* Zhang  _et al._ (2018b) C. Zhang, H. Yuan, N. Zhang, L. Xu, J. Zhang, B. Li, and J. Fang, “Vector magnetometer based on synchronous manipulation of nitrogen-vacancy centers in all crystal directions,” Journal of Physics D: Applied Physics 51, 155102 (2018b).
  * Crosser  _et al._ (2010) M. Crosser, S. Scott, A. Clark, and P. Wilt, “On the magnetic field near the center of helmholtz coils,” Review of Scientific Instruments 81, 084701 (2010).
  * Yang  _et al._ (2020) B. Yang, T. Murooka, K. Mizuno, K. Kim, H. Kato, T. Makino, M. Ogura, S. Yamasaki, M. E. Schmidt, H. Mizuta, _et al._ , “Vector electrometry in a wide-gap-semiconductor device using a spin-ensemble quantum sensor,” Physical Review Applied 14, 044049 (2020).
  * Webb  _et al._ (2020) J. L. Webb, L. Troise, N. W. Hansen, J. Achard, O. Brinza, R. Staacke, M. Kieschnick, J. Meijer, J.-F. Perrier, K. Berg-Sørensen, _et al._ , “Optimization of a diamond nitrogen vacancy centre magnetometer for sensing of biological signals,” Frontiers in Physics 8, 430 (2020).
  * Stürner  _et al._ (2021) F. M. Stürner, A. Brenneis, T. Buck, J. Kassel, R. Rölver, T. Fuchs, A. Savitsky, D. Suter, J. Grimmel, S. Hengesbach, _et al._ , “Integrated and portable magnetometer based on nitrogen-vacancy ensembles in diamond,” Advanced Quantum Technologies 4, 2000111 (2021).
  * Mirams  _et al._ (2013) G. R. Mirams, C. J. Arthurs, M. O. Bernabeu, R. Bordas, J. Cooper, A. Corrias, Y. Davit, S.-J. Dunn, A. G. Fletcher, D. G. Harvey, _et al._ , “Chaste: an open source c++ library for computational physiology and biology,” PLoS computational biology 9, e1002970 (2013).
  * Cao  _et al._ (2013) D.-S. Cao, Q.-S. Xu, Q.-N. Hu, and Y.-Z. Liang, “Chemopy: freely available python package for computational biology and chemoinformatics,” Bioinformatics 29, 1092–1094 (2013).
  * Chapman and Chang (2000) B. Chapman and J. Chang, “Biopython: Python tools for computational biology,” ACM Sigbio Newsletter 20, 15–19 (2000).
  * Prechelt (2000) L. Prechelt, “An empirical comparison of seven programming languages,” Computer 33, 23–29 (2000).
  * Bitter, Mohiuddin, and Nawrocki (2006) R. Bitter, T. Mohiuddin, and M. Nawrocki,  _LabVIEW: Advanced programming techniques_ (Crc Press, 2006).
  * Cahn and Russo (2007) M. H. Cahn and M. F. Russo, “Python and automated laboratory system control,” JALA: Journal of the Association for Laboratory Automation 12, 46–55 (2007).
  * Qud (a) “Qudi original github,” https://github.com/Ulm-IQO/qudi (a).
  * Qud (b) “Qudi documentation,” https://ulm-iqo.github.io/qudi-generated-docs/html-docs/ (b).
  * Our (a) “Qudi github for our new version,” https://github.com/QexSoftware/qudi (a).
  * McGuinness  _et al._ (2011) L. P. McGuinness, Y. Yan, A. Stacey, D. A. Simpson, L. T. Hall, D. Maclaurin, S. Prawer, P. Mulvaney, J. Wrachtrup, F. Caruso, and others, “Quantum measurement and orientation tracking of fluorescent nanodiamonds inside living cells,” Nature nanotechnology 6, 358–363 (2011).
  * Sigaeva  _et al._ (2022b) A. Sigaeva, A. Hochstetter, S. Bouyim, M. Chipaux, M. Stejfova, P. Cigler, and R. Schirhagl, “Single-particle tracking and trajectory analysis of fluorescent nanodiamonds in cell-free environment and live cells,” Small 18, 2201395 (2022b), https://onlinelibrary.wiley.com/doi/pdf/10.1002/smll.202201395 .
  * Qud (c) “Concept of predefined generate methods in qudi,” https://ulm-iqo.github.io/qudi-generated-docs/html-docs/predefined_generate_methods.html (c).
  * Qud (d) “How to use qudi’s jupyter notebooks,” https://ulm-iqo.github.io/qudi-generated-docs/html-docs/jupyterkernel.html (d).
  * (137) “Signals and slots in qt documentation,” https://doc.qt.io/qt-5/signalsandslots.html.
  * (138) “Qudi installation,” https://ulm-iqo.github.io/qudi-generated-docs/html-docs/installation.html.
  * Babashah (2023) H. Babashah,  _High-Precision Optical Magnetometry Using Nitrogen Vacancies in CVD Diamond_ , Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2023), no. 10069 2023.
  * Our (b) “Website for modules tutorial,” https://github.com/QexSoftware/qudi (b).
  * Maertz  _et al._ (2010) B. J. Maertz, A. P. Wijnheijmer, G. D. Fuchs, M. E. Nowakowski, and D. D. Awschalom, “Vector magnetic field microscopy using nitrogen vacancy centers in diamond,” Applied Physics Letters 96, 092504 (2010), https://doi.org/10.1063/1.3337096 .
  * Lenz  _et al._ (2021) T. Lenz, A. Wickenbrock, F. Jelezko, G. Balasubramanian, and D. Budker, “Magnetic sensing at zero field with a single nitrogen-vacancy center,” Quantum Science and Technology 6, 034006 (2021).
  * Wang  _et al._ (2022) N. Wang, C.-F. Liu, J.-W. Fan, X. Feng, W.-H. Leong, A. Finkler, A. Denisenko, J. Wrachtrup, Q. Li, and R.-B. Liu, “Zero-field magnetometry using hyperfine-biased nitrogen-vacancy centers near diamond surfaces,” Phys. Rev. Research 4, 013098 (2022).
  * Jensen  _et al._ (2013) K. Jensen, V. Acosta, A. Jarmola, and D. Budker, “Light narrowing of magnetic resonances in ensembles of nitrogen-vacancy centers in diamond,” Physical Review B 87, 014115 (2013).
  * Robledo  _et al._ (2011) L. Robledo, H. Bernien, T. Van Der Sar, and R. Hanson, “Spin dynamics in the optical cycle of single nitrogen-vacancy centres in diamond,” New Journal of Physics 13, 025013 (2011).
  * Aslam  _et al._ (2013) N. Aslam, G. Waldherr, P. Neumann, F. Jelezko, and J. Wrachtrup, “Photo-induced ionization dynamics of the nitrogen vacancy defect in diamond investigated by single-shot charge state detection,” New Journal of Physics 15, 013064 (2013).
  * Giri  _et al._ (2018) R. Giri, F. Gorrini, C. Dorigoni, C. E. Avalos, M. Cazzanelli, S. Tambalo, and A. Bifone, “Coupled charge and spin dynamics in high-density ensembles of nitrogen-vacancy centers in diamond,” Phys. Rev. B 98, 045401 (2018).
  * Jarmola  _et al._ (2012) A. Jarmola, V. Acosta, K. Jensen, S. Chemerisov, and D. Budker, “Temperature-and magnetic-field-dependent longitudinal spin relaxation in nitrogen-vacancy ensembles in diamond,” Physical review letters 108, 197601 (2012).
  * Matheoud, Sahin, and Boero (2018) A. V. Matheoud, N. Sahin, and G. Boero, “A single chip electron spin resonance detector based on a single high electron mobility transistor,” Journal of Magnetic Resonance 294, 59–70 (2018).


# Inner functions, invariant subspaces and cyclicity in
$\mathcal{P}^{t}(\mu)$-spaces

Adem Limani and Bartosz Malman

###### Abstract

We study the invariant subspaces generated by inner functions for a class of
$\mathcal{P}^{t}(\mu)$-spaces which can be identified as spaces of analytic
functions in the unit disk $\mathbb{D}$, where $\mu$ is a measure supported in
the closed unit disk and $\mathcal{P}^{t}(\mu)$ is the span of analytic
polynomials in the usual Lebesgue space $L^{t}(\mu)$. Our measures define a
range of spaces somewhere in between the Hardy and the Bergman spaces, and our
results are thus a mixture of results from these two theories. For a large
class of measures $\mu$ we characterize the cyclic inner functions, and
exhibit some interesting properties of invariant subspaces generated by non-
cyclic inner functions. Our study is motivated by a connection with the
problem of smooth approximations in de Branges-Rovnyak spaces.

## 1 Introduction

Let $t>0$ and $\mu$ be a compactly supported positive finite Borel measure in
the complex plane $\mathbb{C}$. Denote by $\mathcal{P}^{t}(\mu)$ the closure
of the analytic polynomials in the usual Lebesgue space $L^{t}(\mu)$. The
classical Hardy spaces $H^{t}$ can be defined as the closure of analytic
polynomials in $L^{t}(d\textit{m})$, where $d\textit{m}$ is the Lebesgue
measure on the unit circle $\mathbb{T}$. The Bergman spaces can likewise be
defined as the closures of analytic polynomials in spaces $L^{t}(dA)$, where
$dA$ is the area measure on the unit disk $\mathbb{D}$. In both cases, it so
happens that these closures can be identified with spaces of analytic
functions, but for general measures $\mu$ this is no longer true. For
instance, if $\mu$ is a measure supported on $\mathbb{T}$ and contains a
singular part $\mu_{s}$, then $\mathcal{P}^{2}(\mu)$ contains $L^{2}(\mu_{s})$
as a direct summand, which is certainly not a space of analytic functions. For
absolutely continuous measures supported on $\mathbb{T}$ this phenomenon is
explained by a famous theorem of Szegő. Namely, if $\omega$ is a positive
integrable function on $\mathbb{T}$, then the space $L^{2}(\omega
d\textit{m})$ can be identified with a space of analytic functions if and only
if $\int_{\mathbb{T}}\log(\omega)d\textit{m}>-\infty$. Moreover, if the log-
integrability condition fails, then we actually have that
$\mathcal{P}^{2}(\omega dm)=L^{2}(\omega dm)$.

In the present paper, we shall introduce a class of measures $\mu$ supported
on the closed unit disk $\overline{\mathbb{D}}$ for which the space
$\mathcal{P}^{t}(\mu)$ can be identified with a sufficiently well-behaved
space of analytic functions on $\mathbb{D}$. We will stay within the realm of
Bergman-type spaces, but we will be working with spaces considerably larger
than the usual Hardy spaces. Our purpose is to study subspaces invariant for
the forward shift operator $\mathcal{M}_{z}:f(z)\to zf(z)$ which are generated
by inner functions. Our work is motivated by a connection with certain
approximation problems in the theory of de Branges-Rovnyak spaces which we
will briefly explain below.

