular, no non-trivial singular inner
divisor of a function in $BMOA_{w}$ can be cyclic in $\mathcal{B}$.

An important class of singular inner factors of $BMOA_{w}$-functions are in
fact provided by singular inner factors of $A_{w}$-functions, which turn out
to be intimately connected to sets of finite $w$-entropy due to a deep result
by Shirokov in [30]. For the moment being, we shall primarily establish
THM:PPBMOAw and then illustrate how THM:wsets can be derived as a corollary.
In the pursuit towards proving THM:PPBMOAw, we shall need a couple of
preparatory results. The first lemma can be found in a slight greater
generality in [18] (see Proposition 2.6 therein for further details).

######  Lemma 3.2.

LemBMOw Let $w$ be a majorant satisfying condition (6). Then the $BMO_{w}$
semi-norm is equivalent to the Garsia-type semi-norm, that is,

$\left\lVert
h\right\rVert_{BMO_{w}}\asymp\sup_{z\in\mathbb{D}}\frac{1}{w(1-|z|)}\int_{\partial\mathbb{D}}\left\lvert
h(\zeta)-P(h)(z)\right\rvert P_{z}(\zeta)dm(\zeta)$

where $P(h)$ denotes the Poisson extension of $h$ on $\mathbb{D}$ and
$P_{z}(\zeta)=(1-|z|^{2})|1-\overline{z}\zeta|^{-2}$, $z\in\mathbb{D}$,
$\zeta\in\partial\mathbb{D}$, is the Poisson kernel on $\mathbb{D}$.

With LemBMOw at hand, we may derive the following result on the Cauchy
projection.

######  Lemma 3.3.

BMOwintoW11

Let $w$ be a majorant satisfying condition (6) and the Dini condition (5).
Then the Cauchy projection $P_{+}$ maps $BMO_{w}(\partial\mathbb{D})$
continuously into $W^{1}$.

###### Proof.

Fix an arbitrary $h\in BMO_{w}(\partial\mathbb{D})$ and observe that

$P_{+}(h)^{\prime}(z)=\int_{\partial\mathbb{D}}\frac{\overline{\zeta}h(\zeta)}{(1-\overline{\zeta}z)^{2}}dm(\zeta)=\int_{\partial\mathbb{D}}\frac{\overline{\zeta}\left(h(\zeta)-P(h)(z)\right)}{(1-\overline{\zeta}z)^{2}}dm(\zeta),\qquad
z\in\mathbb{D}.$

According to LemBMOw, we have

$\left\lvert
P_{+}(h)^{\prime}(z)\right\rvert\leq\int_{\partial\mathbb{D}}\frac{\left\lvert
h(\zeta)-P(h)(z)\right\rvert}{|1-\overline{\zeta}z|^{2}}dm(\zeta)\lesssim\left\lVert
h\right\rVert_{BMO_{w}}\frac{w(1-|z|)}{1-|z|},\qquad z\in\mathbb{D}.$

It now readily follows that

$\int_{\mathbb{D}}\left\lvert P_{+}(h)^{\prime}(z)\right\rvert
dA(z)\lesssim\left\lVert h\right\rVert_{BMO_{w}}\int_{0}^{1}\frac{w(t)}{t}dt,$

which shows that $P_{+}:BMO_{w}(\partial\mathbb{D})\to W^{1}$ continuously. ∎

The following result on division and multiplication by inner functions on
weighted $BMOA$ spaces due to K. Dyakonov in [18] plays a crucial role in our
developments.

######  Theorem 3.4 (Dyakonov).

THM:Dyakonov Let $w$ be a majorant satisfying condition (6). Let $g\in
BMOA_{w}$ and let $\Theta$ be an inner function. Then $g\overline{\Theta}$
belongs to $BMO_{w}(\partial\mathbb{D})$ if and only if $g\Theta$ belongs to
$BMOA_{w}$.

It follows from the work in [18] that if $w$ satisfies condition (6), then
$BMOA_{w}$ enjoys the factorization property. More specifically, whenever
$g\in BMOA_{w}$ and $\Theta$ is an inner function with $g/\Theta\in
H^{\infty}$, then in fact $g/\Theta\in BMOA_{w}$.

###### Proof of THM:PPBMOAw.

Let $\Theta:=S_{\mu}$ be a singular inner factor of a function $g\in
BMOA_{w}$. By means of applying the factorization property of $BMOA_{w}$, we
may without loss of generality assume that $g=G\Theta$, where $G$ is an outer
function in $BMOA_{w}$. Now applying THM:Dyakonov to the function $G$, we
conclude that $G\overline{\Theta}$ belongs to $BMO_{w}(\partial\mathbb{D})$
and thus so does $\overline{G}\Theta$. Hence if $k_{\lambda}$ denotes the
Cauchy kernel at the point $\lambda\in\mathbb{D}$, then the function
$g_{\lambda}:=\overline{G}\Theta k_{\lambda}$ also remains in
$BMO_{w}(\partial\mathbb{D})$. Applying BMOwintoW11, we conclude that
$P_{+}(g_{\lambda})\in W^{1}$. Given $H\in L^{\infty}(\partial\mathbb{D},dm)$,
we denote by $T_{H}$ the Toeplitz operator with symbol $H$ defined as
$T_{H}(f):=P_{+}(Hf)$. If $\kappa_{\Theta}(\cdot,\lambda)$ denotes the
reproducing kernel of the model space $K_{\Theta}$ at the point
$\lambda\in\mathbb{D}$, we have that

$T_{\overline{G}}\left(\kappa_{\Theta}(\cdot,\lambda)\right)(z)=T_{\overline{G}}(k_{\lambda})(z)-\overline{\Theta(\lambda)}P_{+}(g_{\lambda})(z),\quad
z\in\mathbb{D},$

belongs to $W^{1}\cap K_{\Theta}$. This follows from the following properties
of Toeplitz operators with co-analytic symbols:

  1. (a)

$T_{\overline{H}}:W^{p}\to W^{p}$ for any $H\in H^{\infty}$ and $1<p<\infty$.

  2. (b)

$T_{\overline{H}}:K_{\Phi}\to K_{\Phi}$ for any $H\in H^{\infty}$ and any
inner function $\Phi$.

For instance, see [14]. Property $(a)$ is actually more than what is needed
here, but it certainly guarantees that $T_{\overline{f}}(k_{\lambda})$ belongs
to $W^{1}$ for each $\lambda\in\mathbb{D}$, while property $(b)$ shows that
$T_{\overline{G}}\left(\kappa_{\Theta}(\cdot,\lambda)\right)$ belongs to
$K_{\Theta}$. Consequently, we conclude that
$T_{\overline{G}}\left(\kappa_{\Theta}(\cdot,\lambda)\right)$ belong to
$W^{1}\cap K_{\Theta}$. We now proceed by verifying that the linear span
$\mathcal{M}$ of the set
$\\{T_{\overline{G}}(\kappa_{S_{\Theta}}(\cdot,\lambda))\\}_{\lambda\in\mathbb{D}}$
is dense in $K_{\Theta}$. To this end, let $f\in K_{\Theta}$ be an element
which annihilates $\mathcal{M}$. Then

$0=\int_{\partial\mathbb{D}}T_{\overline{G}}(\kappa_{\Theta}(\cdot,\lambda))(\zeta)\overline{f(\zeta)}dm(\zeta)=\int_{\partial\mathbb{D}}\kappa_{\Theta}(\zeta,\lambda)\overline{G(\zeta)f(\zeta)}dm(\zeta),\qquad\lambda\in\mathbb{D}.$

Since the linear span of the reproducing kernels are dense in $K_{\Theta}$, we
conclude that there exists a function $h\in H^{2}$ such that $Gf=\Theta h$.
Now recalling that $G$ was outer, we conclude that $f\in\Theta H^{2}\cap
K_{\Theta}=\\{0\\}$, and thus $\mathcal{M}$ is dense in $K_{\Theta}$. Now
since $K_{\Theta}$ is dense in $K^{1}_{\Theta}$, we also conclude that
$\mathcal{M}$ is dense in $K^{1}_{\Theta}$. Recalling that $W^{1}$ is the
Cauchy pre-dual of $\mathcal{B}$, it now follows from Pprinc that
$\left[\Theta\right]_{\mathcal{B}}\cap H^{\infty}\subseteq\Theta H^{\infty}$,
hence the proof is complete.

∎

We now turn our attention to the proof of THM:wsets.

###### Proof of THM:wsets.

We shall primarily assume that the positive finite singular measure $\mu$ on
$\partial\mathbb{D}$ is supported on a single (closed) set $E$ of finite
$w$-entropy. According to Shirokov’s Theorem in [30], there exists an outer
function $f\in A_{w}$ such that the product $fS_{\mu}$ belongs to
$A_{w}\subset BMOA_{w}$. Hence by THM:PPBMOAw $S_{\mu}$ satisfies the
permanence property in $\mathcal{B}$. Now assume that $\mu$ is concentrated on
countable union of sets $\\{E_{n}\\}_{n}$ having finite $w$-entropy, which we
may assume are increasing: $E_{n}\subseteq E_{n+1}$ for any $n$. Consider
$\mu_{n}:=\mu\lvert_{E_{n}}$ and observe that $\mu_{n}$ converges to $\mu$ in
the weak-star topology of finite complex Borel measures on
$\partial\mathbb{D}$. Since each $S_{\mu_{n}}$ satisfies the permanence
property in $\mathcal{B}$, so does $S_{\mu}$ by the monotonicity principle in
Ppropmonprinc. The proof is now complete.

