# Thomson decompositions of measures in the disk

Bartosz Malman

###### Abstract

We study the classical problem of identifying the structure of
$\mathcal{P}^{2}(\mu)$, the closure of analytic polynomials in the Lebesgue
space $L^{2}(\mu)$ of a compactly supported Borel measure $\mu$ living in the
complex plane. In his influential work, Thomson showed that the space
decomposes into a full $L^{2}$-space and other pieces which are essentially
spaces of analytic functions on domains in the plane. For a family of measures
$\mu$ supported on the closed unit disk $\overline{\mathbb{D}}$ which have a
part on the open disk $\mathbb{D}$ which is similar to the Lebesgue area
measure, and a part on the unit circle $\mathbb{T}$ which is the restriction
of the Lebesgue linear measure to a general measurable subset $E$ of
$\mathbb{T}$, we extend the ideas of Khrushchev and calculate the exact form
of the Thomson decomposition of the space $\mathcal{P}^{2}(\mu)$. It turns out
that the space splits according to a certain decomposition of measurable
subsets of $\mathbb{T}$ which we introduce. We highlight applications to the
theory of the Cauchy integral operator and de Branges-Rovnyak spaces.

##  1 Introduction and main results

###  1.1 Background

Let $\mu$ be finite positive Borel measure in the complex plane $\mathbb{C}$
which is compactly supported, and consider the usual Lebesgue space
$L^{2}(\mu)$ of equivalence classes of Borel measurable functions which are
square-integrable with respect to $\mu$. The set of analytic polynomials
$\mathcal{P}$ consists of functions $p$ of the form
$p(z)=\sum_{n=0}^{N}p_{n}z^{n}$, where the coefficients $p_{n}$ are complex
numbers, and $z=x+iy$ is the complex variable. The norm closure of
$\mathcal{P}$ in $L^{2}(\mu)$ is denoted by $\mathcal{P}^{2}(\mu)$. The goal
of this paper is to determine the structure of $\mathcal{P}^{2}(\mu)$ for a
family of relatively simple, but important in the applications, measures $\mu$
living in the closed unit disk
$\overline{\mathbb{D}}:=\\{z\in\mathbb{C}:|z|\leq 1\\}$.

We recall two basic examples of such polynomial closures that appear in
analysis which turn out to very accurately illustrate what occurs in general.

  1. (i)

Let $\mu$ be the restriction to the interval $[0,1]$ of the linear Lebesgue
measure $dx$ on the real line $\mathbb{R}$. Then a consequence of the
classical Weierstrass polynomial approximation theorem is that the polynomials
$p(x)=\sum_{n=0}^{N}p_{n}x^{n}$ are dense in space of square-integrable
function on $[0,1]$. Thus $\mathcal{P}^{2}(\mu)=L^{2}(\mu)$ in this case.

  2. (ii)

Instead now, let $\mu=\textit{m}$ be the linear Lebesgue measure on the unit
circle $\mathbb{T}:=\\{z\in\mathbb{C}:|z|=1\\}$. The corresponding closure
$\mathcal{P}^{2}(\mu)$ is well-known to not coincide with the full space
$L^{2}(\mu)=L^{2}(\mathbb{T})$, but instead with the Hardy space $H^{2}$,
consisting of functions on $\mathbb{T}$ with vanishing negative portion of
their Fourier series. The space contains no non-trivial characteristic
functions (the trivial ones correspond to the constant functions $0$ and $1$).
Each function $f\in H^{2}=\mathcal{P}^{2}(\mu)$ determines an analytic
function in the open unit disk $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$, and
appropriately defined non-tangential boundary values of this extension
coincide with $f$ on $\mathbb{T}$. The magnitude of the values in $\mathbb{D}$
of the extended function is controlled by the size of $f$ on the carrier set
of the defining measure, this carrier set being in this case $\mathbb{T}$.
More precisely, for each $\lambda\in\mathbb{D}$, there exists a constant
$C_{\lambda}>0$ such that

$|f(\lambda)|\leq C_{\lambda}\|f\|_{L^{2}(\mu)}.$ (1)

It is well-known that, in fact, we have
$C_{\lambda}=(1-|\lambda|^{2})^{-1/2}$.

  3. (iii)

Let us now restrict the Lebesgue measure $m$ on $\mathbb{T}$ to proper subset
$A$ which is an arc of the circle: $d\mu=1_{A}\,d\textit{m}$, where $1_{A}$ is
the characteristic function of $A$. In this case,
$\mathcal{P}^{2}(\mu)=L^{2}(\mu)$, similarly to $(i)$ above.