In the case $t=2$, the spaces $\mathcal{P}^{2}(\mu)$ and the shift operator
$\mathcal{M}_{z}$ are universal models for the class of subnormal operators
(see [5]). A subnormal operator is the restriction to an invariant subspace of
a normal operator on a Hilbert space, and any such operator which also has a
cyclic vector is unitarily equivalent to $\mathcal{M}_{z}$ on
$\mathcal{P}^{2}(\mu)$ for some measure $\mu$ supported in the complex plane.
The spaces $\mathcal{P}^{2}(\mu)$ which will fall into our framework here will
be the ones on which $M_{z}$ acts as a completely non-isometric contraction,
in the sense that the ability to identify our spaces as spaces of analytic
functions is equivalent to non-existence of invariant subspaces of $M_{z}$ on
which it acts as an isometry.

We will now survey a number of results on $\mathcal{P}^{t}(\mu)$-spaces as a
way of justifying our choice of the class of measures $\mu$ which will be
presented below. If $\mu$ is supported on the closed unit disk
$\overline{\mathbb{D}}$, then a necessary condition for $\mathcal{P}^{t}(\mu)$
to be a space of analytic functions is that the space contains no
characteristic functions $1_{F}$ for any measurable subset
$F\subset\overline{\mathbb{D}}$. This property is sometimes called
irreducibility, as it was for instance in [3]. The case we will be studying is
when the restriction $\mu|\mathbb{D}$ of $\mu$ to the open disk $\mathbb{D}$
already makes $\mathcal{P}^{t}(\mu|\mathbb{D})$ into a space of analytic
functions on $\mathbb{D}$. Then the possibility to identify
$\mathcal{P}^{t}(\mu)$ with a space of analytic functions is equivalent to
non-existence of a sequence of analytic polynomials $p_{n}$ such that

$\lim_{n\to\infty}\int_{\mathbb{D}}|p_{n}|^{t}d\mu|\mathbb{D}=0$

and

$\lim_{n\to\infty}\int_{\mathbb{T}}|p_{n}-1_{F}|^{t}d\mu|\mathbb{T}=0$

for some measurable subset $F\subseteq\mathbb{T}$, $\mu|\mathbb{T}$ being the
restriction of $\mu$ to $\mathbb{T}$. We will see that this condition is
equivalent to $M_{z}:\mathcal{P}^{t}(\mu)\to\mathcal{P}^{t}(\mu)$ admitting no
invariant subspace on which it acts isometrically. In the literature, the
phenomenon above is referred to as splitting (at least when $F=\mathbb{T}$).
Vast amount of research has been carried out investigating when splitting can
occur, but the exact mechanism is still not understood. For instance,
splitting has been shown to occur if the measure $\mu|\mathbb{D}$ decays
extremely fast close to the boundary and $\mu|\mathbb{T}=\omega d\textit{m}$
with $\log(\omega)\not\in L^{1}(\mathbb{T})$, see for instance [14]. However,
there exists an example in [11] which produces a compact subset $F$ of
$\mathbb{T}$ such that $\mathcal{P}^{2}(dA+1_{F}d\textit{m})$ splits, even
though, clearly, the measure $dA$ does not decay at all. An important result
is [14, Theorem D]: if $\mu|\mathbb{D}$ does not decay too fast and if
$\mu|\mathbb{T}=\omega d\textit{m}$ with
$\int_{I}\log(\omega)d\textit{m}>-\infty$ for some circular arc $I$, then the
situation above, with the two limits simultaneously being zero, cannot occur
for any Borel set $F$ contained in $I$. Thus in this case, if $\omega$ is
supported on $I$, then $\mathcal{P}^{2}(\mu)$ becomes a space of analytic
functions.

The weight $\omega$ will be living on sets more general than intervals, as in
the already cited result from [14]. For a closed set $E$ on $\mathbb{T}$, we
consider the complement $\mathbb{T}\setminus E=\cup_{k}A_{k}$, where $A_{k}$
are disjoint open circular arcs. The set $E$ is a Beurling-Carleson set if the
following condition is satisfied:

$\displaystyle\sum_{k}|A_{k}|\log\left(\frac{1}{|A_{k}|}\right)<\infty.$ (1)

Notation $|A_{k}|$ stands for the Lebesgue measure of the arc $A_{k}$. The
sets satisfying (1) will appear in two contexts below, and we will need to
consider both Beurling-Carleson of positive Lebesgue measure, and of zero
Lebesgue measure. Based on the discussion and results stated above, we will
introduce the following conditions on the positive finite measure $\mu$
supported on $\overline{\mathbb{D}}$.

  1. (A)

The measure $d\mu|\mathbb{T}=\omega d\textit{m}$ is absolutely continuous and
non-zero only on a set $E$ which can be written as a countable union
$E=\cup_{k}E_{k}$ of Beurling-Carleson sets of positive measure for which we
have that

$\int_{E_{k}}\log(\omega)d\textit{m}>-\infty.$

  2. (B)

The measure $\mu|\mathbb{D}$ is a Carleson and a reverse Carleson measure for
some standard weighted Bergman spaces, and thus satisfies the following
Carleson embeddings:

$\mathcal{P}^{t}\left((1-|z|)^{\beta}dA\right)\hookrightarrow\mathcal{P}^{t}(\mu|\mathbb{D})\hookrightarrow\mathcal{P}^{t}\left((1-|z|)^{\alpha}dA\right)$

for some $-1<\beta\leq\alpha$.

The condition (A) is present primarily to make $\mathcal{P}^{t}(\mu)$ into a
space of analytic functions, but is alone not sufficient without some
assumption on $\mu|\mathbb{D}$, and indeed it is an interplay between the
assumption (A) and (B) that ensures that $\mathcal{P}^{t}(\mu)$ is a space of
analytic functions. On the other hand, a remarkable result by Khrushchev says
that if $\omega$ is concentrated on a set which contains no Beurling-Carleson
set of positive measure, then $\mathcal{P}^{t}(d\mu|\mathbb{D}+\omega dm)$
splits for any standard weighted Bergman space
$\mathcal{P}^{t}(\mu|\mathbb{D})$ (see [12], Theorem 4.1). This indicates that
Beurling-Carleson sets of positive measure are the appropriate carriers of
weights on the unit circle in order to ensure $\mathcal{P}^{t}(\mu)$ being a
space of analytic functions contained in the standard weighted Bergman spaces.
Beyond this first condition, we shall also implicitly assume that
$\log(\omega)$ is not integrable on the entire unit circle $\mathbb{T}$, since
integrability of this quantity leads to a space $\mathcal{P}^{t}(\mu)$ which
is essentially a Hardy space. The key point of assumption (B) is that it is
readily satisfied by any standard Bergman weight
$d\mu|\mathbb{D}=(1-|z|)^{\alpha}dA$, for $\alpha>-1$, and the reader can
think of $\mu|\mathbb{D}$ as such a weight. Our techniques allow however to
replace the standard Bergman weight by such a two-sided Carleson measure,
which is a somewhat more general condition.

We move on to the presentation of our main results. As a starting point, we
must of course establish that $\mathcal{P}^{t}(\mu)$ can genuinely be
identified as a space of analytic functions. This is a consequence of
irreducibilitypropcarleson which is proved in Section 2. In fact, as soon as
this is established, this result can be used in combination with the work of
Aleman, Richter and Sundberg in [3] to infer further properties of the space
$\mathcal{P}^{t}(\mu)$. This combination establishes the following result.

######  Theorem 1.1.

irredTheorem Let $\mu=\mu|\mathbb{D}+\mu|\mathbb{T}$ be a positive finite
measure on $\overline{\mathbb{D}}$ satisfying the assumptions $(A)$ and $(B)$.
For any $t\in(0,\infty)$, the space $\mathcal{P}^{t}(\mu)$ is a genuine space
of analytic functions: the restriction $f|\mathbb{D}$ of any element
$f\in\mathcal{P}^{t}(\mu)$ is an analytic function on $\mathbb{D}$ and
$f|\mathbb{D}$ extends uniquely to the element $f$ in $\mathcal{P}^{t}(\mu)$.
The space satisfies the following additional properties:

  1. (i)

Each $f\in\mathcal{P}^{t}(\mu)$ has non-tangential limits
$\mu|\mathbb{T}$-almost everywhere on $\mathbb{T}$.

  2. (ii)

The index of each $M_{z}$-invariant subspace of $\mathcal{P}^{t}(\mu)$ is
equal to 1.

The index notion in the theorem refers to the co-dimension of the image of
$M_{z}$ when restricted to an invariant subspace. Our main motivation for this
work was the study of the closed $M_{z}$-invariant subspaces of
$\mathcal{P}^{t}(\mu)$-spaces generated by inner functions. For a positive
singular measure $\nu$ supported on $\mathbb{T}$, the singular inner function
$S_{\nu}$ is defined by the formula

$S_{\nu}(z)=\exp\Big{(}-\int_{\mathbb{T}}\frac{\zeta+z}{\zeta-z}d\mu(\zeta)\Big{)}.$
(2)

A general inner function $\theta$ is the product $\theta=BS_{\nu}$ of a
singular inner function $S_{\nu}$ and a convergent Blaschke product $B$ of the
form

$B(z)=\prod_{n=1}^{\infty}\frac{|a_{n}|}{a_{n}}\frac{a_{n}-z}{1-\overline{a_{n}}z}.$

One of the main questions we want to answer is: for which inner functions
$\theta$ are the polynomial multiples of $\theta$ dense in the space
$\mathcal{P}^{t}(\mu)$? We compare to the situation in Hardy and Bergman
spaces. A classical result of operator theory characterizes the
$M_{z}$-invariant subspaces of the Hardy space $H^{p}$ with $0<p<\infty$ as
subspaces of the form $\theta H^{2}$. In particular, every inner function
$\theta$ generates an invariant subspace properly contained in $H^{p}$. A
description of invariant subspaces generated by inner functions exists also
for the Bergman spaces, and it involves the notion Beurling-Carleson sets of
zero measure, defined as above in (1). Every positive singular measure $\nu$
on $\mathbb{T}$ can be uniquely decomposed into a sum

$\nu=\nu_{\mathcal{C}}+\nu_{\mathcal{K}}$ (3)

of mutually orthogonal measures where $\nu_{\mathcal{C}}$ is concentrated on a
countable union of Beurling-Carlesons sets of measure zero, while
$\nu_{\mathcal{K}}$ vanishes on every such Beurling-Carleson set. For a
measure $\nu$, by concentrated on a set $F$, we mean that $\mu(B)=\mu(B\cap
F)$ for every Borel set $B$. The decomposition in (3) corresponds to a
factorization $S_{\nu}=S_{\nu_{\mathcal{C}}}S_{\nu_{\mathcal{K}}}$. We will
call $\nu_{\mathcal{C}}$ for the Beurling-Carleson part of $\nu$ and
$\nu_{\mathcal{K}}$ for the Korenblum-Roberts part of $\nu$, with similar
designations for the singular inner functions constructed from these measures.
The works of Roberts in [17] and Korenblum in [13] independently establish
that the singular inner function $S_{\nu_{\mathcal{K}}}$ is a cyclic vector
for the $M_{z}$-operator on the standard weighted Bergman spaces, while
$S_{\nu_{\mathcal{C}}}$ generates a proper invariant subspace. The subspace
generated by a non-cyclic inner function satisfies a very special property
which we will below denote by (P) and describe next.