∎

##  4 The permanence property and $W^{1}$

###  4.1 The permanence property and invisibility

Let $\mu$ be a positive finte Borel singular measure on $\partial\mathbb{D}$.
It was already mentioned in the introduction that the permanence property
holds for singular inner functions in $W^{1}$. Our first observation in this
section provides a simple way for inducing the permanence property in
$\mathcal{B}$ of singular inner functions. Let $\mu$ be a positive finite
singular Borel measure on $\partial\mathbb{D}$ and assume

$\sup\left\\{\mu(E):S_{\mu\lvert_{E}}\in W^{1}\right\\}>0.$ (7)

We may pick a sequence of Borel sets $\\{E_{N}\\}_{N}$ with the property that
each $S_{\mu\lvert_{E_{N}}}\in W^{1}$ and such that $\mu(E_{N})$ converges to
the supremum above. Observe that whenever $\phi,\psi$ are bounded functions in
$W^{1}$, so is their product $\phi\psi$, and thus we may without loss of
generality, assume that the $E_{N}\subseteq E_{N+1}$ for any $N$. Now let
$E:=\cup_{N}E_{N}$ and set $\mu_{0}:=\mu-\mu\lvert_{E}$ and observe that by
maximality, there exists no non-trivial $\nu_{0}\leq\mu_{0}$ with
$S_{\nu_{0}}\in W^{1}$. Measures satisfying the above condition are declared
to be $W^{1}$ invisible. As previously observed in the introduction, each
$S_{\mu_{E_{N}}}$ satisfies the permanence property in $\mathcal{B}$ and thus
according to the monotonicity principle in Ppropmonprinc, so does
$S_{\mu\lvert_{E}}$. Our discussion can thus be summarized in the following
result.

######  Proposition 4.1.

PPW11 Let $\mu$ be a positive finite singular Borel measure on
$\partial\mathbb{D}$ and $\mu\lvert_{E}$ denote the corresponding piece in the
decomposition of $\mu$ appearing in the previous paragraph. Then
$S_{\mu\lvert_{E}}$ satisfies the permanence property in $\mathcal{B}$, that
is, $\left[S_{\mu\lvert_{E}}\right]_{\mathcal{B}}\cap H^{\infty}\subseteq
S_{\mu\lvert_{E}}H^{\infty}.$ As a consequence, if $S_{\mu}$ is cyclic in
$\mathcal{B}$, then $\mu$ is $W^{1}$ invisible.

###  4.2 Singular inner functions in $W^{1}$

This section is devoted to study singular inner functions in $W^{1}$. For any
arc $I\subseteq\partial\mathbb{D}$, we denote its associated Carleson square
by $Q_{I}=\left\\{z\in\mathbb{D}:z/|z|\in I,\,\,1-|z|\leq|I|\right\\},$ and we
let $T_{I}=\\{z\in Q_{I}:1-|z|\geq|I|/2\\}$ be the top-half of $Q_{I}$.

######  Lemma 4.2.

QnotELemma There exists a universal constant $C>0$, such that for any singular
measure $\mu$ on $\partial\mathbb{D}$ and any arc $I\subset\partial\mathbb{D}$
with $\mu(I)=0$, the associated singular inner function $S_{\mu}$ satisfies

$\int_{Q_{I}}\left\lvert S_{\mu}^{\prime}(z)\right\rvert dA(z)\leq C|I|.$

###### Proof.

Let $\\{I_{k}\\}_{k\in\mathbb{Z}}$ denote the Whitney decomposition of $I$
satisfying

$\text{dist}(I_{k},\partial I)=|I_{k}|=\frac{1}{3\cdot 2^{|k|}}|I|,\qquad
k\in\mathbb{Z}.$

We first observe that there exists an absolute constant $C>0$ such that

$\int_{Q_{I}\setminus\bigcup_{k}Q_{I_{k}}}|S^{\prime}_{\mu}(z)|dA(z)\leq\int_{Q_{I}\setminus\bigcup_{k}Q_{I_{k}}}\frac{dA(z)}{1-|z|^{2}}\leq
C|I|.$ (8)

It remains to estimate the integral of $|S^{\prime}_{\mu}|$ on
$\cup_{k}Q_{I_{k}}$. To this end, for each Whitney arc $I_{k}$, we denote by
$\xi_{k}$ its center and let $z_{k}=(1-|I_{k}|)\xi_{k}$. Note that there
exists an absolute constant $c>0$ such that $c^{-1}|\zeta-
z_{k}|\leq|\zeta-z|\leq c|\zeta-z_{k}|$ for any $z\in Q_{I_{k}}$ and any
$\zeta\in\partial\mathbb{D}\setminus I$. Since $\mu(I)=0$ the Poisson
extension $P(\mu)$ of $\mu$ satisfies

$P(\mu)(z)=\frac{1-|z|^{2}}{1-|z_{k}|^{2}}\int_{\partial\mathbb{D}\setminus
I}\frac{1-|z_{k}|^{2}}{|\zeta-z|^{2}}d\mu(\zeta)\asymp(1-|z|)\frac{P(\mu)(z_{k})}{|I_{k}|},\qquad
z\in Q_{I_{k}}.$

Now using this in conjunction with the obvious estimate

$(1-|z|^{2})|S^{\prime}_{\mu}(z)|\leq 2P(\mu)(z)\exp(-P(\mu)(z)),\qquad
z\in\mathbb{D},$

it follows that there exists a universal constant $C>0$ such that

$\int_{Q_{I_{k}}}|S^{\prime}_{\mu}(z)|dA(z)\lesssim\frac{P(\mu)(z_{k})}{|I_{k}|}\int_{Q_{I_{k}}}\exp\left(-C(1-|z|)\frac{P(\mu)(z_{k})}{|I_{k}|}\right)dA(z).$

A straightforward computation of the integral above gives

$\int_{Q_{I_{k}}}|S^{\prime}_{\mu}(z)|dA(z)\lesssim|I_{k}|.$

Summing over $k$ completes the proof. ∎

We shall now locate the critical domain of integration in order for $S_{\mu}$
to belong to $W^{1}$, which will be convenient for our further developments.

######  Corollary 4.3.

eqcondW11 Let $\mu$ be a positive singular Borel measure supported on a closed
set $E\subset\partial\mathbb{D}$. Then $S_{\mu}$ belongs to $W^{1}$ if and
only if

$\sum_{\begin{subarray}{c}I\in\mathcal{D}\\\ I\cap
E\neq\emptyset\end{subarray}}\int_{T_{I}}\left\lvert
S^{\prime}_{\mu}(z)\right\rvert dA(z)<\infty,$

where $\mathcal{D}$ denotes the collection of dyadic arcs on
$\partial\mathbb{D}$.

###### Proof.

It is sufficient to show that the above condition implies that $S_{\mu}\in
W^{1}$. To this end, let $\mathcal{G}$ denote the collection of maximal dyadic
arcs which do not intersect $E$. Observe that we can write

$\int_{\mathbb{D}}\left\lvert S^{\prime}_{\mu}(z)\right\rvert
dA(z)=\sum_{\begin{subarray}{c}I\in\mathcal{D}\\\ I\cap
E\neq\emptyset\end{subarray}}\int_{T_{I}}\left\lvert
S^{\prime}_{\mu}(z)\right\rvert
dA(z)+\sum_{I\in\mathcal{G}}\int_{Q_{I}}\left\lvert
S^{\prime}_{\mu}(z)\right\rvert dA(z)$

Note that QnotELemma gives

$\sum_{I\in\mathcal{G}}\int_{Q_{I}}\left\lvert S^{\prime}_{\mu}(z)\right\rvert
dA(z)\leq C\sum_{I\in\mathcal{G}}|I|<\infty,$

which is enough to prove the desired claim. ∎

As a consequence, we obtain the following sufficient condition for membership
of $S_{\mu}$ in $W^{1}$.

######  Corollary 4.4.

suffW11cond Let $\mu$ be a positive finite singular Borel measure supported on
a Beurling Carleson set of the unit circle. Then $S_{\mu}$ belongs to $W^{1}$.

###### Proof.

It is well known that a closed set $E\subset\partial\mathbb{D}$ of Lebesgue
measure zero is a Beurling Carleson set if and only if

$\sum_{\begin{subarray}{c}I\in\mathcal{D}\\\ I\cap
E\neq\emptyset\end{subarray}}|I|<\infty,$

see [10]. The claim now follows from Schwarz Lemma and eqcondW11.

∎

###  4.3 No estimate from below implies cyclicity in $\mathcal{B}$

This subsection is devoted to THM:NoMOC, which essentially asserts that no
condition on the modulus of continuity on $\mu$ alone is an impediment for
$S_{\mu}$ to be a member in $W^{1}$. According to the discussion in the
introduction, this implies that there cannot be any estimate from below of a
singular inner function which ensures it to be cyclic in $\mathcal{B}$.

###### Proof of THM:NoMOC.

We shall divide the proof in three different steps.