If (1) holds for some point $\lambda$, then the operation taking
$f\in\mathcal{P}^{2}(\mu)$ to $f(\lambda)$ is called a bounded point
evaluation. The operation is called an analytic bounded point evaluation if an
entire open neighbourhood of $\lambda$ admits a uniform such bound. A long
line of research on existence of bounded point evaluations culminated in the
work of Thomson in [15], who proved that the condition
$\mathcal{P}^{2}(\mu)=L^{2}(\mu)$ is the only obstruction to existence of
analytic bounded point evaluations for $\mathcal{P}^{2}(\mu)$. He proves that
for any measure $\mu$, there exists a partition $\mu=\mu_{0}+\sum_{n\geq
1}\mu_{i}$, where each $\mu_{i}$ is the restriction of $\mu$ to a subset of
its support, and a decomposition

$\mathcal{P}^{2}(\mu)=L^{2}(\mu_{0})\oplus\Big{(}\oplus_{i\geq
1}\mathcal{P}^{2}(\mu_{i})\Big{)}$ (2)

such that for $i\geq 1$, $\mathcal{P}^{2}(\mu_{i})$ contains no non-trivial
characteristic functions, and there exists an open simply connected set
$U_{i}$ consisting of analytic bounded point evaluations for $\mu_{i}$. For
$i\geq 1$, the pieces $\mathcal{P}^{2}(\mu_{i})$ are thus essentially spaces
of analytic functions on the domain $U_{i}$. The deep work of Aleman, Sundberg
and Richter in [1] complements Thomson’s results, and provides important
information about the growth, zero sets and boundary behaviour of functions in
such ”analytic” $\mathcal{P}^{2}(\mu)$-spaces on the unit disk, i.e., when the
corresponding open set of analytic point evaluations is the unit disk
$\mathbb{D}$.

For clarity, we mention that Thomson and Aleman, Richter and Sundberg, prove
their main results also in the non-Hilbertian context, with exponents $t\neq
2$ and Lebesgue spaces $L^{t}(\mu)$. In this paper, we shall focus on the case
$t=2$, to avoid a few rather inessential technical details. We deal with a
family of measures of the form

$d\mu=(1-|z|)^{\alpha}dA+1_{E}\,d\textit{m},$ (3)

where $dA$ is the Lebesgue area measure of the open unit disk $\mathbb{D}$,
$d\textit{m}$ is the Lebesgue linear measure of the circle $\mathbb{T}$, and
$1_{E}$ is the characteristic (or indicator) function of a general measurable
subset $E$ of $\mathbb{T}$. In order for the measure to be finite, we need the
restriction $\alpha>-1$. The parameter $\alpha$ will not play an important
role, unlike the parameter $E$ which will be at the center of our attention.
For this class of measures, we seek the exact form of the decomposition
appearing in (2). An ”analytic” piece $\mathcal{P}^{2}(\mu_{i})$ will appear
in the decomposition (2) of our measure, since the part
$\mu_{\mathbb{D}}:=(1-|z|)^{\alpha}dA$ of the measure ensures that
$U=\mathbb{D}$ will consist of analytic bounded point evaluations for
$\mathcal{P}^{2}(\mu)$. The point in question is if a full $L^{2}$-piece
appears in the decomposition, and if so, what is its exact structure.

###  1.2 Work of Khrushchev and our main result

To motivate and explain our main theorem, we recall the work of Khrushchev in
[9] which implies some partial results of the type that we are interested in.
We assume that $\mu$ has the form (3), and we will explain the structure of
$\mathcal{P}^{2}(\mu)$ for different special choices of the set $E$.

  1. (iv)

Assume that $E$ in (3) is an arc. In this case, the space
$\mathcal{P}^{2}(\mu)$ will contain no $L^{2}$-summand. To see this, consider
the usual Carleson curvilinear rectangle
$S:=\\{z\in\mathbb{D}:1-|z|>|E|,z/|z|\in E\\},$ and $\phi:\mathbb{D}\to S$ a
conformal mapping. A simple computation involving the harmonic measure of $S$
will show that for any sequence of polynomials bounded in the norm of
$\mathcal{P}^{2}(\mu)$, the sequence $p\circ\phi$ must be bounded in the
Nevanlinna class $\mathcal{N}$ of the unit disk. Then the classical
Khinchin–Ostrowski theorem (see [6] for precise definitions), which connects
the values in $\mathbb{D}$ to values in $\mathbb{T}$ for converging sequences
of functions in $\mathcal{N}$, ensures that any convergent sequence of
polynomials in $\mathcal{P}^{2}(\mu)$ cannot have a non-trivial characteristic
function as its limit.