Let $\mathcal{N}^{+}$ denote the Smirnov class $f=u/v$ of quotients of bounded
analytic functions, with $v$ being outer (see [7]). Consider a pair
$(X,\theta)$, where $X$ is a space of analytic function on $\mathbb{D}$ which
contains bounded functions and on which the $M_{z}$ operator acts
continuously, and $\theta$ be an inner function. Let $[\theta]_{X}$ be the
smallest closed $M_{z}$-invariant subspace of $X$ containing $\theta$.
Consider the following property of the pair $(X,\theta)$.

  1. (P)

The invariant subspace $[\theta]_{X}$ satisfies
$[\theta]_{X}\cap\mathcal{N}^{+}\subseteq\theta\mathcal{N}^{+}$.

In other words, if $\theta h_{n}$ converges to $f$ in the topology of $X$,
$h_{n}$ are polynomials and $f$ is contained in $\mathcal{N}^{+}$, then
$f/\theta\in\mathcal{N}^{+}$. This phenomenon of permanence of the inner
factor has appeared implicitly in less general setting in the work of Roberts
in [17]. When $X$ is a standard weighted Bergman space, then then property (P)
for $(X,\theta)$ is actually equivalent to density of classes of functions
with some degree of regularity on $\mathbb{T}$ in the classical model spaces
$K_{\theta}:=H^{2}\ominus\theta H^{2}$. Indeed, in BCpermanence below, this
permanence property for measures $\mu|\mathbb{D}$ satisfying (B) and Beurling-
Carleson inner functions is deduced from the density result of [15].

To further illustrate the importance of the concept, we note that the
fundamental and useful fact that the functions with continuous extensions to
$\mathbb{T}$ are dense in model spaces, established by Aleksandrov in [1], can
be related to the permanence of the inner factor under weak-star convergence
in the space of Cauchy transforms. The density of the same class of functions
in de Branges-Rovnyak spaces $\mathcal{H}(b)$ established in [2] can be tied
to a similar phenomenon. Establishing such corresponding properties for
$\mathcal{P}^{t}(\mu)$-spaces is related instead to approximations by
functions with certain degree of smoothness in $\mathcal{H}(b)$-spaces, and
the authors elaborate on this in [16]. In short, functions from the smoothness
class $\mathcal{H}^{\tau}$ defined below in (10) are dense in space
$\mathcal{H}(b)$ if and only if for the measure
$d\mu=(1-|z|)^{\tau-1}dA+(1-|b|^{2})d\textit{m}$, the space
$\mathcal{P}^{2}(\mu)$ is a space of analytic functions and the property (P)
holds for $\mathcal{P}^{2}(\mu)$ and the inner factor of $b$. Precisely this
connection is the main motivation for the present research.

Our results are of the kind discussed in the above paragraph. Before stating
them we will need to set some further notations and conventions. The symbols
$\mu|\mathbb{T},\mu|\mathbb{D},\omega,E,E_{k}$ appearing in the assumptions
(A) and (B) will be used throughout the paper to denote the related objects.
Thus, $E=\cup_{k}E_{k}$ denotes the carrier set of the weight
$d\mu|\mathbb{T}=\omega d\textit{m}$ on $\mathbb{T}$ (that is, the set on
which $\omega$ is concentrated), and the sets $E_{k}$ are Beurling-Carleson
sets of positive measure. Further, the support of the measure $\mu|\mathbb{T}$
will be denoted $F$. It could be larger than the carrier $E$, but $E=F$ if $E$
is a single Beurling-Carleson set, keeping in mind that finite unions of
Beurling-Carleson sets, again are Beurling-Carleson sets. The notation
$\|\cdot\|_{\mu,t}$ denotes the norm on the space $L^{t}(\mu)$ for $t\geq 1$,
and for $t\in(0,1)$ the usual translation-invariant metric. All of our results
are valid for $t\in[1,\infty)$, and some are valid also for $t\in(0,\infty)$,
and this will be indicated. For a Borel set $S$ in the closed disk we denote
by $\mu|S$ the restriction of $\mu$ to the set $S$. For a measure $\nu$, the
measures $\nu_{\mathcal{C}}$ and $\nu_{\mathcal{K}}$ denote the Beurling-
Carleson and Korenblum-Roberts parts respectively, as in (3).

The following theorem establishes the inner factor permanence property (P) for
a large class of inner functions in $\mathcal{P}^{t}(\mu)$-spaces.

######  Theorem 1.2.

InnerPermanenceTheorem Let $t\in[1,\infty)$ and $\mu$ satisfy (A) and (B), and
let $\theta=BS_{\nu}$ be an inner function. Let $\nu_{\mathcal{K}}|E$ be the
restriction of $\nu_{\mathcal{K}}$ to $E$. If
$\theta_{0}=BS_{\nu_{\mathcal{C}}}S_{\nu_{\mathcal{K}}|E}$, then the pair
$(\mathcal{P}^{t}(\mu),\theta_{0})$ satisfies property (P), and thus generates
a proper $M_{z}$-invariant subspace.

The difference from the Bergman space case is that in our setting, the
singular inner function constructed from the measure $\nu_{\mathcal{K}}|E$,
which lives on the carrier of $\mu|\mathbb{T}$, satisfies the property (P)
with respect to $\mathcal{P}^{t}(\mu)$. In the Bergman spaces, this inner
function is cyclic.

Next, we present our result on cyclicity of inner functions in
$\mathcal{P}^{t}(\mu)$.

######  Theorem 1.3.

cyclicitytheorem Let $\mu$ satisfy (B) and let $F$ denote the support of the
measure $\mu|\mathbb{T}$. Suppose $S_{\nu}$ is a Korenblum-Roberts inner
function, in the sense that $\nu$ vanishes on Beurling-Carleson sets of zero
Lebesgue measure. If $\nu$ is concentrated on the complement of $F$, then
$S_{\nu}$ is a cyclic vector for $M_{z}$ on $\mathcal{P}^{t}(\mu)$, for all
$t>0$.

Note specifically that the assumption (A) in cyclicitytheorem is not
necessary, and that $F$ refers to the (closed) support of the measure $\mu$,
which can indeed be larger than the set $E=\cup_{n}E_{n}$ appearing in
condition (A). However, as a significant upside, the theorem applies to sets
$F$ which are way more general than Beurling-Carleson sets.

Combining InnerPermanenceTheorem and cyclicitytheorem, we can formulate a
corollary which actually gives us complete description of invariant subspaces
generated by inner functions, according to property (P) and cyclicity, for a
large class of measures $\mu$. In fact, since outer functions are easily seen
to be cyclic vectors in the spaces considered, the below result can be also
considered as a description of invariant subspaces generated by bounded
analytic functions.

######  Corollary 1.4.

maintheorem Let $t\in[1,\infty)$ and $\mu$ satisfy conditions (A) and (B) with
$\mu|\mathbb{T}$ living on a single Beurling-Carleson set $E$. Let
$\theta=BS_{\nu}$ be an inner function and decompose $\nu$ as

$\nu=\nu_{\mathcal{C}}+\nu_{\mathcal{K}}|E+\nu_{\mathcal{K}}|E^{c},$

where $E^{c}$ is the complement of $E$ with respect to $\mathbb{T}$.

  1. (i)

If $\theta_{0}=BS_{\nu_{\mathcal{C}}}S_{\nu_{\mathcal{K}}|E}$, then
$\theta_{0}$ generates a proper $M_{z}$-invariant subspace and the pair
$(\mathcal{P}^{t}(\mu),\theta_{0})$ satisfies property (P).

  2. (ii)

$S_{\nu_{\mathcal{K}}|E^{c}}$ is a cyclic vector for the shift operator
$M_{z}$ on $\mathcal{P}^{t}(\mu)$.

Note that the result allows us to decompose every inner function into two
factors, one satisfying the property (P), and one being cyclic. In particular,
one easily deduces from the above that the invariant subspace generated by a
bounded analytic function $f=\theta F$, with $F$ outer, equals the invariant
subspace $[\theta_{0}]$. The result has an obvious Hardy-Bergman interplay
flavour. Close to the support of the measure $\nu$ on the circle, the
properties of the invariant subspace generated by an inner function $S_{\nu}$
are Hardy-like, while away from this support, the situation resembles what
happens in the Bergman spaces.

The rest of the paper is structured as follows. In Section 2 we will recall
techniques from [14] and [12] and prove that our assumptions (A) and (B) allow
us to identify $\mathcal{P}^{t}(\mu)$ as a space of analytic functions on
$\mathbb{D}$. In Section 3 we develop some duality theory which we will use in
Section 4. Sections 4 and 5 deal with proofs of our main theorems related to
invariant subspaces generated by inner functions.

## 2 Assumption (A), (B), and irreducibility of the shift

The below proposition is a version of a deep result on simultaneous
approximation by Khrushchev, found in [12, Theorem 3.1]. We shall introduce an
alternate and minimal version of the result which is suitable for our further
applications. We also give a sketch of the proof, which is essentially a minor
modification of the original argument in [12]. The proposition will be our
principal technical tool in establishing that
$M_{z}:\mathcal{P}^{t}(\mu)\to\mathcal{P}^{t}(\mu)$ is completely non-
isometric if $\mu$ satisfies assumptions (A) and (B). In this section, we
allow $t$ to be any finite positive number.

######  Proposition 2.1.

hruscevKOtheorem Let $\mu|\mathbb{T}=\omega d\textit{m}$ be a measure
satisfying condition (A) and which is concentrated on a single Beurling-
Carleson set $E$ of positive measure. Let $\mu|\mathbb{D}$ be a measure
satisfying the condition (B). Assume that there exists a sequence of analytic
polynomials $\\{p_{n}\\}_{n\geq 1}$, an analytic function
$f\in\mathcal{P}^{t}(\mu|\mathbb{D})$ and $g\in L^{t}(\omega d\textit{m})$
such that

$\lim_{n\to\infty}\int_{\mathbb{D}}|p_{n}-f|^{t}d\mu|\mathbb{D}=0$

and

$\lim_{n\to\infty}\int_{\mathbb{T}}|p_{n}-g|^{t}\omega d\textit{m}=0.$

If $g$ is non-zero, then $f$ is non-zero.