_Step 1: Construction of $\mu$_

Let $\\{n_{l}\\}_{l}$ be an increasing sequence of positive integers to be
specified later according to the majorant $w$. Set $\mu(\partial\mathbb{D})=1$
and consider the subcollection $G_{1}=\\{I^{(1)}_{j}\\}_{j}$ of every other
dyadic arc of generation $n_{1}$. In other words $|I^{(1)}_{j}|=2^{-n_{1}}$
and $\text{dist}(I^{(1)}_{j},I^{(1)}_{j+1})=2^{-n_{1}}$ for any $j$. On these
arcs, we set $\mu(I^{(1)}_{j})=2|I^{(1)}_{j}|$ for each $j$, and note that

$\sum_{j}\mu(I^{(1)}_{j})=2\sum_{j}|I^{(1)}_{j}|=1.$

Hence $\mu$ spreads its mass precisely on the collection $G_{1}$. Next we
consider the subcollection $G_{2}=\\{I^{2}_{j}\\}_{j}$ of dyadic arcs of
generation $n_{2}$ contained in $\cup I^{(1)}_{j}$, in such a way that inside
each $I^{(1)}_{k}$, we pick every other dyadic arc of generation $n_{2}$ to
include in our subcollection. Thus $|I^{(2)}_{j}|=2^{-n_{2}}$ and
$\text{dist}(I^{(2)}_{j},I^{(2)}_{j+1})\geq 2^{-n_{2}}$ for each $j$. With
this at hand, we now set $\mu(I^{(2)}_{j})=2^{2}|I^{(2}_{j}|$ for each $j$ and
observe that for any arc $I^{(1)}_{k}\in G_{1}$, we have

$\sum_{I^{(2)}_{j}\subset
I^{1}_{k}}\mu(I^{(2)}_{j})=2^{2}\sum_{I^{(2)}_{j}\subset
I^{(1)}_{k}}|I^{(2)}_{j}|=2|I^{(1)}_{k}|=\mu(I^{(1)}_{k}).$

Hence inside each $I^{(1)}_{k}$, we again evenly distribute the mass of $\mu$
on every other dyadic arc of generation $n_{2}$, and denote this joint
collection by $G_{2}=\\{I^{(2)}_{j}\\}$. We proceed by induction. Assume the
first $l-1$ collections of dyadic arcs have been constructed. Then we consider
the collection $G_{l}=\\{I^{(l)}_{k}\\}$ of every other dyadic arc contained
in an arc of $G_{l-1}$ and set $\mu(I^{(l)}_{j})=2^{l}|I^{(l)}_{j}|$, for each
$j$. Hence for each $I^{(l-1)}_{k}\in G_{l-1}$ we have

$\sum_{I^{(l)}_{j}\subset
I^{(l-1)}_{k}}\mu(I^{(l)}_{j})=2^{l}\sum_{I^{(l)}_{j}\subset
I^{(l)}_{k}}|I^{(l)}_{j}|=2^{l-1}|I^{(l-1)}_{k}|=\mu(I^{(l-1)}_{k}).$

Note also that for each $l\geq 1$, we have

$\sum_{k}|I^{(l)}_{k}|=2^{-l}.$ (9)

Now extend $\mu$ to a Borel measure on $\partial\mathbb{D}$. Note that by
construction, $\mu$ is supported inside $E:=\cap_{l}\cup_{j}I^{(l)}_{j}$,
which has Lebesgue measure zero and thus $\mu$ is indeed a positive finite
singular measure.

_Step 2: Verifying condition $(i)$_

We shall now specify a choice for the sequence of positive integers
$\\{n_{l}\\}_{l}$ so that condition $(i)$ holds. To this end, notice that
since $w(t)/t\to\infty$ as $t\to 0^{+}$, for any non-negative integer $l$, we
can pick $n_{l}$ in such a way that

$\frac{w(2^{-n_{l}})}{2^{-n_{l}}}\geq 2^{l}6.$

With this choice, we have

$\mu(I^{(l)}_{j})=2^{l}|I^{(l)}_{j}|=2^{l}2^{-n_{l}}\leq
w(2^{-n_{l}})/6=w(|I_{j}^{(l)}|)/6.$

We first check that condition (i) holds for dyadic arcs $I$. We fix an
arbitrary dyadic arc $I$ of length $2^{-n}$ and pick a positive integer $l$
such that $n_{l-1}\leq n<n_{l}$. We have

$\mu(I)=\sum_{j:I^{l}_{j}\subset I}\mu(I^{(l)}_{j})=\sum_{j:I^{l}_{j}\subset
I}2^{l}\left\lvert I^{(l)}_{j}\right\rvert\leq
2^{l-1}|I|\leq\frac{w(2^{-n_{l-1}})}{2^{-n_{l-1}}}\frac{|I|}{3}\leq\frac{w(|I|)}{3},$

where in the last step we have used the property that $w(s)/s\leq 2w(t)/t$
whenever $s<t$, which easily follows from the sub-additivity of majorants. Let
$I\subset\partial\mathbb{D}$ be an arbitrary arc and pick $n>0$ such that
$2^{-n}<b-a\leq 2^{-(n-1)}$. Since $I$ is contained in at most three dyadic
arcs of length $2^{-n}$, the estimate $(i)$ holds.

_Step 3: Verifying condition $(ii)$_

To this end, for each pair of positive integers $(k,l)$ we consider the set

$\Omega^{(l)}_{k}:=\left\\{z\in\mathbb{D}:2^{-n_{l+1}}<1-|z|\leq
2^{-n_{l}},\,z/|z|\in I^{(l)}_{k}\right\\}$

and note that according to $(iii)$ of eqcondW11 it suffices to prove that

$\sum_{k,l}\int_{\Omega^{(l)}_{k}}\left\lvert S_{\mu}^{\prime}(z)\right\rvert
dA(z)<\infty.$ (10)

For $z\in\mathbb{D}\setminus\\{0\\}$ let $I_{z}$ denote the arc centered at
$z/|z|$ of length $1-|z|$. Observe that there exists a positive constant $c>0$
such that for $z\in\Omega^{(l)}_{k}$, we have that $P(\mu)(z)\geq
c\mu(I_{z})/|I_{z}|\geq c2^{l}$. Using the formula
$S_{\mu}^{\prime}(z)=S_{\mu}(z)H(\mu)^{\prime}(z)$, $z\in\mathbb{D}$, where
$H(\mu)$ denotes the Herglotz transform of $\mu$, we get

$\sum_{k,l}\int_{\Omega^{(l)}_{k}}\left\lvert S_{\mu}^{\prime}(z)\right\rvert
dA(z)\leq\sum_{k,l}e^{-c2^{l}}\int_{\Omega^{(l)}_{k}}\left\lvert
H(\mu)^{\prime}(z)\right\rvert dA(z).$ (11)

The rest of the proof shall be devoted to establishing the following estimate

$\int_{\Omega^{(l)}_{k}}\left\lvert H(\mu)^{\prime}(z)\right\rvert
dA(z)\lesssim 2^{l}\,|I^{(l)}_{k}|=\mu(I^{(l)}_{k}),\quad k,l\geq 1.$ (12)

Indeed, once (12) is established we simply use the fact that
$\sum_{k}\mu(I_{k}^{(l)})=1$ and apply the estimate (11) to deduce that

$\sum_{k,l}\int_{\Omega^{(l)}_{k}}\left\lvert S_{\mu}^{\prime}(z)\right\rvert
dA(z)\lesssim\sum_{l}e^{-c2^{l}}\sum_{k}\mu(I_{k}^{(l)})<\infty,$

which gives (10) and finishes the proof. So it only remains to show that
estimate (12) holds. For $z\in\Omega^{(l)}_{k}$ we write

$H(\mu)^{\prime}(z)=\sum_{j}\int_{I^{(l+1)}_{j}}\frac{2\zeta
d\mu(\zeta)}{(\zeta-z)^{2}}=\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}\int_{I^{(l+1)}_{j}}\frac{2\zeta
d\mu(\zeta)}{(\zeta-z)^{2}}+\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\cap
I^{(l)}_{k}\end{subarray}=\emptyset}\int_{I^{(l+1)}_{j}}\frac{2\zeta
d\mu(\zeta)}{(\zeta-z)^{2}}\\\ =(I)+(II).$

We shall first treat the term $(II)$. Note that if $z\in\Omega^{(l)}_{k}$ and
$\zeta\in I^{(l+1)}_{j}$ with $I^{(l+1)}_{j}\cap I^{(l)}_{k}=\emptyset$, then
$\left\lvert\zeta-z\right\rvert\gtrsim|I^{(l)}_{k}|$. Then

$\left\lvert(II)\right\rvert\lesssim\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\cap
I^{(l)}_{k}\end{subarray}=\emptyset}\frac{\mu(I^{(l+1)}_{j})}{\text{dist}(I^{(l+1)}_{j},I^{(l)}_{k})^{2}}=2^{l+1}\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\cap
I^{(l)}_{k}\end{subarray}=\emptyset}\frac{|I^{(l+1)}_{j}|}{\text{dist}(I^{(l+1)}_{j},I^{(l)}_{k})^{2}}\lesssim\frac{2^{l}}{|I^{(l)}_{k}|}.$

This implies

$\int_{\Omega^{(l)}_{k}}\left\lvert(II)\right\rvert dA\lesssim
2^{l}|I^{(l)}_{k}|,$

and hence we may now devote our attention to estimating $(I)$. To this end, we
shall decompose it into two pieces by rewriting

$(I)=\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}\int_{I^{(l+1)}_{j}}\frac{2\zeta}{(\zeta-z)^{2}}2^{l+1}dm(\zeta)+\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}\int_{I^{(l+1)}_{j}}\frac{2\zeta}{(\zeta-z)^{2}}\left(d\mu(\zeta)-2^{l+1}dm(\zeta)\right)\\\
=A+B.$

Denote by $\xi^{(l+1)}_{j}$ the center of $I^{(l+1)}_{j}$ and observe that
since $\mu(I^{(l+1)}_{j})=2^{l+1}|I^{(l+1)}_{j}|$, we can add a cancellative
term to $B$ and write