  2. (v)

Khrushchev in [9] and Kegejan in [8] extend the idea in the previous point to
sets more general than intervals. If $E$ is a closed subset of $\mathbb{T}$,
and $\mathbb{T}\setminus E=\cup_{n}\ell_{n}$ is the decomposition of its
complement into disjoint open arcs $\ell_{n}$, then the set $E$ is called a
Beurling-Carleson set if

$\sum_{n}|\ell_{n}|\log(1/|\ell_{n}|)<\infty.$ (4)

For such a set, Khrushchev’s approach in [9] is to carefully estimate the
harmonic measure of a certain domain $\Omega\subset\mathbb{D}$ constructed
from the set $E$ and conclude, as before, that any sequence of analytic
polynomials bounded in the $\mathcal{P}^{2}(\mu)$-norm cannot simultaneously
converge pointwise to zero on $\mathbb{D}$ and a non-zero quantity on
$\mathbb{T}$ (or vice versa). Thus, if $E$ in (3) is a Beurling-Carleson set,
then again $\mathcal{P}^{2}(\mu)$ contains no non-trivial characteristic
functions (we note that Khrushchev in [9] does not work with
$\mathcal{P}^{2}(\mu)$-spaces, but with certain Banach spaces, but his results
are directly applicable to our situation).

  3. (vi)

Conversely, assume that $E$ in (3) is a measurable subset of $\mathbb{T}$
which contains no Beurling-Carleson subsets of positive Lebesgue measure.
Under this assumption, Khrushchev constructs a sequence of analytic
polynomials that simultaneously converges pointwise to $1$ on $\mathbb{D}$, to
$0$ on $E$, and is bounded in the norm of $\mathcal{P}^{2}(\mu)$, with $\mu$
given by (3). It follows that $\mathcal{P}^{2}(\mu)$ contains $L^{2}(E)$ as a
summand.

Consideration of the above examples leads us to the following definition. Let
$E$ be a measurable subset of $\mathbb{T}$, and consider the quantity

$C:=\sup_{G}\Big{\\{}|G|:|G\setminus E|=0,G\text{ is a Beurling-Carleson set
}\Big{\\}}.$ (5)

That is, we compute the supremum of the Lebesgue measures of Beurling-Carleson
sets $G$ which are almost contained in $E$, in the sense that the set
difference $|G\setminus E|$ has Lebesgue measure zero. Let $\\{G_{n}\\}_{n}$
be a sequence of such Beurling-Carleson sets for which the supermum in (5) is
attained: $\lim_{n\to\infty}|G_{n}|=C$. We define

$\text{core}(E):=\cup_{n}G_{n},$ (6)

and call this set for the core of $E$. The set $\text{core}(E)$ is the biggest
subset of $E$ which can be synthesized from Beurling-Carleson sets. We define
also

$\text{res}(E):=E\setminus\text{core}(E),$ (7)

which we call for the residual of $E$. Then $E$ equals the union of its core
and its residual, up to a set $N$ of Lebesgue measure zero. This partition of
$E$ will be called for the core-residual decomposition of $E$. It is not hard
to see that the decomposition is unique, up to sets of Lebesgue measure zero.

The main result of this paper is that the spaces $\mathcal{P}^{2}(\mu)$ under
consideration split up according to the core-residual decomposition.

######  Theorem A.

Let $d\mu=(1-|z|)^{\alpha}dA+1_{E}\,d\textit{m}$, where $\alpha>-1$ and $E$ is
any measurable subset of $\mathbb{T}$. Let

$d\mu_{1}:=d\mu-1_{\text{res}(E)}d\textit{m}=(1-|z|)^{\alpha}dA+1_{\text{core}(E)}\,d\textit{m}.$
(8)

Then

$\mathcal{P}^{2}(\mu)=L^{2}(\text{res}(E))\oplus\mathcal{P}^{2}(\mu_{1}),$ (9)

where $\mathcal{P}^{2}(\mu_{1})$ contains no non-trivial characteristic
functions.