###### Proof.

As indicated above, we will follow closely the proof of [12, Theorem 3.1].
There, an auxiliary domain $G$ is constructed. If $\mathbb{T}\setminus
E=\cup_{n}A_{n}$ are the complementary arcs to $E$, we associate to each arc
$A_{n}$ its usual Carleson square $C_{n}$, and then the domain $G$ is obtained
by removing the squares $C_{n}$ from $\mathbb{D}$. It is not difficult to see
that the boundary $\partial G$ is a rectifiable curve, and that $\partial
G\cap\mathbb{T}=E$. Let $\nu$ be the harmonic measure on $\partial G$
associated with the point $0\in G$. Since assumption (B) is satisfied and thus
$\mathcal{P}^{t}(\mu|\mathbb{D})$ is continuously contained in a standard
weighted Bergman space, it follows that we can find constants $C,K>0$,
independent of $n$, such that

$|p_{n}(z)|\leq C\|p_{n}\|_{\mu_{\mathbb{D}},t}(1-|z|)^{-K}.$

Thus

$\displaystyle\int_{\partial G}\log^{+}|p_{n}|d\nu=\int_{\partial
G\cap\mathbb{D}}\log^{+}(|p_{n}|)d\nu+\int_{E}\log^{+}(|p_{n}|)d\nu$
$\displaystyle\lesssim 1+\int_{\partial
G\cap\mathbb{D}}\log^{+}\Big{(}\frac{1}{1-|z|}\Big{)}d\nu+\int_{E}\log^{+}(|p_{n}|)d\nu.$
(4)

The main technical difficulty of the proof of [12, Theorem 3.1] is using the
assumption that $E$ is a Beurling-Carleson in order to show that the integral
over $\partial G\cap\mathbb{D}$ is finite. For the proof of this fact, we
refer the reader to the cited paper. Proceeding, assumption (A) ensures that
there exists an outer function $h$ which satisfies $|h|^{t}=\omega$ on $E$,
and from this we obtain

$\int_{E}\log^{+}(|p_{n}|)d\nu\lesssim\int_{E}\log^{+}(|p_{n}h|)d\nu+\int_{E}\log^{+}(|h|^{-1})d\nu.$
(5)

We will now show that the Lebesgue measure $dm$ dominates $d\nu$ on $E$. For
any positive continuous function $\psi$ on $\mathbb{T}$, let $P\psi$ denote
its Poisson extension to $\mathbb{D}$, and note that

$\int_{\mathbb{T}}\psi d\textit{m}=P\psi(0)=\int_{\partial G}P\psi
d\nu\geq\int_{E}P\psi d\nu=\int_{E}\psi d\nu.$

Thus $d\textit{m}-d\nu|E$ is a non-negative measure, and so (5) implies that

$\int_{E}\log^{+}(|p_{n}|)d\nu\lesssim 1+\int_{E}|p_{n}|^{t}\omega
d\textit{m}+\int_{E}|\log\omega|d\textit{m}$ (6)

which, by condition (A) and the convergence of the sequence $\\{p_{n}\\}_{n}$
in $L^{t}(\omega d\textit{m})$, is a uniformly bounded quantity. Combining
(4), (5) and (6), we get that

$\sup_{n}\int_{\partial G}\log^{+}(|p_{n}|)d\nu<\infty.$

Let $\phi:\mathbb{D}\to G$ be a Riemann mapping satisfying $\phi(0)=0$. It
extends to a homeomorphism between $\mathbb{T}$ and $\partial G$. A change of
variable on the previous expression implies that

$\sup_{n}\int_{\mathbb{T}}\log^{+}(|p_{n}\circ\phi|)d\textit{m}<\infty.$ (7)

Because $p_{n}\circ\phi$ is continuous up to the boundary of $\mathbb{D}$, (7)
implies that $p_{n}\circ\phi$ is a bounded sequence in the Nevanlinna class of
the unit disk (see [7, Chapter II, Section 5]). By modifying $g$ on a set of
Lebesgue measure zero, we can assume that $g$ is a non-zero Borel measurable
function. Extracting a subsequence, we may assume that the $p_{n}\to g$ almost
everywhere on $E$, and by taking away a set of Lebesgue measure zero, we can
construct a set $E_{0}$ which is a Borel set, $|E|=|E_{0}|$ and $p_{n}\to g$
everywhere on $E_{0}$. Note that $E_{0}$ is a subset of
$\mathbb{T}\cap\partial G$ and so both $\nu(E_{0})$ and $|E_{0}|$ make sense.
Since $\partial G$ is rectifiable, it follows that $\nu$ is absolutely
continuous with respect to the Lebesgue measure, and thus $\nu(E_{0})>0$.

The function $g\circ\phi$ is measurable since $g$ is assumed to be a Borel
function and $\phi$ is a continuous function. Thus $p_{n}\circ\phi\to
g\circ\phi$ everywhere on the Borel set $\tilde{E_{0}}:=\phi^{-1}(E_{0})$
which satisfies $|\tilde{E_{0}}|=\nu(E_{0})>0$. Of course, $p_{n}\circ\phi\to
f\circ\phi$ uniformly on compact subsets of $\mathbb{D}$, and $f\circ\phi$ is
in the Nevanlinna class by (7) and the subsequent remark. We are in the
situation of the classical Khinchin-Ostrowski theorem (see [8, Section 2 of
Chapter 3]), and its application leads us to the following conclusion: the
sequence $p_{n}\circ\phi$ converges on $\tilde{E_{0}}$ to the boundary values
of $f\circ\phi$. Thus $f$ cannot be zero, since $p_{n}\circ\phi$ converges on
$\tilde{E_{0}}$ to $g\circ\phi$, a non-trivial function on $\tilde{E_{0}}$. ∎

With the above technical result at hand, we now prove the main result of this
section.

######  Proposition 2.2.

irreducibilitypropcarleson If $\mu$ satisfies properties (A) and (B), then the
shift operator $M_{z}$ defined on $\mathcal{P}^{t}(\mu)$ is a completely non-
isometric operator, in the sense that there exists no non-trivial closed
invariant subspace of $X\subset\mathcal{P}^{t}(\mu)$ with the property that
$f\in X$ implies that $\|M_{z}f\|_{\mu,t}=\|f\|_{\mu,t}$.

###### Proof.

Since $\mathcal{P}^{t}(\mu)\subseteq\mathcal{P}^{t}(\mu|\mathbb{D})\oplus
L^{t}(\mu|\mathbb{T})$, we can represent $f\in\mathcal{P}^{t}(\mu)$ as a tuple
$(f_{\mathbb{D}},f_{\mathbb{T}})$, where
$f_{\mathbb{D}}\in\mathcal{P}^{t}(\mu|\mathbb{D})$ and $f_{\mathbb{T}}\in
L^{t}(\mu|\mathbb{T})$. Note that if $\|M^{n}_{z}f\|_{\mu,t}=\|f\|_{\mu,t}$
for all integers $n\geq 1$, then we must necessarily have that
$f_{\mathbb{D}}\equiv 0$ and thus $f=(0,f_{\mathbb{T}})$. Since the tuple
$(0,f_{\mathbb{T}})$ is a limit of polynomial tuples of the form $(p,p)$ in
the norm of $\mathcal{P}^{t}(\mu|\mathbb{D})\oplus L^{t}(\mu|\mathbb{T})$, it
is also a limit of such tuples in the space
$\mathcal{P}^{t}(\mu|\mathbb{D})\oplus L^{t}(\mu|E)$, where $\mu|E$ is the
restriction to a Beurling-Carleson set (of positive Lebesgue measure)
$E\subseteq\mathbb{T}$ of the measure $\mu|\mathbb{T}$. It follows by
hruscevKOtheorem that $f_{\mathbb{T}}$ is zero almost everywhere with respect
to $\mu|E$ for every Beurling-Carleson set. Thus, since $\mu|\mathbb{T}$ is
concentrated on a union of Beurling-Carleson sets, $f_{\mathbb{T}}$ also is
zero almost everywhere with respect to $\mu|\mathbb{T}$. Consequently, $M_{z}$
is completely non-isometric on $\mathcal{P}^{t}(\mu)$. ∎

As a consequence, $\mathcal{P}^{t}(\mu)$ can be identified with a space of
analytic functions on $\mathbb{D}$ and every function in
$\mathcal{P}^{t}(\mu)$ is uniquely determined by its analytic restriction to
$\mathbb{D}$. irredTheorem of Section 1 now follows.

## 3 Cauchy duals of $\mathcal{P}^{t}(\mu)$-spaces

Let $\mu$ be a positive measure supported on $\overline{\mathbb{D}}$ and $G\in
L^{1}(\mu)$. The Cauchy transform $C_{G\mu}$ is the analytic function on
$\mathbb{D}$ given by the formula

$C_{G\mu}(z)=\int_{\overline{\mathbb{D}}}\frac{G(\zeta)d\mu(\zeta)}{1-\overline{\zeta}z},\quad
z\in\mathbb{D}.$

The following realization of the dual space of $\mathcal{P}^{t}(\mu)$ is
particularly well-suited for our purposes. Here and throughout this section
$t\in[1,\infty)$ and $t^{\prime}$ is the Hölder conjugate exponent such that
$1/t+1/{t^{\prime}}=1$, of course interpreting $t^{\prime}$ as infinity if
$t=1$.

######  Proposition 3.1.

CauchyDualChar Let $g\in Hol(\mathbb{D})$ and $f$ be a polynomial. Then

$\Lambda_{g}(f):=\lim_{r\to
1}\int_{\mathbb{T}}f(r\zeta)\overline{g(r\zeta)}d\textit{m}(\zeta)$ (8)

extends to a bounded linear functional on $\mathcal{P}^{t}(\mu)$ if and only
if $g$ is of the form $g=C_{G\mu}$ for some $G\in L^{t^{\prime}}(\mu)$. If
this is the case, then we have that

$\left\lvert\Lambda_{g}(f)\right\rvert=\inf\\{\|G\|_{\mu,t^{\prime}}:g=C_{G\mu}\\}.$
(9)

Moreover, every bounded linear functional on $\mathcal{P}^{t}(\mu)$ can be
represented in this way.