$B=\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}\int_{I^{(l+1)}_{j}}\left(\frac{2\zeta}{(\zeta-z)^{2}}-\frac{2\xi^{(l+1)}_{j}}{(\xi^{(l+1)}_{j}-z)^{2}}\right)\left(d\mu(\zeta)-2^{l+1}dm(\zeta)\right).$

Now using the mean value Theorem together and the fact that
$|\zeta-z|\gtrsim|\xi^{(l+1)}_{j}-z|$ whenever $\zeta\in I^{(l+1)}_{j}$ and
$z\in\Omega_{k}^{(l)}$, we get

$|B|\lesssim\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}\frac{\mu(I^{(l+1)}_{j})+2^{l+1}|I^{(l+1)}_{j}|}{|\xi^{(l+1)}_{j}-z|^{3}}|I^{(l+1)}_{j}|=\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}\frac{2^{l+2}|I^{(l+1)}_{j}|^{2}}{|\xi^{(l+1)}_{j}-z|^{3}}.$
(13)

A straightforward calculation shows that

$\int_{\Omega_{k}^{(l)}}\frac{dA(z)}{|\xi^{(l+1)}_{j}-z|^{3}}\lesssim\frac{1}{|I^{(l+1)}_{j}|}.$
(14)

Combining this with (13), we conclude

$\int_{\Omega_{k}^{(l)}}|B|dA\lesssim
2^{l}\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}|I^{(l+1)}_{j}|\leq 2^{l}|I^{(l)}_{k}|.$

Hence it only remains to estimate the quantity $A$. Denote by $I+|I|$ the arc
$I$ rotated by $|I|$-units and observe that by construction of the dyadic arcs
$I^{(l+1)}_{j}$, we have

$\bigcup_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}I^{(l+1)}_{j}\cup\left(I^{(l+1)}_{j}+|I^{(l+1)}_{j}|\right)=I^{(l)}_{k}.$

Applying the mean value Theorem as before, we get

$\left\lvert\int_{I^{(l+1)}_{j}}\frac{\zeta
dm(\zeta)}{(\zeta-z)^{2}}-\int_{I^{(l+1)}_{j}+|I^{(l+1)}_{j}|}\frac{\zeta
dm(\zeta)}{(\zeta-z)^{2}}\right\rvert\lesssim\frac{|I^{(l+1)}_{j}|^{2}}{|z-\xi^{(l+1)}_{j}|^{3}},\quad
z\in\Omega_{k}^{(l)},\quad j,l,k\geq 1.$

Hence

$\int_{\Omega^{(l)}_{k}}\left\lvert A-2^{l}\int_{I^{(l)}_{k}}\frac{2\zeta
dm(\zeta)}{(\zeta-z)^{2}}\right\rvert dA(z)\lesssim\\\
\int_{\Omega^{(l)}_{k}}2^{l}\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}\left\lvert\int_{I^{(l+1)}_{j}}\frac{\zeta
dm(\zeta)}{(\zeta-z)^{2}}-\int_{I^{(l+1)}_{j}+|I^{(l+1)}_{j}|}\frac{\zeta
dm(\zeta)}{(\zeta-z)^{2}}\right\rvert dA(z)\lesssim\\\
2^{l}\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}|I^{(l+1)}_{j}|^{2}\int_{\Omega^{(l)}_{k}}\frac{dA(z)}{\left\lvert\xi^{l+1}_{j}-z\right\rvert^{3}}\lesssim
2^{l}\sum_{\begin{subarray}{c}j:I^{(l+1)}_{j}\subseteq
I^{(l)}_{k}\end{subarray}}|I^{(l+1)}_{j}|\leq 2^{l}|I^{l}_{k}|.$

In the penultimate step, we have used (14). In order to finish the proof, it
just remains to verify that

$\int_{\Omega^{(l)}_{k}}\left\lvert\int_{I^{(l)}_{k}}\frac{\zeta
dm(\zeta)}{(\zeta-z)^{2}}\right\rvert dA(z)\lesssim|I^{(l)}_{k}|.$ (15)

To this end, let $I^{(l)}_{k}=(a^{(l)}_{k},b^{(l)}_{k})$ and observe that
fixed $z\in\mathbb{D}$, the primitive of $\zeta/(\zeta-z)^{2}$ is explicitly
given by $\log(\zeta-z)-z/(\zeta-z)$, which allows us to compute

$\int_{I^{(l)}_{k}}\frac{\zeta
dm(\zeta)}{(\zeta-z)^{2}}=\log\left(\frac{b^{(l)}_{k}-z}{a_{k}^{(l)}-z}\right)+\frac{z|I^{(l)}_{k}|}{(b^{(l)}_{k}-z)(a_{k}^{(l)}-z)}.$

Then

$\left\lvert\int_{I^{(l)}_{k}}\frac{\zeta
dm(\zeta)}{(\zeta-z)^{2}}\right\rvert\lesssim\frac{|I^{(l)}_{k}|}{|b^{(l)}_{k}-z||a_{k}^{(l)}-z|},\qquad
z\in\Omega^{(l)}_{k},$

and thus

$\int_{\Omega^{(l)}_{k}}\left\lvert\int_{I^{(l)}_{k}}\frac{\zeta
dm(\zeta)}{(\zeta-z)^{2}}\right\rvert
dA(z)\lesssim|I^{(l)}_{k}|\int_{\Omega_{k}^{(l)}}\frac{dA(z)}{|b^{(l)}_{k}-z||a_{k}^{(l)}-z|}.$

Since either $|z-b_{k}^{(l)}|\geq|I^{(l)}_{k}|/2$ or
$|z-a_{k}^{(l)}|\geq|I^{(l)}_{k}|/2$ for any $z\in\Omega^{(l)}_{k}$, we deduce

$\int_{\Omega_{k}^{(l)}}\frac{dA(z)}{|b^{(l)}_{k}-z||a_{k}^{(l)}-z|}\lesssim
1.$

This gives (15) and finishes the proof. ∎

###  4.4 The permanence property discriminates no compact set

Here we devote our attention to the proof of THM:Nosupp, which roughly asserts
that any compact set on the unit circle of Lebesgue measure zero can support a
singular measure $\mu$ for which the associated singular inner function is a
member of $W^{1}$. This implies that no condition on the carrier set of a
singular measure $\mu$ alone can describe the permanence property of $S_{\mu}$
in the Bloch space.

###### Proof of THM:Nosupp.

The proof is divided in three steps.

_Step 1: Constructing a covering of $E$_

We first construct by induction an appropriate sequence of coverings of $E$ by
dyadic arcs. Set $\mathcal{G}_{0}:=\\{\partial\mathbb{D}\\}$ and assume that a
covering $\mathcal{G}_{k}=\\{I^{(k)}_{j}:j\geq 1\\}$ of $E$ by dyadic arcs has
already been constructed and $E\cap I^{(k)}_{j}\neq\emptyset$ for any $j\geq
1$. For any $I\in\mathcal{G}_{k}$ we denote by $\widetilde{\mathcal{G}}(I)$
the collection of maximal dyadic subarcs $\widetilde{J}$ of $I$ such that at
least one of the two dyadic children of $\widetilde{J}$ does not intersect
$E$. Note that $\widetilde{G}(I)=\\{I\\}$ if one of the two dyadic children of
$I\in\mathcal{G}_{k}$ does not intersect $E$. Moreover, by maximality of the
collection $\widetilde{G}(I)$, for each $\widetilde{J}\in\widetilde{G}(I)$
there exists exactly one dyadic child $J$ of $\widetilde{J}$ with $J\cap
E\neq\emptyset$. Now consider

$\mathcal{G}(I):=\left\\{J:J\,\,\text{dyadic child of
some}\,\,\widetilde{J}\in\widetilde{\mathcal{G}}(I),\text{ with }J\cap
E\neq\emptyset\right\\}.$

The set $\mathcal{G}_{k+1}:=\bigcup_{I\in\mathcal{G}_{k}}\mathcal{G}(I)$ is
again a covering of $E$ that will be denoted by
$\mathcal{G}_{k+1}=\\{I^{(k+1)}_{j}:j\geq 1\\}.$ Since $E$ is a closed set of
Lebesgue measure zero, for any arc $I\in\mathcal{G}_{k}$, the union of the
arcs of the collection $\widetilde{G}(I)$ covers almost every point of $I$.
Hence the collection $\\{\mathcal{G}_{k}\\}_{k}$ satisfies the following
packing condition: for any $I^{(k)}_{l}\in\mathcal{G}_{k}$ we have

$\sum_{\begin{subarray}{c}J\in\mathcal{G}_{k+1}\\\ J\subset
I^{(k)}_{l}\end{subarray}}|J|=\sum_{\begin{subarray}{c}j\geq 1\\\
I^{(k+1)}_{j}\subseteq
I^{(k)}_{l}\end{subarray}}|I^{(k+1)}_{j}|=\frac{|I^{(k)}_{l}|}{2},\qquad
k,l\geq 1.$ (16)

Observe that by construction $E=\cap_{k}\cup_{J\in\mathcal{G}_{k}}J$. Indeed,
the inclusion $\subseteq$ follows from the fact that each $\mathcal{G}_{k}$ is
a covering of $E$, while if $\zeta_{0}\notin E$, then there exists a dyadic
arc $I_{0}$ containing $\zeta_{0}$ with $I_{0}\cap E=\emptyset$. Hence for
every sufficiently large $k$, we must have $\zeta_{0}\in
I_{0}\subset\cap_{J\in\mathcal{G}_{k}}\partial\mathbb{D}\setminus J$, which
establishes the reverse inclusion $\supseteq$.

_Step 2: Constructing the measure $\mu$:_

We now construct $\mu$ by declaring its mass on dyadic arcs. Initially, we set
$\mu(\partial\mathbb{D})=1$. Now let $I\subset\partial\mathbb{D}$ be a dyadic
arc and assume, by means of induction, that $\mu(I)$ has already been defined
in such a way that $\mu(I)=0$ whenever $I\cap E=\emptyset$. Denote by
$I_{+},I_{-}$ the dyadic children of $I$. If both $I_{+},I_{-}$ happens to
meet $E$, then we distribute the mass evenly by declaring

$\frac{\mu(I_{+})}{|I_{+}|}=\frac{\mu(I_{-})}{|I_{-}|}=\frac{\mu(I)}{|I|}.$

If not, say $I_{+}\cap E=\emptyset$, then we distribute all the mass to
$I_{-}$ by setting

$\frac{\mu(I_{-})}{|I_{-}|}=2\frac{\mu(I)}{|I|},\qquad\mu(I_{+})=0.$

Indeed, this is consistent since

$\frac{\mu(I)}{|I|}=\frac{1}{2}\frac{\mu(I_{+})}{|I_{+}|}+\frac{1}{2}\frac{\mu(I_{-})}{|I_{-}|}.$

Observe that by construction, the support of $\mu$ is equal to
$E=\cap_{k}\cup_{J\in\mathcal{G}_{k}}J$ which has Lebesgue measure zero. Hence
$\mu$ is a non-trivial singular measure. Moreover, it is clear that for any
dyadic arc $I$ with $I\cap E\neq\emptyset$, we have $\mu(I)\geq|I|$. Observe
also that by construction, for any $I^{(k)}_{j}\in\mathcal{G}_{k}$, we have

$\frac{\mu(I^{(k)}_{j})}{|I^{(k)}_{j}|}\geq 2^{k},\quad j,k\geq 1.$ (17)

_Step 3: Verifying condition $(ii)$_

It now remains to prove that the associated singular inner function $S_{\mu}$
belongs to $W^{1}$. To this end, for each $k\geq 1$ and
$I^{(k)}_{j}\in\mathcal{G}_{k}$ consider the region

$\Omega^{(k)}_{j}:=Q_{I^{(k)}_{j}}\setminus\bigcup_{\widetilde{J}\in\widetilde{\mathcal{G}}(I^{(k)}_{j})}Q_{\widetilde{J}}.$

According to (iii) of eqcondW11, it is sufficient to establish that

$\sum_{k,j}\int_{\Omega^{(k)}_{j}}\left\lvert S_{\mu}^{\prime}\right\rvert
dA<\infty.$

Next we shall make yet another reduction. Let $I=I^{(k)}_{j}$ and denote by
$\mu_{I}$ the restriction of $\mu$ to $I$. According to QnotELemma, there
exists an absolute constant $C>0$ such that

$\int_{Q_{I}}|S^{\prime}_{\mu_{\partial\mathbb{D}\setminus I}}(z)|dA(z)\leq
C|I|.$

Hence using (16), we get

$\sum_{k,j}\int_{\Omega^{(k)}_{j}}|S^{\prime}_{\mu_{\partial\mathbb{D}\setminus
I^{(k)}_{j}}}(z)|dA(z)\leq C\sum_{k,j}|I^{(k)}_{j}|=\sum_{k\geq 1}2^{-k}=C.$

With this observation at hand, it then suffices to show that

$\sum_{k,j}\int_{\Omega^{(k)}_{j}}|S_{\mu_{I^{(k)}_{j}}}^{\prime}(z)|dA(z)<\infty.$
(18)

Using (17) in conjunction with standard estimates of Poisson kernels, there
exists a universal constant $c>0$ such that

$|S_{\mu_{I^{(k)}_{j}}}^{\prime}(z)|\leq
e^{-c2^{k}}|H(\mu_{I^{(k)}_{j}})^{\prime}(z)|,\quad
z\in\Omega^{(k)}_{j},k,j\geq 1.$ (19)

Thus (18) follows from the estimate

$\int_{\Omega^{(k)}_{j}}|H(\mu_{I^{(k)}_{j}})^{\prime}(z)|dA(z)\lesssim\mu(I^{(k)}_{j})\quad
j,k\geq 1.$

whose proof is quite similar to the proof of estimate (12). We omit the
details.

∎

##  5 Cyclicity and invertibility

###  5.1 A sufficient condition for cyclicity

This subsection is devoted to proving THM:SuffcycB. A finite complex Borel
measure $\nu$ on $\partial\mathbb{D}$ (not necessarily singular) is said to be
a _Zygmund measure_ if

$\left\lVert\nu\right\rVert_{*}:=\,\sup\,\left\lvert\frac{\nu(I)}{|I|}-\frac{\nu(I^{\prime})}{|I^{\prime}|}\right\rvert<\infty,$

where the supremum is taken over all pairs of contiguous arcs $I,I^{\prime}$
of the same length. For the sake of future references, we shall state the
following result due to Duren, Shapiro and Shields in [16] below, see also
[15].

######  Lemma 5.1.

ZygBloch A finite complex Borel measure $\nu$ on $\partial\mathbb{D}$ is a
Zygmund measure if and only if its Herglotz transform defined as

$H(\nu)(z)=\int_{\partial\mathbb{D}}\frac{\zeta+z}{\zeta-z}d\nu(\zeta),\qquad
z\in\mathbb{D},$

belongs to $\mathcal{B}$. In this case, there exists a universal constant
$C>0$ such that

$\left\lvert H(\nu)(z)-H(\nu)(w)\right\rvert\leq
C\left\lVert\nu\right\rVert_{*}\beta(z,w),\quad z,w\in\mathbb{D},$

where $\beta(z,w)$ denotes the hyperbolic distance on $\mathbb{D}$ between the
points $z$ and $w$. Moreover, the following asymptotic relation holds:

$P(\nu)(z)=\frac{\nu(I_{z})}{|I_{z}|}+\mathcal{O}\left(1\right)\left\lVert\nu\right\rVert_{*},\qquad
z\in\mathbb{D}.$ (20)

where $I_{z}\subset\partial\mathbb{D}$ denotes the arc centered at $z/|z|$ of
length $1-|z|$.

We will also use an auxiliary result that has appeared in the setting of
$\mathbb{R}^{d}$ in [15]. We shall need it in the context of the unit circle
where its proof is an exercise left to the reader.

######  Lemma 5.2.

Herg’est Let $\nu$ be a finite Borel measure on $\partial\mathbb{D}$. Then
there exists an absolute constant $C>0$, such that

$(1-|z|)\left\lvert H(\nu)^{\prime}(z)\right\rvert\leq
C\int_{0}^{2\pi}\frac{(1-|z|)}{|e^{it}-|z||^{3}}\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert
dt,\quad z\in\mathbb{D},$

where $I(z,t)$ denotes the smallest arc joining $z/|z|$ to $ze^{it}/|z|$ if
$z\neq 0$ and $I(0,t)$ is the arc joining 1 and $e^{it}$.

Given an arc $I\subset\partial\mathbb{D}$ and $C>0$ let $CI$ denotethe arc
with the same center as $I$ and length $C|I|$. We will also need the following
technical result.

######  Lemma 5.3.

Let $\nu$ be a positive finite Borel measure on $\partial\mathbb{D}$. Assume
there exists a constant $C=C(\nu)>0$ such that

$\left\lvert\frac{\nu(I)}{|I|}-\frac{\nu(I^{\prime})}{|I^{\prime}|}\right\rvert\leq
Ce^{-\nu(I)/|I|},$ (21)

for any pair of contiguous arcs $I,I^{\prime}\subset\partial\mathbb{D}$ of the
same length. Then for any $\varepsilon>0$, there exists $\delta>0$ such that
for any integer $n$ with $|n|\leq\delta e^{\nu(I)/|I|}$ we have

$\left\lvert\frac{\nu(2^{n}I)}{|2^{n}I|}-\frac{\nu(I)}{|I|}\right\rvert\leq\min\\{\varepsilon
n,1\\}.$ (22)

###### Proof.

Note that the assumption gives that $\nu$ is a Zygmund measure. Moreover (21)
gives that there exists a constant $C_{1}>0$ such that

$\left\lvert\frac{\nu(2I)}{2I}-\frac{\nu(I)}{|I|}\right\rvert\leq
C_{1}e^{-\max\\{\nu(2I)/|2I|,\nu(I)/|I|\\}},$

for any arc $I\subset\partial\mathbb{D}$. Using the elementary estimate
$\left\lvert e^{x}-e^{y}\right\rvert\leq e^{\max\\{x,y\\}}|x-y|$,
$x,y\in\mathbb{R}$, we deduce

$\left\lvert e^{\nu(2I)/|2I|}-e^{\nu(I)/|I|}\right\rvert\leq C_{1},$

for any arc $I\subset\partial\mathbb{D}$. Hence for any integer $k$ we have

$\left\lvert e^{\nu(2^{k}I)/|2^{k}I|}-e^{\nu(I)/|I|}\right\rvert\leq
C_{1}|k|.$ (23)

Now if $|n|\leq(2C_{1})^{-1}e^{\nu(I)/|I|}$, we obtain

$\left\lvert
e^{\nu(2^{n}I)/|2^{n}I|}-e^{\nu(I)/|I|}\right\rvert\leq\frac{1}{2}e^{\nu(I)/|I|}.$
(24)

Hence

$\left\lvert\frac{\nu(2^{n}I)}{|2^{n}I|}-\frac{\nu(I)}{|I|}\right\rvert\leq
1.$

Note that to check (22) we can assume that $\nu(I)/|I|$ is large. Fix
$\varepsilon>0$. Let $M=M(\varepsilon)>0$ be a large number to be fixed later.
Use (23) to pick $\delta>0$ such that $\nu(2^{k}I)/|2^{k}I|>M$ if $k$ is an
integer with $|k|<\delta e^{\nu(I)/|I|}$. Then assumption (21) gives that

$\left\lvert\frac{\nu(2^{k}I)}{|2^{k}I|}-\frac{\nu(2^{k-1}I)}{|2^{k-1}I|}\right\rvert\lesssim
e^{-M},\quad\text{ if }|k|\leq\delta e^{\nu(I)/|I|}.$

We deduce that

$\left\lvert\frac{\nu(2^{n}I)}{|2^{n}I|}-\frac{\nu(I)}{|I|}\right\rvert\leq\sum_{k=1}^{n}\left\lvert\frac{\nu(2^{k}I)}{|2^{k}I|}-\frac{\nu(2^{k-1}I)}{|2^{k-1}I|}\right\rvert\lesssim
e^{-M}n,\quad\text{ if }|n|\leq\delta e^{\nu(I)/|I|}.$

and taking $M\lesssim\log(1/\varepsilon)$, estimate (22) follows. ∎

We are now ready to prove THM:SuffcycB.