###### Proof.

First assume that $g$ equals the Cauchy transform $C_{G\mu}$ with $G\in
L^{t^{\prime}}(\mu)$. If $f$ is a polynomial, then in particular $f(rz)\to
f(z)$ uniformly on $\overline{\mathbb{D}}$ as $r\to 1-$, and from this it
follows that

$\displaystyle\Lambda_{g}(f)=\lim_{r\to
1-}\int_{\mathbb{T}}f(r\zeta)\overline{g(r\zeta)}d\textit{m}(\zeta)$
$\displaystyle=\lim_{r\to
1-}\int_{\mathbb{T}}\int_{\overline{\mathbb{D}}}f(r\zeta)\frac{\overline{G(z)}}{1-r\overline{\zeta}z}d\mu(z)d\textit{m}(\zeta)$
$\displaystyle=\lim_{r\to
1-}\int_{\overline{\mathbb{D}}}\overline{G(z)}\Big{(}\int_{\mathbb{T}}\frac{f(r\zeta)}{1-r\overline{\zeta}z}dm(\zeta)\Big{)}d\mu(z)$
$\displaystyle=\lim_{r\to
1-}\int_{\overline{\mathbb{D}}}\overline{G(z)}f(r^{2}z)d\mu(z)$
$\displaystyle=\int_{\overline{\mathbb{D}}}\overline{G(z)}f(z)d\mu(z).$

This shows that any $g=C_{G\mu}$ with $G\in L^{t^{\prime}}(\mu)$ induces a
bounded linear functional $\Lambda_{g}$, with

$\left\lvert\Lambda_{g}(f)\right\rvert\leq\|G\|_{\mu,t^{\prime}}\|f\|_{\mu,t}.$

Conversely, given any bounded linear functional $\Lambda$ on
$\mathcal{P}^{t}(\mu)$, the Hahn-Banach theorem and the usual duality between
$L^{t}(\mu)$ and $L^{t^{\prime}}(\mu)$ implies that $\Lambda$ can be
represented in the form

$\Lambda(f)=\int_{\overline{\mathbb{D}}}f(z)\overline{G(z)}d\mu(z)$

for some $G\in L^{t^{\prime}}(\mu)$. Following the steps backward in the above
computation, we easily deduce that

$\displaystyle\int_{\overline{\mathbb{D}}}f(z)\overline{G(z)}d\mu(z)=\lim_{r\to
1-}\int_{\mathbb{T}}f(r\zeta)\overline{\int_{\overline{\mathbb{D}}}\frac{G(z)}{1-r\overline{z}\zeta}d\mu(z)}\,d\textit{m}(\zeta).$

Thus $\Lambda(f)=\Lambda_{g}(f)$, with $g=C_{G\mu}$. The norm equality in (9)
follows in a standard way. ∎

Our assumption (B) on the measure $\mu|\mathbb{D}$ implies that functions with
sufficiently smooth extensions to the boundary correspond to bounded linear
functionals on $\mathcal{P}^{t}(\mu|\mathbb{D})$ under Cauchy duality.
Consequently, they are also bounded on the smaller space
$\mathcal{P}^{t}(\mu)$ (if the latter is a space of analytic functions). This
observation is the content of our next corollary. For $\tau>0$, let
$\mathcal{H}^{\tau}$ denote the so-called analytic Sobolev spaces:

$\mathcal{H}^{\tau}=\\{f\in
Hol(\mathbb{D}):\|f\|^{2}_{\mathcal{H}^{\tau}}:=\sum_{n=0}^{\infty}{(n+1)}^{\tau}|f_{n}|^{2}<\infty\\}.$
(10)

It is easy to see that given any positive integer $N$, there exists a $\tau>0$
such that $\mathcal{H}^{\tau}$ will consist entirely of functions for which
the first $N$ derivatives extend continuously to the boundary of $\mathbb{D}$.

Recall the quantity $\alpha>-1$ defined in property (B).

######  Corollary 3.2.

CauchySmoothSolutions If $\mu$ satisfies the second of the embeddings in (B),
then for all $g\in\mathcal{H}^{\tau}$ with $\tau=4+2\alpha$, the functional
$\Lambda_{g}$ is bounded on $\mathcal{P}^{t}(\mu|\mathbb{D})$. In particular,
for any $g\in\mathcal{H}^{\tau}$ there exists a solution $G\in
L^{t^{\prime}}(\mu|\mathbb{D})$ to the equation $g=C_{G\mu|\mathbb{D}}$.

###### Proof.

If $f\in\mathcal{P}^{t}(\mu|\mathbb{D})$, then a straightforward computation
gives the following estimate for the Taylor coefficients of $f$:

$\displaystyle\Big{|}\frac{f_{n}}{(n+1)^{1+\alpha}}\Big{|}\simeq\Big{|}\int_{\mathbb{D}}f\overline{z^{n}}(1-|z|^{2})^{\alpha}dA(z)\Big{|}\leq
C\|f\|_{\mu|\mathbb{D},t},$

where $C>0$ is independent of $f$ and $n$. In the last step we used Hölder’s
inequality and the reverse Carleson measure assumption
$\mathcal{P}^{t}(\mu|\mathbb{D})\hookrightarrow\mathcal{P}^{t}\left((1-|z|)^{\alpha}dA\right)$
of (B). It follows that

$\frac{|f_{n}|^{2}}{(n+1)^{2+2\alpha}}\leq C^{2}\|f\|^{2}_{\mu|\mathbb{D},t}.$

In particular, this implies

$\sum_{n=0}^{\infty}\frac{|f_{n}|^{2}}{(n+1)^{4+2\alpha}}\lesssim\|f\|^{2}_{\mu|\mathbb{D},t}.$

If $\tau=4+2\alpha$ and $g\in\mathcal{H}^{\tau}$, then it follows from Cauchy-
Schwarz inequality and the above computation that

$\sum_{n=0}^{\infty}|f_{n}\overline{g_{n}}|\leq\|g\|_{\mathcal{H}^{\tau}}\|f\|_{\mu|\mathbb{D},t}.$

Consequently, the limit in

$\Lambda_{g}(f)=\lim_{r\to
1-}\sum_{n=0}^{\infty}f_{n}\overline{g_{n}}r^{2n}=\lim_{r\to
1-}\int_{\mathbb{T}}f(r\zeta)\overline{g(r\zeta)}d\textit{m}(\zeta)$

exists and moreover we have
$|\Lambda_{g}(f)|\leq\|g\|_{\mathcal{H}^{\tau}}\|f\|_{\mu|\mathbb{D},t}.$ The
existence of a solution $G$ of the equation $g=C_{G\mu|\mathbb{D}}$ is now an
immediate consequence of CauchyDualChar. ∎

## 4 Permanence of inner factors under norm convergence

In this section, we shall use results of Section 3 to establish the property
(P) for the situation described in InnerPermanenceTheorem of Section 1. This
claim will be a consequence of the two propositions which are presented below.
Here, we require that $t\in[1,\infty)$.

######  Proposition 4.1.

BCpermanence Assume that the measure $\mu|\mathbb{D}$ satisfies (B) and
$\theta=BS_{\nu}$, where $B$ is a Blaschke product and $\nu$ is a Beurling-
Carleson singular measure, that is, $\nu$ is supported on a countable union of
Beurling-Carleson sets of Lebesgue measure zero. If $\theta h_{n}\to f$ in the
norm of $\mathcal{P}^{t}(\mu|\mathbb{D})$, where $h_{n}$ are bounded analytic
functions and $f\in\mathcal{N}^{+}$, then $f/\theta\in\mathcal{N}^{+}$.

###### Proof.

Let $f=u/v$ be a factorization of $f$ into $u,v\in H^{\infty}$ with $v$ being
outer. By multiplying the sequence by the bounded function $v$, we get that
$\theta p_{n}v\to u$ in $\mathcal{P}^{t}(\mu|\mathbb{D})$. Let
$\mathcal{A}^{\infty}$ denote the algebra of analytic functions in
$\mathbb{D}$ with smooth extensions to the boundary $\mathbb{T}$. For any
$s\in\mathcal{A}^{\infty}\cap K_{\theta}$, we have

$\int_{\mathbb{T}}u\overline{s}\,d\textit{m}=\lim_{n\to\infty}\int_{\mathbb{T}}\theta
p_{n}v\overline{s}\,d\textit{m}=0.$

Indeed, the passage to the limit is justified by the fact that
$s\in\mathcal{A}^{\infty}$ and CauchySmoothSolutions and the integrals in the
limit vanish because $s\in K_{\theta}=(\theta H^{2})^{\perp}$. It was shown in
[15] that $\mathcal{A}^{\infty}\cap K_{\theta}$ is dense in $K_{\theta}$ (if
$\theta$ is a Beurling-Carleson inner function), hence we conclude that
$u=\theta u_{0}$, with $u_{0}\in H^{\infty}$, and consequently
$f=\theta(u_{0}/v)\in\theta\mathcal{N}^{+}$. ∎

The following lemma isolates a technical argument needed in the proof of
Kpermanence below. The proof of this lemma is similar to Carleson’s
construction in [4], but here we instead deal with Beurling-Carleson sets of
positive measure.

######  Lemma 4.2.

smoothcutofflemma Let $E$ be a Beurling-Carleson set of positive measure and
$\\{A_{k}\\}$ be the arcs complementary to $E$ on $\mathbb{T}$, where $A_{k}$
is extending between points $a_{k}$ and $b_{k}$ on $\mathbb{T}$. Define the
function

$\phi(\zeta)=\begin{cases}|(\zeta-a_{k})(\zeta-b_{k})|,&\hskip
14.22636pt\zeta\in A_{k}\\\ 1,&\hskip 14.22636pt\zeta\in E.\end{cases}$

Then $\log(\phi)\in L^{1}(\mathbb{T})$. Let $N$ be some very large positive
integer and $s$ be the outer function with modulus $|s|=|\phi|^{N}$ on
$\mathbb{T}$. Let $g_{1},g_{2}$ be any two functions analytic in $\mathbb{D}$
which are of the form

$g_{l}(z)=\exp\Big{(}-\int_{E}\frac{\zeta+z}{\zeta-z}d\nu_{l}(\zeta)\Big{)},\quad
z\in\mathbb{D},\qquad l=1,2,$

where $\nu_{1},\nu_{2}$ are positive finite Borel measures, both concentrated
on the set $E$. If $1_{\mathbb{T}\setminus E}$ denotes the indicator function
of the set $\mathbb{T}\setminus E$, then the $2\pi$-periodic function $F$
defined by

$F(t)=\overline{g_{1}(e^{it})}g_{2}(e^{it})s(e^{it})1_{\mathbb{T}\setminus
E}(e^{it})$

is in $\mathbb{C}^{n}(\mathbb{R})$, as long as $N=N(n)$ is chosen large
enough.

###### Proof.

Ideas here will be borrowed from the elegant proof of Carleson’s result cited
above that appears in [6, Theorem 4.4.3]. It follows easily from the Beurling-
Carleson condition on $E$ that $\log(\phi)\in L^{1}(\mathbb{T})$. For any
fixed $k$, the function $(1-z/a_{k})^{N}(1-z/b_{k})^{N}$ is outer, and
therefore we can reproduce it using the formula

$(1-z/a_{k})^{N}(1-z/b_{k})^{N}=\exp\Big{(}N\int_{\mathbb{T}}\frac{\zeta+z}{\zeta-z}\log(|(\zeta-
a_{k})(\zeta-b_{k})|d\textit{m}(\zeta)\Big{)}.$

The function $s$ is the outer function with $|s|=|\phi|^{N}$ on $\mathbb{T}$,
thus

$s(z)=\exp\Big{(}N\int_{\mathbb{T}}\frac{\zeta+z}{\zeta-z}\log(\phi(\zeta))d\textit{m}(\zeta)\Big{)}.$

Note that the moduli of the two above functions coincide on $A_{k}$.
Consequently

$s(z)=(1-z/a_{k})^{N}(1-z/b_{k})^{N}\exp\Big{(}Nh_{k}(z)\Big{)}$

where

$h_{k}(z)=\int_{\mathbb{T}\setminus
A_{k}}\frac{\zeta+z}{\zeta-z}\log\Big{(}\frac{\phi(\zeta)}{|(\zeta-
a_{k})(\zeta-b_{k})|}\Big{)}d\textit{m}(\zeta).$ (11)

This formula shows that $s$ extends analytically across each of the arcs
$A_{k}$ (because $h_{k}$ does).