###### Proof of THM:SuffcycB.

It is sufficient to find a sequence of bounded analytic functions $h_{n}$ such
that:

  1. (i)

$\sup_{n}\left\lVert h_{n}f\right\rVert_{\mathcal{B}}<\infty$,

  2. (ii)

$h_{n}(z)f(z)\to 1$ for any $z\in\mathbb{D}$.

Set $r_{n}:=1-2^{-n}$ and consider $h_{n}(z):=1/f(r_{n}z)$, $z\in\mathbb{D}$.
Note that condition $(ii)$ is trivially satisfied, hence it remains to verify
$(i)$. Observe that we can write

$(1-|z|)\left(\frac{f(z)}{f(r_{n}z)}\right)^{\prime}=(1-|z|)\left(\frac{f^{\prime}(z)}{f(r_{n}z)}\right)-(1-|z|)\left(\frac{r_{n}f(z)f^{\prime}(r_{n}z)}{f^{2}(r_{n}z)}\right)=:A+B.$

We proceed by estimating both terms separately.

_Step 1: Estimation of B_

Set $f=\exp(-H(\nu))$ where $H(\nu)$ is the Herglotz transform of the
Herglotz-Nevanlinna measure $\nu$ of $f$. Observing that $\nu$ is necessarily
a Zygmund measure, we may apply ZygBloch to deduce

$(1-|r_{n}z|)\left\lvert\frac{r_{n}f^{\prime}(r_{n}z)}{f(r_{n}z)}\right\rvert\leq\left\lVert
H(\nu)\right\rVert_{\mathcal{B}}\lesssim\left\lVert\mu\right\rVert_{*},\qquad
z\in\mathbb{D},$

and thus

$\left\lvert B\right\rvert\lesssim\frac{1-|z|}{1-|r_{n}z|}\frac{\left\lvert
f(z)\right\rvert}{\left\lvert
f(r_{n}z)\right\rvert}=\frac{1-|z|}{1-|r_{n}z|}\exp\left(P(\nu)(r_{n}z)-P(\nu)(z)\right),\quad
z\in\mathbb{D},$ (25)

where $P(\nu)$ denotes the Poisson extension of $\nu$. Now for $|z|\leq
r_{n}$, we clearly have $\beta(z,r_{n}z)\lesssim 1$. An application of
ZygBloch then gives $\left\lvert P(\nu)(r_{n}z)-P(\nu)(z)\right\rvert\lesssim
1$ if $|z|\leq r_{n}$. and hence

$\sup_{|z|\leq r_{n}}|B|\lesssim 1.$

We may thus fix $z\in\mathbb{D}$ with $|z|\geq r_{n}$. Note that in order to
estimate the right hand side of (25), we may without loss of generality assume
that $P(\nu)(r_{n}z)$ is large, which according to ZygBloch is equivalent to
$\nu(I_{r_{n}z})/|I_{r_{n}z}|$ being large. Doing so, we pick a large number
$R>0$ for which $\nu(I_{r_{n}z})/|I_{r_{n}z}|>R$, to be specified momentarily.
Observe that condition $\eqref{expzyginfcond}$, then implies that

$\left\lvert\frac{\nu(I)}{|I|}-\frac{\nu(I/2)}{|I/2|}\right\rvert\leq Ce^{-R}$
(26)

for any arc $I\subseteq I_{r_{n}z}$. Let $N:=N(n,z)$ denote the largest
positive integer satisfying

$N\leq\log\left(\frac{|I_{r_{n}z}|}{|I_{z}|}\right)=\log\left(\frac{1-|r_{n}z|}{1-|z|}\right).$

By means of iterating the estimate (26) $N$ times, we obtain

$\left\lvert\frac{\nu(I_{z})}{|I_{z}|}-\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}\right\rvert\leq
CNe^{-R}\leq Ce^{-R}\log\left(\frac{1-|r_{n}z|}{1-|z|}\right).$

Yet another application of ZygBloch allows us to express

$P(\nu)(r_{n}z)-P(\nu)(z)=\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}-\frac{\nu(I_{z})}{|I_{z}|}+\mathcal{O}(\left\lVert\nu\right\rVert_{*})$

and thus we get

$\left\lvert P(\nu)(r_{n}z)-P(\nu)(z)\right\rvert\leq
Ce^{-R}\log\left(\frac{1-|r_{n}z|}{1-|z|}\right)+\mathcal{O}(\left\lVert\nu\right\rVert_{*}).$

Now it is just a matter of choosing $R>0$ sufficiently large so that
$Ce^{-R}<1$, which ultimately yields

$|B|\lesssim\left(\frac{1-|z|}{1-|r_{n}z|}\right)^{1-Ce^{-R}}\lesssim 1.$

_Step 2: Estimation of A_

By definition, we readily have

$\left\lvert A\right\rvert\lesssim(1-|z|)\left\lvert
H(\nu)^{\prime}(z)\right\rvert\exp\left(P(\nu)(r_{n}z)-P(\nu)(z)\right),\qquad
z\in\mathbb{D}.$ (27)

As observed previously $\beta(z,r_{n}z)\lesssim 1$ whenever $|z|\leq r_{n}$.
Hence using that $\nu$ is a Zygmund measure, we have
$(1-|z|)|H(\mu)^{\prime}(z)|\lesssim 1$ and
$|P(\nu)(r_{n}z)-P(\nu)(z)|\lesssim 1$ if $|z|\leq r_{n}$. This gives

$\sup_{|z|\leq r_{n}}\left\lvert A\right\rvert\lesssim 1.$

In what follows, we shall thus fix $z\in\mathbb{D}$ with $|z|>r_{n}$ and
moreover, we may also assume that $P(\nu)(r_{n}z)$ is sufficiently large.
Denote by $I(z,t)$ the smallest arc on $\partial\mathbb{D}$ joining $z/|z|$
with $ze^{it}/|z|$. An application of Herg’est gives

$(1-|z|)\left\lvert
H(\nu)^{\prime}(z)\right\rvert\lesssim\int_{0}^{2\pi}\frac{(1-|z|)}{|e^{it}-|z||^{3}}\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert
dt.$

Decomposing the integral according to the intervals
$I(k)=\\{t\in[0,2\pi]:|t|\leq 2^{k}(1-|z|)\\}$, we get

$(1-|z|)\left\lvert
H(\nu)^{\prime}(z)\right\rvert\lesssim(1-|z|)^{-2}\int_{I(1)}\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert
dt\\\ +\sum_{k>0}2^{-3k}(1-|z|)^{-2}\int_{I(k+1)\setminus
I(k)}\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert dt.$

Let $T=T(n,z)$ denote the integer part of $\log_{2}(2^{-n}/(1-|z|))$. Observe
that if $t\in I(k)$ with $k\leq T$, then $I(z,t)$ is contained in a fixed
multiple of $I_{r_{n}z}$. This in conjunction with the assumption (1) yields

$\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert\lesssim|t|\exp\left(-\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}\right).$

Hence we get

$\sum_{0\leq
k<T}2^{-3k}(1-|z|)^{-2}\int_{I(k)}\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert
dt\lesssim\exp\left(-\frac{\nu(I_{r_{n}z}}{|I_{r_{n}z}|}\right).$

Fix $t\in I(k+1)\setminus I(k)$ with $k>T$. Note that the arc $I_{r_{n}z}$ is
contained in a fixed multiple of $I(z,t)\cup I(z,-t)$. According to Lemma 5.3,
there exists $\delta>0$ such that

$\left\lvert\frac{\nu(I(z,t))}{|I(z,t)|}-\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}\right\rvert\lesssim
1,\quad\text{ if }\log_{2}(\frac{2^{k}(1-|z|)}{2^{-n}})\leq\delta
e^{\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}}.$ (28)

The assumption (1) implies in this case

$\left\lvert\frac{\nu(I(z,t))}{|I(z,t)|}-\frac{\nu(I(z,-t))}{|I(z,-t)|}\right\rvert\lesssim\exp\left(-\frac{\nu(I(z,t)}{|I(z,t)|}\right).$

Using estimate (28) in the right hand side term, we obtain

$\left\lvert\frac{\nu(I(z,t))}{|I(z,t)|}-\frac{\nu(I(z,-t))}{|I(z,-t)|}\right\rvert\lesssim\exp\left(-\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}\right),\quad\text{
if }\log_{2}(\frac{2^{k}(1-|z|)}{2^{-n}})\leq\delta
e^{\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}}.$

Let $T^{*}$ be the largest integer such that
$T^{*}+\log_{2}(\frac{(1-|z|)}{2^{-n}})\leq\delta
e^{\nu(I_{r_{n}z})/|I_{r_{n}z}|}$. With this choice, we get

$\sum_{k=T}^{T^{*}}2^{-3k}(1-|z|)^{-2}\int_{I(k+1)}\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert
dt\lesssim\exp\left(-\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}\right).$

On the other hand, using that $\nu$ is a Zygmund measure we arrive at

$\sum_{k>T^{*}}2^{-3k}(1-|z|)^{-2}\int_{I(k+1)}\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert
dt\lesssim\exp\left(-\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}\right).$

Finally, combining all terms in the sum, we obtain

$(1-|z|)|H^{\prime}(\nu)(z)|\lesssim\exp\left(-\frac{\nu(I_{r_{n}z})}{|I_{r_{n}z}|}\right),\qquad|z|>r_{n}.$

With these estimates at hand, we may now return back to (27), deducing that
$|A|\lesssim\exp\left(-P(\nu)(z)\right)\leq 1$. This completes the proof. ∎

###  5.2 A characterization of invertibility in Bloch

###### Proof of THM:Invertibility B.

In order to prove the sufficiency, note that by means of writing
$(1/f)^{\prime}=H(\nu)^{\prime}/f$, it suffices to show that
$(1-|z|^{2})|H(\nu)^{\prime}(z)|\lesssim|f(z)|$, $z\in\mathbb{D}.$ Again,
condition (2) readily implies that $\nu$ is a Zygmund measure. In view of the
estimate (20) in Lemma LABEL:ZygBloch, it suffices to prove that

$(1-|z|^{2})|H(\nu)^{\prime}(z)|\lesssim e^{-\nu(I_{z})/|I_{z}|},\quad
z\in\mathbb{D}.$ (29)

For $z\in\mathbb{D}$ and $n\in\mathbb{Z}$, consider the intervals
$I_{n}=\\{t\in[0,2\pi]:2^{n-1}(1-|z|)\leq t<2^{n}(1-|z|)\\}$. According to
Herg’est we have

$(1-|z|^{2})|H(\nu)^{\prime}(z)|\lesssim\sum_{n\in\mathbb{Z}}\int_{I_{n}}\frac{1-|z|}{|e^{it}-|z||^{3}}\left\lvert\nu(I(z,t))-\nu(I(z,t))\right\rvert
dt,\qquad z\in\mathbb{D}.$

We now estimate each term in the sum. Assumption (2) implies

$\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert\leq
C|t|e^{-\nu(I_{t}(z))/|I_{t}(z)|},$ (30)

where as before, $I_{t}(z)$ denotes the arc $I_{t}(z)=I(z,t)\cup I(z,-t)$. Let
$\varepsilon>0$ be a small number to be fixed later. Applying Lemma 5.3, we
obtain a number $\delta>0$ such that

$\left\lvert\frac{\nu(I_{t}(z))}{|I_{t}(z)|}-\frac{\nu(I_{z})}{|I_{z}|}\right\rvert\leq\varepsilon
n,\quad t\in I_{n},\quad|n|\leq T:=\delta e^{\nu(I_{z})/|I_{z}|}.$ (31)

It thus follows from (30) and (31) that

$\left\lvert\frac{\nu(I(z,t))}{t}-\frac{\nu(I(z,-t)}{t}\right\rvert\lesssim
e^{\varepsilon n}e^{-\nu(I_{z})/|I_{z}|},\quad t\in I_{n},\quad|n|\leq T.$

Now if $0<\varepsilon<\log 2$, we deduce that

$\sum_{n:|n|\leq
T}\int_{I_{n}}\frac{1-|z|}{|e^{it}-|z||^{3}}\left\lvert\nu(I(z,t))-\nu(I(z,-t))\right\rvert
dt\lesssim e^{-\nu(I_{z})/|I_{z}|}.$

The tail of the sum is estimated using the fact that $\nu$ is a Zygmund
measure. In fact, one obtains

$\sum_{n:|n|>T}\int_{I_{n}}\frac{1-|z|}{|e^{it}-|z||^{3}}\left\lvert\nu(I(z,t))-\nu(I(z,t))\right\rvert
dt\lesssim 2^{-T}\leq C(\delta)e^{-\nu(I_{z})/|I_{z}|},$

where $C(\delta)>0$ is a constant depending on $\delta$. This proves the
estimate (29) and therefore completes the proof of the sufficiency. We now
turn our attention to the necessity part of the proof. Rewrite
$f^{\prime}=H(\nu)^{\prime}f$, we see that our assumption reads

$(1-|z|^{2})|H(\nu)^{\prime}(z)|\lesssim e^{-P(\nu)(z)},\quad z\in\mathbb{D}.$
(32)

It turns out that condition (32) implies that

$P(\nu)(z)=\frac{\nu(I_{z})}{|I_{z}|}+\mathcal{O}\left(1\right)e^{-\nu(I_{z})/|I_{z}|},\quad
z\in\mathbb{D}.$ (33)

This is completely analogous to the estimate (20) stated in Lemma
LABEL:ZygBloch which holds for Zygmund measures. Actually one can prove (33)
mimicking the proof of (20) given in [15]. The details are omitted. Note that
(32) gives that $H(\nu)\in\mathcal{B}$. According to Lemma LABEL:ZygBloch,
$\nu$ is a Zygmund measure. Using this, we see that the estimate (32) implies

$\left\lvert P(\nu)(z_{I})-P(\nu)(z_{I^{\prime}})\right\rvert\leq\left\lvert
H(\nu)(z_{I})-H(\nu)(z_{I^{\prime}})\right\rvert\lesssim e^{-\nu(I)/|I|},$

for any pair of contiguous arcs $I,I^{\prime}\subset\partial\mathbb{D}$ of the
same length. This in conjunction with (33) then shows that condition (2)
holds, which finishes the proof. ∎

###  5.3 Invertibility does not imply cyclicity in $\mathcal{B}$

This section is devoted to a self-contained proof of THM:CycvsInv.

###### Proof of THM:CycvsInv.

Let $E$ be a sequence of complex numbers containing the origin such that both
$\mathbb{C}\setminus E$ and $\mathbb{C}\setminus E^{-1}$ are open sets which
do not contain arbitrarily large discs, that is, there exists $R_{0}>0$ such
that no disc of radius $R_{0}$ is contained in either $\mathbb{C}\setminus E$
or $\mathbb{C}\setminus E^{-1}$. Here $E^{-1}=\\{w\in\mathbb{C}:1/w\in E\\}$.
Let $f:\mathbb{D}\rightarrow\mathbf{C}\setminus E$ be an analytic universal
covering map. According to Bloch’s Theorem, we have that both $f$ and $1/f$
belong to $\mathcal{B}$. Next we shall show that $f$ is not cyclic in
$\mathcal{B}$.

Assume $f$ has non-tangential limit, say $L$, at a point of the unit circle.
Since $f$ is a covering map, $L$ must be either infinity or a point in $E$.
Since $E$ is countable, Privalov’s Theorem gives that $f$ can only have non-
tangential limit at a set of Lebesgue measure zero of points of the unit
circle. By Plessner’s Theorem (for instance, see p. 205 in [21]), its non-
tangential cluster set at almost every point of the unit circle is the whole
extended plane, that is,

$\overline{{\cap}_{0<r<1}f(\Gamma_{r}(\xi))}=\mathbb{C}\cup\\{\infty\\},\quad
m\text{- a.e. }\xi\in\partial\mathbb{D}.$ (34)

Here $\Gamma_{r}(\xi)=\\{z\in\mathbb{D}:|z-\xi|<M(1-|z|),|z|>r\\}$ denotes the
truncated Stolz angle of some fixed aperture $M>1$ with vertex at the point
$\xi\in\partial\mathbb{D}$.

Next we will show that $\mathcal{A}(f)=\\{fh\in\mathcal{B}:h\in H^{\infty}\\}$
is weak-star closed in $\mathcal{B}$. Since $\mathcal{A}(f)$ is convex, it
suffices by the Krein-Smulian Theorem to show that it is weak-star
sequentially closed. To this end, let $h_{n}\in H^{\infty}$ such that $fh_{n}$
tends to $F$ weak-star in $\mathcal{B}$. Then there exists a constant $C>0$
such that

$\sup_{z\in\mathbb{D}}\left\lvert(1-|z|^{2})(f^{\prime}(z)h_{n}(z)+f(z)h^{\prime}_{n}(z))\right\rvert\leq
C.$ (35)

Fix $a\in\mathbf{C}\setminus E$ and use (34) to show that for almost every
$\xi\in\partial\mathbb{D}$ there exist points $z_{k}(\xi)\in\mathbb{D}$
tending non-tangentially to $\xi$ such that $f(z_{k}(\xi))\to a$. Since
$(1-|z|^{2})|f^{\prime}(z)|\geq\text{dist}(f(z),E)$, $z\in\mathbb{D}$ (see
[5]), we deduce that there exists a constant $c>0$ such that

$\liminf_{k\to\infty}(1-|z_{k}(\xi)|^{2})|f^{\prime}(z_{k}(\xi))|>c$ (36)

Note that Bloch’s Theorem shows that if an analytic function $F$ in the unit
disc has non-tangential limit at a given point $\zeta\in\partial\mathbb{D}$,
then $(1-|z|)|F^{\prime}(z)|\to 0$ as $z\in\mathbb{D}$ tends non-tangentially
to $\zeta$. Since $h_{n}$ has non-tangential limit along almost every radius,
for each integer $n$ we have

$\lim(1-|z|)|h^{\prime}_{n}(z)|=0,\quad m\text{- a.e.
}\xi\in\partial\mathbb{D},$

where the limit is taken as $z\in\mathbb{D}$ tends to $\xi$ non-tangentially.
Applying (35) and (36) we deduce that

$\limsup_{k\to\infty}|h_{n}(z_{k}(\xi))|\leq C,\quad m\text{- a.e.
}\xi\in\partial\mathbb{D},\quad n=1,2,\ldots.$

Hence $\|h_{n}\|_{\infty}\leq C$, $n=1,2,\ldots$ and we may extract a
subsequence of $\\{h_{n}\\}$ which converges pointwise in $\mathbb{D}$ to a
function $h\in H^{\infty}$ and thus $F=fh$. We conclude that $\mathcal{A}(f)$
is weak-star closed in $\mathcal{B}$. Now
$\left[f\right]_{\mathcal{B}}\subseteq\mathcal{A}(f)$ and it is clear that
$\mathcal{A}(f)$ is a proper subspace of $\mathcal{B}$. We conclude that $f$
is not cyclic in $\mathcal{B}$.