Now let $z\in A_{k}$ and denote by
$\text{dist}(z,E)=\min(|z-a_{k}|,|z-b_{k}|)$ the distance from $z$ to the
closed Beurling-Carleson set $E$. We have that $s$ is analytic in a
neighbourhood of $z$, and if (without loss of generality) we assume that $z$
is closer to $a_{k}$ than it is to $b_{k}$, then

$\displaystyle|s(z)|=|(z-a_{k})(z-b_{k})|^{N}\leq\text{dist}(z,E)^{N}2^{N}.$

We will now estimate the derivatives. It is clear from the formula (11) that

$|h^{(j)}_{k}(z)|\leq C\text{dist}(z,E)^{-j-1},\quad z\in A_{k}$

where the constant $C$ depends on $j$ but not on $k$. The formulas for $g_{1}$
and $g_{2}$ easily imply that these functions extend analytically across each
arc $A_{k}$, and their modulus equals 1 on each such arc. Moreover, it is
again easy to see that

$|g_{l}^{(j)}(z)|\leq C\text{dist}(z,E)^{-j-1},\qquad l=1,2,$

with constant $C$ depending on number of derivatives $j$ taken. By the product
rule, it follows from these observations that

$\Big{|}\Big{(}\frac{d}{dt}\Big{)}^{j}(\overline{g_{1}}g_{2}s)(e^{it})\Big{|}\leq
C_{j}\text{dist}(e^{it},E)^{N^{\prime}},\quad e^{it}\in A_{k}$

for some positive integer $N^{\prime}$ and constant $C_{j}$ depending only on
$j$, but not on $k$, as long as $N$ is much larger than $j$. Note that we have
used the fact that $|\frac{d}{dt}g(e^{it})|=|g^{\prime}(e^{it})|$ if $g$ is
analytic. We have thus verified that any finite number of derivatives of
$\overline{g_{1}}g_{2}s$ on $\mathbb{T}\setminus E$ tend to zero rapidly as
$z\in\mathbb{T}\setminus E$ approaches $E$, as long as the integer $N$ is
fixed large enough. It remains now to show that the derivatives of
$\overline{g_{1}}g_{2}s1_{\mathbb{T}\setminus E}$ on $E$ are zero. If
$w_{0}=e^{it_{0}}$ belongs to $E$ and $z=e^{it}$, then for the first
derivative we have

$\displaystyle\lim_{t\to
t_{0}}\Big{|}\frac{\overline{g_{1}(e^{it})}g_{2}(e^{it})s(e^{it})1_{\mathbb{T}\setminus
E}(e^{it})}{t-t_{0}}\Big{|}\leq\limsup_{t\to
t_{0}}\frac{C\text{dist}(e^{it},E)^{N^{\prime}}}{\left\lvert
t-t_{0}\right\rvert}$ $\displaystyle\lesssim\limsup_{t\to
t_{0}}C\text{dist}(e^{it},E))^{N^{\prime}-1}=0.$

Higher order derivatives are handled in the exact same way. The proof is
complete.

∎

######  Remark.

Using more refined techniques from [18] or from Lemma 7.11 in [9], one can
actually construct a smooth $C^{\infty}$-function. However, the current form
of smoothcutofflemma is enough for our purposes.

The next result extends BCpermanence to certain measures for which the
Korenblum-Roberts part does not vanish.

######  Proposition 4.3.

Kpermanence Let $\mu|\mathbb{T}=\omega d\textit{m}$ be a measure satisfying
condition (A) and is concentrated on a single Beurling-Carleson set $E$ of
positive Lebesgue measure, and let $\mu|\mathbb{D}$ be a measure satisfying
the condition (B). Suppose $\theta$ is an inner function with corresponding
singular measure which is supported on $E$. If $\theta p_{n}\to f$ in the norm
of $\mathcal{P}^{t}(\mu)$, where $p_{n}$ are bounded analytic functions and
$f\in\mathcal{N}^{+}$, then $f/\theta\in\mathcal{N}^{+}$.

###### Proof.

Let $H$ be the outer function with boundary values which satisfy
$|H|=\min(1,\omega)$ on $E$ and $|H|=1$ on $\mathbb{T}\setminus E$. We note
that $H$ extends analytically across $\mathbb{T}\setminus E$. Let $s$ be a
bounded outer function, $p$ be an analytic polynomial, and consider the family
of functions

$F_{p}(z)=\int_{\mathbb{T}}\frac{\theta(\zeta)\overline{p(\zeta)s(\zeta)\zeta
H(\zeta)}}{1-\overline{\zeta}z}d\textit{m}(\zeta).$

The function $F_{p}$ is clearly the orthogonal projection of the bounded
function $\theta(\zeta)\overline{p(\zeta)s(\zeta)\zeta H(\zeta)}\in
L^{2}(d\textit{m})$ to the Hardy space $H^{2}$. These functions belong to
$K_{\theta}$. To see this, let $\big{\langle}\cdot,\cdot\big{\rangle}_{L^{2}}$
denote the inner product on $L^{2}$ and let $P_{+}:L^{2}(\mathbb{T})\to H^{2}$
be the orthogonal projection. Then, if $q$ is any analytic polynomial, we have

$\big{\langle}\theta q,F_{p}\big{\rangle}_{L^{2}}=\big{\langle}\theta
q,\theta\overline{\zeta
psH}\big{\rangle}_{L^{2}}=\big{\langle}q,\overline{\zeta
psH}\big{\rangle}_{L^{2}}=0.$

Thus $F_{p}\in(\theta H^{2})^{\perp}=K_{\theta}$. Moreover, the functions
$F_{p}$ form a dense subset of $K_{\theta}$. Indeed, assume that $f\in
K_{\theta}$ is orthogonal to $F_{p}$ for all analytic polynomials $p$. Recall
that $f=\theta\overline{f_{0}}$ on $\mathbb{T}$ for some $f_{0}\in H^{2}$ with
$f_{0}(0)=0$. We obtain

$0=\big{\langle}f,F_{p}\big{\rangle}_{L^{2}}=\big{\langle}\theta\overline{f_{0}},\theta\overline{\zeta
psH}\big{\rangle}_{L^{2}}=\big{\langle}\overline{f_{0}},\overline{\zeta
psH}\big{\rangle}_{L^{2}}.$

Since $sH$ is outer, it follows that $\zeta psH$ is dense in the set of
functions in $H^{2}$ which vanish at $0$, and so $f_{0}\equiv 0$.
Consequently, $f=0$ and the functions $F_{p}$ are dense in $K_{\theta}$.

We now decompose $F_{p}$ as

$\displaystyle F_{p}(z)=\int_{E}\frac{\theta(\zeta)\overline{\zeta
p(\zeta)s(\zeta)H(\zeta)}}{1-\overline{\zeta}z}d\textit{m}(\zeta)$
$\displaystyle+\int_{\mathbb{T}\setminus E}\frac{\theta(\zeta)\overline{\zeta
p(\zeta)s(\zeta)H(\zeta)}}{1-\overline{\zeta}z}d\textit{m}(\zeta).$ (12)

Note that on $E$ we have that $|H|\leq\omega$, so that $H/\omega:=U\in
L^{\infty}(E)$. Thus the first integral in the sum above can be re-written as

$\int_{E}\frac{\theta(\zeta)\overline{\zeta
p(\zeta)s(\zeta)U(\zeta)}}{1-\overline{\zeta}z}\omega(\zeta)d\textit{m}(\zeta).$

This is a Cauchy transform of a function in $L^{t^{\prime}}(\mu|\mathbb{T})$.
For the second of the integrals, the choice of $s=s(N)$ as in
smoothcutofflemma implies that

$\zeta\mapsto\theta(\zeta)\overline{\zeta
p(\zeta)s(\zeta)H(\zeta)}1_{\mathbb{T}\setminus E}(\zeta)$

belongs to $C^{n}(\mathbb{T})$, if only $N=N(n)$ is chosen large enough. For
an appropriate choice of $n$, the Fourier coefficients of the function will
decay rapidly enough so that the integral (12) will define a function in
$\mathcal{H}^{\tau}$ (see (10) to recall the definition), for any fixed choice
of $\tau>0$. These choices will ensure that the second integral is a Cauchy
transform of a function in $L^{t^{\prime}}(\mu|\mathbb{D})$ by
CauchySmoothSolutions, and consequently by CauchyDualChar the functionals
$\Lambda_{F_{p}}$ are bounded on $\mathcal{P}^{t}(\mu)$.

Now assume that $\|\theta p_{n}-f\|_{\mu,t}\to 0$ and $f\in\mathcal{N}^{+}$.
Then there exists a factorization $f=u/v$, where $u,v\in H^{\infty}$. It
follows that $\|\theta vp_{n}-u\|_{\mu,t}\to 0$. The functionals
$\Lambda_{F_{p}}$ are bounded on $\mathcal{P}^{t}(\mu)$ and $F_{p}\in
K_{\theta}$, so

$\displaystyle 0=\lim_{n\to\infty}\int_{\mathbb{T}}\theta
vp_{n}\overline{F_{p}}d\textit{m}=\lim_{n\to\infty}\Lambda_{F_{p}}(\theta
vp_{n})=\Lambda_{F_{p}}(u)=\int_{\mathbb{T}}u\overline{F_{p}}d\textit{m}.$

By density of the functions $F_{p}$ inside $K_{\theta}$, we deduce that
$u\in(K_{\theta})^{\perp}=\theta H^{2}$, and consequently
$f/\theta\in\mathcal{N}^{+}$. ∎

It follows easily from Kpermanence that the permanence property holds for
general measure $\mu|\mathbb{T}$ concentrated on a countable union
$\cup_{k}E_{k}$ of Beurling-Carleson sets $E_{k}$ of positive measure, and
$\theta$ being an inner function with corresponding singular measure
concentrated on $\cup_{k}E_{k}$. Indeed, if $\theta=BS_{\nu}$, then we simply
apply the above proposition to $\theta_{E_{k}}=BS_{\nu|E_{k}}$, where
$\nu|E_{k}$ is the restriction of $\nu$ to the set $E_{k}$, for each $k$. This
observation completes the proof of InnerPermanenceTheorem of Section 1.