∎

###  5.4 A problem of weak-star sequential closure in $\mathcal{B}$

In the proof of THM:Seqcyc we will need to construct functions in
$\mathcal{B}$ which enjoy maximal radial growth at considerable large subset
of $\mathbb{D}$.

######  Proposition 5.4.

unbddnoncyc There exists $f\in\text{BMOA}\subset\mathcal{B}$ satisfying the
following properties:

  1. (i)

The function $1/f$ belongs to $H^{\infty}$.

  2. (ii)

For some $\delta>0$, the set $E(f):=\left\\{z\in\mathbb{D}:\left\lvert
f(z)\right\rvert\geq\delta\log(\frac{1}{1-|z|})\right\\}$ accumulates
everywhere on the unit circle $\partial\mathbb{D}$, that is,
$\partial\mathbb{D}\subset\overline{E(f)}$.

Observe that such an $f$ must necessarily be outer, hence it is cyclic
$\mathcal{B}$. The set of points for which the Bloch function $f$ attains its
maximal growth are located in the set $E(f)$. In the context of cyclic vectors
in growth spaces, Hedenmalm and Borichev observed, roughly speaking that
cyclic functions therein cannot grow ”maximally” on too ”massive” sets. See
[9]. Our construction below is deeply inspired from this idea. Taking
unbddnoncyc for granted, we first deduce THM:Seqcyc.

###### Proof of THM:Seqcyc.

Let $f\in BMOA$ be given by Proposition LABEL:unbddnoncyc. Let
$\\{Q_{n}\\}_{n}$ be a sequence of analytic polynomials with
$\sup_{n}\left\lVert fQ_{n}\right\rVert_{\mathcal{B}}<\infty$. It follows that

$\sup_{n}\sup_{z\in\mathbb{D}}\log^{-1}\left(\frac{e}{1-|z|}\right)\left\lvert
f(z)Q_{n}(z)\right\rvert<\infty.$

However, this implies

$\sup_{n}\sup_{z\in E(f)}\left\lvert Q_{n}(z)\right\rvert<\infty.$

Now the assumption on $E(f)$ in conjunction with the maximum principle implies
that $\sup_{n}\left\lVert Q_{n}\right\rVert_{H^{\infty}}<\infty.$ By Helly’s
selection theorem, we can extract a subsequence $\\{Q_{n_{k}}\\}_{k}$ which
converges pointwise on $\mathbb{D}$ to some function $h\in H^{\infty}$. Hence
any weak star sequential limit of $\\{fQ_{n}\\}$ is of the form $fh$ with
$h\in H^{\infty}$. But functions of that form have finite radial limits at
$m$-a.e on $\partial\mathbb{D}$, hence the set $\\{fh\in\mathcal{B}:h\in
H^{\infty}\\}$ is a proper subspace of $\mathcal{B}$, which completes the
proof.

∎

The proof of Proposition LABEL:unbddnoncyc is based on the following
construction.

######  Lemma 5.5.

ACmeasure Let $0<\delta<1$. Then there exists a positive finite Borel measure
$\nu$ on $\partial\mathbb{D}$ satisfying the following property: For any arc
$I\subset\partial\mathbb{D}$ there exists a dyadic subarc $J\subset I$, such
that $\nu(J)\geq|J|^{1-\delta}.$

###### Proof.

Fix $0<\eta<1$ to be specifiec later. We shall define the measure $\nu$ by
declaring its mass on dyadic arcs. First we set $\nu(\partial\mathbb{D})=1$.
Inductively, assume that $\nu(J)$ has already been defined for some dyadic arc
$J\subset\partial\mathbb{D}$. We denote by $J_{+},J_{-}$ the right and left
dyadic children of $J$, respectively. We shall assign a larger portion of the
available mass to the right child than to the left by declaring

$\frac{\nu(J_{\pm})}{|J_{\pm}|}=(1\pm\eta)\frac{\nu(J)}{|J|}.$

Now it is a standard fact in measure theory that this construction gives rise
to a uniquely defined positive finite Borel measure $\nu$ on
$\partial\mathbb{D}$. Hence it suffices to verify that $\nu$ has the required
property. We may without loss of generality assume that $I$ is a dyadic arc.
Now let $J_{n}\subset I$ be the rightmost dyadic subarc of $I$ with
$|J_{n}|=2^{-n}|I|$. Then by construction, we have

$\frac{\nu(J_{n})}{|J_{n}|}=(1+\eta)^{n}\frac{\nu(I)}{|I|}\geq\frac{1}{|J_{n}|^{\delta}}=\frac{2^{\delta
n}}{|I|^{\delta}}$

whenever $(1+\eta)^{n}/2^{n\delta}\geq|I|^{1-\delta}/\nu(I)$. However,
choosing $0<\eta<1$ so that $2^{\delta}<1+\eta$, the above inequality will
certainly hold for sufficiently large $n$. This finishes the proof. ∎

###### Proof of unbddnoncyc.

Let $\nu$ be the positive finite Borel measure given by ACmeasure. Consider an
analytic selfmap $b=b_{\nu}$ of $\mathbb{D}$ defined via the Herglotz
transform of $\nu$:

$\frac{1+b(z)}{1-b(z)}=\int_{\partial\mathbb{D}}\frac{\zeta+z}{\zeta-z}d\nu(\zeta),\qquad
z\in\mathbb{D}.$

In other words, $\nu$ is the so-called Aleksandrov-Clark measure for $b$.
Consider the function

$f(z)=\log\left(\frac{e}{1-b(z)}\right),\qquad z\in\mathbb{D}.$

Note that since $f$ has bounded imaginary part, it follows that $f$ belongs to
$\text{BMOA}\subset\mathcal{B}$. Moreover, the trivial fact that $\left\lvert
1-b(z)\right\rvert<2$ for $z\in\mathbb{D}$, implies that $1/f\in H^{\infty}$.
It remains to show that the closure of the corresponding set $E(f)$ contains
the entire unit circle. Indeed, the reason for our choice of $f$ stems from
the observation that the set $E(f)$ contains

$\widetilde{E}(b):=\left\\{z\in\mathbb{D}:\operatorname{Re}\left(\frac{1+b(z)}{1-b(z)}\right)\geq(1-|z|)^{-\delta}\right\\},$

thus it suffices to check that the closure of the smaller set
$\widetilde{E}(b)$ contains $\partial\mathbb{D}$. To this end, fix a small
$\varepsilon>0$ and let $\zeta\in\partial\mathbb{D}$ be an arbitrary point. If
$0<\delta^{\prime}<1$ is fixed, then an application of ACmeasure shows that
for any arc $I\subset\partial\mathbb{D}$ containing $\zeta$ with length at
most $\varepsilon/2$, there exists a small dyadic arc $J\subset I$ (not
necessarily containing $\zeta$), such that
$\nu(J)\geq|J|^{1-\delta^{\prime}}$. Let $\xi_{J}$ denote the center of the
arc $J$ and $z_{J}:=(1-|J|)\xi_{J}$, and observe that a trivial estimate of
the Poisson kernel implies that

$\operatorname{Re}\left(\frac{1+b(z_{J})}{1-b(z_{J})}\right)=P(\nu)(z_{J})\geq
c\frac{\nu(J)}{|J|}\geq\frac{c}{|J|^{\delta^{\prime}}}=c(1-|z_{J}|)^{-\delta^{\prime}}\geq(1-|z_{J}|)^{-\delta},$

where $c>0$ is an absolute constant and $0<\delta<\delta^{\prime}$
sufficiently small. But this shows that $z_{J}\in\widetilde{E}(b)$ with
$\left\lvert z_{J}-\zeta\right\rvert\leq 2|I|<\varepsilon$. Hence it follows
that for sufficiently small $\delta>0$, the corresponding set $E(f)$
accumulates to any point $\zeta\in\partial\mathbb{D}$, thus we conclude the
proof of unbddnoncyc. ∎

### Acknowledgements

The authors would like to thank Konstantin Dyakonov, Oleg Ivrii and Bartosz
Malman for fruitful discussions.

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Adem Limani,
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Barcelona
<EMAIL_ADDRESS>
Artur Nicolau,
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Centre de Recerca Matemàtica
08193 Barcelona
<EMAIL_ADDRESS>