## 5 Cyclic inner functions in $\mathcal{P}^{t}(\mu)$

In this final section, we will establish our results on cyclicity. Here we may
assume that $t\in(0,\infty)$. The notation $\|f\|_{\mu,t}$ should in case
$t<1$ be interpreted in the usual way which makes it into a translation-
invariant metric on $\mathcal{P}^{t}(\mu)$, thus for any $t>0$, we have

$\left\lVert
f\right\rVert_{\mu,t}=\left(\int_{\overline{\mathbb{D}}}\left\lvert
f(z)\right\rvert^{t}d\mu(z)\right)^{\min\\{1,1/t\\}}.$

The relevant assumptions on $\mu$ that we shall need in this section are that
$\mathcal{P}^{t}(\mu)$ can be identified as a space of analytic functions and
that condition (B) holds. In fact, we may actually relax condition (B)
slightly, by requiring that the measure $\mu|\mathbb{D}$ is such that the
norms of monomials do not decrease too slowly: There exists a constant
$\beta>0$, such that

  1. (B’)

$\left\lVert z^{n}\right\rVert_{\mu|\mathbb{D},t}\lesssim n^{-\beta}$

for all positive integers $n$. Our main task in this section is to prove
cyclicitytheorem. The idea behind the proof is very much inspired by the
beautiful techniques of Roberts [17], which establishes cyclicity of
Korenblum-Roberts inner functions in the case of Bergman-type spaces. However,
due to the fact that our measure $\mu$ lives on $\mathbb{T}$, these techniques
require certain adaptations to our setting. In this section, we shall denote
by $\mathcal{P}$ the set of analytic polynomials and fix $\rho$ to be the
distance between the closed sets $\text{supp}({\nu})$ and
$F:=\text{supp}({\mu|\mathbb{T}})$, where $\nu$ is a Korenblum-Roberts measure
concentrated on the complement of $F$. The following observation will allow us
to assume that $\rho>0$.

######  Lemma 5.1.

lemreduc Let $S_{\nu}$ be a singular inner function as in the statement of
cyclicitytheorem. Let $\nu_{n}$ be the restriction of $\nu$ to the set
$\\{\zeta\in\mathbb{T}:\text{dist}(\zeta,F)>1/n\\}$ and $S_{\nu_{n}}$ be the
corresponding inner function. If $S_{\nu_{n}}$ is a cyclic vector in
$\mathcal{P}^{t}(\mu)$ for all sufficiently large $n\geq 1$, then so is
$S_{\nu}$.

###### Proof.

For any analytic polynomial $p$ and integer $n\geq 1$, we have

$\displaystyle\left\lVert S_{\nu}p-1\right\rVert_{\mu,t}\leq$
$\displaystyle\left\lVert
S_{\nu}p-S_{(\nu-\nu_{n})}\right\rVert_{\mu,t}+\left\lVert
S_{(\nu-\nu_{n})}-1\right\rVert_{\mu,t}$ $\displaystyle{}\leq\left\lVert
S_{\nu_{n}}p-1\right\rVert_{\mu,t}+\left\lVert
S_{(\nu-\nu_{n})}-1\right\rVert_{\mu,t}.$

By our assumption, we can for any sufficiently large $n\geq 1$ find an
analytic polynomial $p_{n}$ making the quantity $\left\lVert
S_{\nu_{n}}p_{n}-1\right\rVert_{\mu,t}$ arbitrarily small, thus

$\inf_{p\in\mathcal{P}}\left\lVert
S_{\nu}p-1\right\rVert_{\mu,t}\leq\left\lVert
S_{(\nu-\nu_{n})}-1\right\rVert_{\mu,t}.$

Now since $\nu$ is a singular continuous measure, we have that $\nu_{n}\to\nu$
in total variation norm, as $n\to\infty$. From this it follows that
$S_{(\nu-\nu_{n})}(z)\to 1$ for every $z\in\mathbb{D}$. Moreover, we also have
that

$\int_{\mathbb{T}}\left\lvert
1-S_{(\nu-\nu_{n})}(\zeta)\right\rvert^{2}dm(\zeta)=2\left(1-e^{-(\nu-\nu_{n})(\mathbb{T})}\right)\to
0,$

as $n\to\infty$. This means that we can extract a subsequence
$\\{n_{k}\\}_{k}$ with $n_{k}\to\infty$, such that
$S_{(\nu-\nu_{n_{k}})}(\zeta)\to 1$ for $m$-a.e every $\zeta\in\mathbb{T}$.
Now combining these observations and using the fact that
$d\mu=d\mu\lvert\mathbb{D}+\omega dm$ is a finite measure on
$\overline{\mathbb{D}}$, we may apply the dominated convergence theorem to
obtain

$\lim_{k\to\infty}\left\lVert S_{(\nu-\nu_{n_{k}})}-1\right\rVert_{\mu,t}=0.$

It follows now that $S_{\nu}$ is a cyclic vector in $\mathcal{P}^{t}(\mu)$. ∎

With lemreduc at hand, we may fix $\rho>0$. Next we shall use a decomposition
result for singular measures which implicitly appears in the work of Roberts
in [17]. This result is also carefully treated in [10], where Ivrii clarifies
some technical points in the construction. Before we present this result we
recall the notion of modulus of continuity of a positive measure $\nu$,
defined by

$\omega_{\nu}(\delta)=\sup_{|I|<\delta}\nu(I),$

where the supremum is taken over all arcs $I\subset\mathbb{T}$ of Lebesgue
measure $|I|<\delta$.

######  Proposition 5.2 (Roberts, [17]).

Robthm

Let $\nu$ be a Korenblum-Roberts measure and let $N>1$ be a positive integer
and $c>0$ be given. Then there exists a sequence of positive measures
$\\{\nu_{k}\\}_{k}$, satisfying the following conditions on their modulus of
continuity

$\omega_{\nu_{k}}\left(\frac{1}{n_{k}}\right)\leq\frac{c\log(n_{k})}{n_{k}},\qquad
n_{k}=2^{2^{N+k}},\,k=0,1,2,\dots,$

such that $\nu$ decomposes as

$\nu=\sum_{k=1}^{\infty}\nu_{k}.$

In the similar spirit as [17], we will utilize the following quantitative form
of the Corona theorem.

######  Lemma 5.3 (The Corona Theorem).

There exists a universal constant $K>0$, such that whenever $f_{1},f_{2}\in
H^{\infty}$ with $\|f_{j}\|_{\infty}\leq 1$, for $j=1,2$, and

$\left\lvert f_{1}(z)\right\rvert+\left\lvert
f_{2}(z)\right\rvert\geq\delta,\qquad z\in\mathbb{D},$

where $0<\delta\leq 1/2$, then there exists functions $g_{1},g_{2}\in
H^{\infty}$ with $\|g_{j}\|_{\infty}\leq\delta^{-K}$, so that

$f_{1}g_{1}+f_{2}g_{2}=1.$

The point of using the Corona theorem is to find bounded analytic functions
$g_{k}$, such that $S_{\nu_{k}}g_{k}-1$ has very small
$\mathcal{P}^{t}(\mu)$-norm, where $\nu_{k}$ is a singular measure appearing
in the decomposition of Proposition LABEL:Robthm.

Our next quest is to establish that singular inner functions associated to
singular measures $\nu_{k}$ are bounded below on considerably large parts of
the unit disc $\mathbb{D}$. To this end, we shall now fix $\nu$ to be a
Korenblum-Roberts measure with the property that the distance between
$\text{supp}({\nu})$ and $F:=\text{supp}({\mu|\mathbb{T}})$ is $\rho>0$.
Associated to $\nu$, we introduce the following sets

$\Omega_{n}:=D_{n}\cup\\{z\in\mathbb{D}:\text{dist}(z,\text{supp}({\nu}))\geq\rho/2\\},$

where $D_{n}$ denotes the disc centered at origin of radius $1-1/n$.

######  Lemma 5.4.

S0below There exists a constant $c_{1}>0$, such that whenever $\nu_{0}$ is a
proper restriction of the Korenblum-Roberts measure $\nu$ and $n_{0}$ is a
sufficiently large positive integer such that $\omega_{\nu_{0}}(1/n_{0})\leq
c\log(n_{0})/n_{0}$, for some $c>0$, then the associated singular inner
function $S_{\nu_{0}}$ satisfies

$\left\lvert S_{\nu_{0}}(z)\right\rvert\geq n_{0}^{-cc_{1}},\quad
z\in\Omega_{n_{0}}.$

###### Proof.

The bound from below on $D_{n_{0}}$, for some $c_{1}>0$, follows from the
assumption on the modulus of continuity of $\nu_{0}$ and holds for any
$n_{0}>2$, see [17, Lemma 2.2]). The function $S_{\nu}$ extends analytically
across $\\{z\in\mathbb{T}:\text{dist}(z,\text{supp}({\nu}))\geq\rho/2\\}$ with
$|S_{\nu}|=1$ there, and since $\nu_{0}$ is a restriction of the measure
$\nu$, we have $|S_{\nu_{0}}|\geq|S_{\nu}|$ on $\mathbb{D}$. Thus if $n_{0}$
is large enough, then $|S_{\nu_{0}}|>1/2$ on $\Omega_{n_{0}}\setminus
D_{n_{0}}$, which completes the proof. ∎

We shall now proceed by constructing an appropriate companion to the singular
inner function $S_{\nu_{0}}$ appearing in S0below, which will allow us to
apply the Corona theorem. Indeed, we shall construct a sequence of analytic
functions in the closed unit ball of $H^{\infty}$, which are bounded from
below on $\mathbb{D}\setminus\Omega_{n}$, but have comparatively smaller
$\mathcal{P}^{t}(\mu)$-norms. The lemma below is technical and revolves around
choosing specific parameters in a careful way.

######  Lemma 5.5.

h_n Let $c_{1}>0$ denote the constant in S0below, $K>0$ denote the constant in
the Corona theorem above and fix a small $c>0$, so that $3cc_{1}K<\beta$,
where $\beta>0$ is as in (B’). Then there exists a positive integer $N_{0}$
and a sequence of functions $\\{f_{n}\\}_{n\geq N_{0}}$ inside the closed unit
ball of $H^{\infty}$, satisfying the estimates

  1. (i)

$\left\lvert f_{n}(z)\right\rvert\geq n^{-cc_{1}},\quad
z\in\mathbb{D}\setminus\Omega_{n},$

  2. (ii)

$\left\lVert f_{n}\right\rVert_{\mu,t}\leq n^{-3cc_{1}K}.$

###### Proof.

Note that according to assumption (B’) and $3cc_{1}K<\beta$, we see that for
all sufficiently large $n$, the monomials $z^{n}$ will satisfy the estimate

$\left\lVert
z^{n}\right\rVert_{\mu|\mathbb{D},t}\leq\frac{1}{2}n^{-3cc_{1}K}.$ (13)

Let $\sigma>0$ be a constant to be specified later and define the outer
function

$h_{n}(z)=\exp\left(-\sigma
n\log(n)\int_{F}\frac{\zeta+z}{\zeta-z}dm(\zeta)\right),\quad z\in\mathbb{D}.$

Then $\left\lvert h_{n}\right\rvert=n^{-\sigma n}$ $m$-a.e on $F$ and
$\left\lvert h_{n}\right\rvert=1$ on $\mathbb{T}\setminus F$. We now set
$f_{n}(z):=z^{n}h_{n}(z)$ and claim that $\\{f_{n}\\}$ is our candidate, that
will eventually satisfy both $(i)$ and $(ii)$. In light of (13), we note that
in order for $\\{f_{n}\\}$ to satisfy $(ii)$, it suffices to find a positive
integer $N_{0}$, such that

$\left(\int_{F}\left\lvert
h_{n}(\zeta)\right\rvert^{t}\omega(\zeta)dm(\zeta)\right)^{\min(1,1/t)}\leq\frac{1}{2}n^{-3cc_{1}K},\quad
n\geq N_{0}.$ (14)

However, since $\left\lvert h_{n}\right\rvert=n^{-\sigma n}$ $m$-a.e on $F$
and $\omega\in L^{1}(\mathbb{T},m)$, it readily follows that for any fixed
$\sigma>0$, we can choose $N_{0}>0$ (depending on $\sigma$) such that (14)
holds for all $n\geq N_{0}$. It now remains to check that $\\{f_{n}\\}$ also
eventually satisfies $(i)$. Recall that by definition of $\rho>0$, every
$z\in\mathbb{D}\setminus\Omega_{n}$ must necessarily satisfy
$\text{dist}(z,F)\geq\rho/2$, hence

$-\log\left\lvert h_{n}(z)\right\rvert=\sigma
n\log(n)\int_{F}\frac{1-|z|^{2}}{\left\lvert\zeta-z\right\rvert^{2}}dm(\zeta)\leq\frac{8\sigma}{\rho^{2}}\log(n),\quad
z\in\mathbb{D}\setminus\Omega_{n}.$

This implies that

$\left\lvert f_{n}(z)\right\rvert=\left\lvert z\right\rvert^{n}\left\lvert
h_{n}(z)\right\rvert\geq(1-1/n)^{n}n^{-8\sigma/\rho^{2}}\geq
4^{-1}n^{-8\sigma/\rho^{2}},\quad z\in\mathbb{D}\setminus\Omega_{n}.$

In the last step, we used the elementary inequality $(1-1/n)^{n}\geq 1/4$, for
all $n\geq 1$. Thus in order to satisfy condition $(i)$, we only need the
simple inequality

$n^{cc_{1}}\geq 4n^{8\sigma/\rho^{2}}$

to hold for all large $n$. However, choosing a fixed
$0<\sigma<cc_{1}\rho^{2}/8$, it follows that the inequality above is easily
met, thus $(i)$ holds for all sufficiently large $n$. Combining with our
previous observations about $\\{h_{n}\\}$ eventually satisfying (14), we
conclude that there exists an integer $N_{0}>0$, such that the sequence
$\\{f_{n}\\}_{n\geq N_{0}}$ belongs to the closed unit ball of $H^{\infty}$
and satisfies both conditions $(i)$ and $(ii)$. ∎

We now turn to the final proof of cyclicitytheorem. We shall follow the ideas
in [17] rather closely, but there are some minor differences, mainly
attributed to the presence of h_n and the fact that a part of our measure
$\mu$ lives on $\mathbb{T}$. For the reader’s convenience, we shall include a
proof.

###### Proof of cyclicitytheorem.

Recall the definition of $\rho$ as the distance between $\text{supp}({\nu})$
and $F$, which according to lemreduc can be assumed positive and fixed. Let
$c>0$ according to h_n and let $n_{0}$ and $N_{0}$ denote the constants from
S0below and h_n, respectively. We now pick an arbitrary integer
$N>\max(n_{0},N_{0})$ and apply Robthm with parameters $c$ and $N$ as above,
in order to decompose our Korenblum-Roberts measure
$\nu=\sum_{j=1}^{\infty}\nu_{j}$, where each measure $\nu_{j}$ satisfies

$\omega_{\nu_{j}}\left(\frac{1}{n_{j}}\right)\leq\frac{c\,\log n_{j}}{n_{j}}$

with $n_{j}=2^{2^{N+j}}$. Note that according to S0below, we can find a
constant $c_{1}>0$, such that for any $j$, we have

$\left\lvert S_{\nu_{j}}(z)\right\rvert\geq n_{j}^{-cc_{1}},\quad
z\in\Omega_{n_{j}}.$

We may also apply h_n, in order to obtain a sequence of functions
$\\{f_{n_{j}}\\}_{j}\subset H^{\infty}$ with $\left\lVert
f_{n_{j}}\right\rVert_{\infty}\leq 1$, such that

$\left\lvert f_{n_{j}}(z)\right\rvert\geq n_{j}^{-cc_{1}},\quad
z\in\mathbb{D}\setminus\Omega_{n_{j}}.$

Combining, we have that $\left\lvert S_{\nu_{j}}(z)\right\rvert+\left\lvert
f_{n_{j}}(z)\right\rvert\geq n_{j}^{-cc_{1}}$ for all $z\in\mathbb{D}$.
Invoking the Corona theorem for each $j$, we obtain functions $g_{j},h_{j}\in
H^{\infty}$ with $\left\lVert g_{j}\right\rVert_{\infty},\left\lVert
h_{j}\right\rVert_{\infty}\leq n_{j}^{cc_{1}K}$, that solve the equation
$S_{\nu_{j}}g_{j}+f_{n_{j}}h_{j}=1$ on $\mathbb{D}$. Using this and $(ii)$ of
h_n, we get

$\left\lVert S_{\nu_{j}}g_{j}-1\right\rVert_{\mu,t}\leq\left\lVert
f_{n_{j}}\right\rVert_{\mu,t}\cdot\left\lVert h_{j}\right\rVert_{\infty}\leq
n_{j}^{-3cc_{1}K}\cdot n_{j}^{cc_{1}K}=n_{j}^{-2cc_{1}K}.$ (15)

With this observation at hand, we now combine:

$\displaystyle\left\lVert
S_{\nu_{1}+\nu_{2}}g_{1}g_{2}-1\right\rVert_{\mu,t}\leq\left\lVert
S_{\nu_{1}}g_{1}\right\rVert_{\infty}\left\lVert
S_{\nu_{2}}g_{2}-1\right\rVert_{\mu,t}+\left\lVert
S_{\nu_{1}}g_{1}-1\right\rVert_{\mu,t}$
$\displaystyle\leq\left(n_{1}/n^{2}_{2}\right)^{cc_{1}K}+n_{1}^{-2cc_{1}K}.$

Setting $\nu^{M}:=\sum_{1\leq j\leq M}\nu_{j}$ and iterating this procedure,
we actually get for any positive integer $M>1$:

$\inf_{p\in\mathcal{P}}\,\left\lVert
S_{\nu^{M}}p-1\right\rVert_{\mu,t}\leq\frac{1}{n_{1}^{2cc_{1}K}}+\sum_{j=2}^{M}\left(\frac{n_{1}\dots
n_{j-1}}{n^{2}_{j}}\right)^{cc_{1}K}\leq C2^{-Ncc_{1}K}.$

Here $C>0$ is an absolute constant, independent of $M,N$. With this at hand,
we deduce that for any $M>1$:

$\displaystyle\inf_{p\in\mathcal{P}}\left\lVert
S_{\nu}p-1\right\rVert_{\mu,t}\leq\left\lVert
S_{(\nu-\nu^{M})}-1\right\rVert_{\mu,t}+\inf_{p\in\mathcal{P}}\left\lVert
S_{\nu^{M}}p-1\right\rVert_{\mu,t}$ $\displaystyle\leq\left\lVert
S_{(\nu-\nu^{M})}-1\right\rVert_{\mu,t}+C2^{-Ncc_{1}K}.$ (16)

An argument similar to that of the proof of lemreduc, allows us to extract a
subsequence $\\{M_{j}\\}_{j}$, such that

$\lim_{j\to\infty}\left\lVert S_{(\nu-\nu^{M_{j}})}-1\right\rVert_{\mu,t}=0.$

Returning back to (16) with $M=M_{j}$ and taking $j\to\infty$, we obtain

$\inf_{p\in\mathcal{P}}\left\lVert S_{\nu}p-1\right\rVert_{\mu,t}\leq
C2^{-Ncc_{1}K}.$

Since the parameter $N>1$ in the decomposition of Robthm can be chosen
arbitrary large, we finally conclude that $S_{\nu}$ is a cyclic vector for the
shift operator on $\mathcal{P}^{t}(\mu)$.

∎

## References

  * [1] A. B. Aleksandrov, Invariant subspaces of shift operators. An axiomatic approach, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 113 (1981), pp. 7–26, 264.
  * [2] A. Aleman and B. Malman, Density of disk algebra functions in de Branges–Rovnyak spaces, C. R. Math. Acad. Sci. Paris, 355 (2017), pp. 871–875.
  * [3] A. Aleman, S. Richter, and C. Sundberg, Nontangential limits in ${\mathcal{P}}^{t}(\mu)$-spaces and the index of invariant subspaces, Annals of mathematics, (2009), pp. 449–490.
  * [4] L. Carleson, Sets of uniqueness for functions regular in the unit circle, Acta mathematica, 87 (1952), pp. 325–345.
  * [5] J. B. Conway, The theory of subnormal operators, no. 36, American Mathematical Soc., 1991.
  * [6] O. El-Fallah, K. Kellay, J. Mashreghi, and T. Ransford, A primer on the Dirichlet space, vol. 203 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2014.
  * [7] J. Garnett, Bounded analytic functions, vol. 236, Springer Science & Business Media, 2007.
  * [8] V. P. Havin and B. Jöricke, The uncertainty principle in harmonic analysis, vol. 72 of Encyclopaedia Math. Sci., Springer, Berlin, 1995.
  * [9] H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, vol. 199 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
  * [10] O. Ivrii, Prescribing inner parts of derivatives of inner functions, Journal d’Analyse Mathématique, 139 (2019), pp. 495–519.
  * [11] E. M. Kegejan, Simultaneous approximation in a disc, Izv. Akad. Nauk Armjan. SSR Ser. Mat, (1966), pp. 317–330.
  * [12] S. V. Khrushchev, The problem of simultaneous approximation and of removal of the singularities of Cauchy type integrals, Trudy Matematicheskogo Instituta imeni VA Steklova, 130 (1978), pp. 124–195.
  * [13] B. Korenblum, Cyclic elements in some spaces of analytic functions, Bulletin of the American Mathematical Society, 5 (1981), pp. 317–318.
  * [14] T. L. Kriete and B. D. MacCluer, Mean-square approximation by polynomials on the unit disk, Transactions of the American Mathematical Society, 322 (1990), pp. 1–34.
  * [15] A. Limani and B. Malman, On model spaces and density of functions smooth on the boundary, arXiv preprint arXiv:2101.01746, (2021).
  * [16] , On the problem of smooth approximations in $\mathcal{H}(b)$ and connections with subnormal operators, preprint, (2021).
  * [17] J. Roberts, Cyclic inner functions in the Bergman spaces and weak outer functions in $H^{p}$, $0<p<1$, Illinois Journal of Mathematics, 29 (1985), pp. 25–38.
  * [18] B. A. Taylor and D. L. Williams, Ideals in rings of analytic functions with smooth boundary values, Canadian Journal of Mathematics, 22 (1970), pp. 1266–1283.

Adem Limani,
Centre for Mathematical Sciences, Lund University,
Lund, Sweden
<EMAIL_ADDRESS>
Bartosz Malman,
KTH Royal Institute of Technology,
Stockholm, Sweden
<EMAIL_ADDRESS>