By $L^{2}(\text{res}(E))$ above we mean the Lebesgue space of square-
integrable functions on the set $\text{res}(E)$. It should be interpreted as
the zero space if $E$ is residual-free: $|\text{res}(E)|=0$. A consequence of
the fact that $\mathcal{P}^{2}(\mu_{1})$ contains no non-trivial
characteristic functions is that every function $f\in\mathcal{P}^{2}(\mu_{1})$
has an analytic restriction to $\mathbb{D}$, and this restriction is a one-to-
one mapping from $\mathcal{P}^{2}(\mu_{1})$ to analytic functions in
$\mathbb{D}$.

Our result here is a generalization of Khrushchev’s (when specialized to the
Hilbertian setting). Indeed, the two examples $(v)$ and $(vi)$ above
correspond, more or less, to the cases $|\text{core}(E)|=|E|$ and
$|\text{res}(E)|=|E|$, respectively. In the proofs, we do use Khrushchev’s
general strategy, so our approach to his results is not new. Rather we extend
his ideas to the general situation in which the residual and the core of the
set $E$ are simultaneously of positive measure.

We should make a comment on the nature of the set $\text{res}(E)$ to better
explain the difficulty that arises in our setting. Indeed, this set has the
property found in example $(vi)$ above: it contains no Beurling-Carleson
subsets of positive Lebesgue measure. This property of a set $F$ alone allows
Khrushchev to construct a sequence of polynomials which simultaneously
converges to 1 on $\mathbb{D}$, to 0 on $F$ and is bounded in the
$\mathcal{P}^{2}(\mu)$-norm if $F=E$ in (3). But the case $F$ being a proper
subset of $E$ is different, and the mentioned property of $F$ alone (i.e.,
containing no Beurling-Carleson subsets of positive measure) cannot in general
single-handedly allow for the existence of a non-trivial $L^{2}(F)$-summand in
the space $\mathcal{P}^{2}(\mu)$. Indeed, consider a rather banal example
$E=\mathbb{T}=F\cup(\mathbb{T}\setminus F)$. In this case,
$\mathcal{P}^{2}(\mu)$ certainly contains no $L^{2}$-summands, and
$\mathcal{P}^{2}(\mu)=H^{2}$ as sets, with equality of norms. This example
highlights the perhaps obvious fact that $F$ being a residual set is not an
intrinsic property of $F$, but is tightly related to the surrounding set $E$
that the residual is produced with respect to. The ”splitting off” of the
piece $L^{2}(\text{res}(E))$ from $\mathcal{P}^{2}(\mu)$ is in fact related to
the geometric way in which $\text{res}(E)$ is contained in $E$, see
mainCRestimate below and the remark following it.

The theorem presented here is a specialization to the simplest and most
important case of our most general result, which is p2musplittheorem below.
For most of the paper, we adopt a convention similar to Khrushchev’s in [9]
and work in the more general case of sets which we call $h$-Carleson (where
(25) replaces the definition (4) presented above), generalized core-residual
decompositions, and measures $\mu$ which are compatible with the new
definition. The generalization is not deeper than replacing the function
$t\log(1/t)$, which is doing the work in (4), by any function $h(t)$ with
similar properties. This change adds no technical troubles, and we would say
that it rather helps to emphasize the essential parts in the proofs.

###  1.3 Application I: functions with sparse support and strong unilateral
decay of Fourier coefficients

One of Khrushchev’s main applications of his results, which we described in
the examples $(v)$ and $(vi)$ above, was in the study of the existence
question for the following type of functions. Consider
$f(\zeta)=\sum_{n\in\mathbb{Z}}f_{n}\zeta^{n}\in L^{2}(\mathbb{T})$ which
vanishes on the measurable subset $\mathbb{T}\setminus E$;

$f(\zeta)=0,\text{ for almost every }\zeta\in\mathbb{T}\setminus E,$ (10)

and yet satisfies a unilateral rapid decay condition on its Fourier
coefficients:

$\lim_{n\to+\infty}|f_{n}|n^{K}=0,\quad\text{ for all }K\geq 0.$ (11)

In other words, we look at of those square-integrable functions $f$ which live
only on $E$, and their Cauchy transforms

$C_{f}(z):=\int_{\mathbb{T}}\frac{f(\zeta)}{1-z\overline{\zeta}}d\textit{m}(\zeta)=\sum_{n\geq
0}f_{n}z^{n},\quad z\in\mathbb{D}$ (12)

are members of the algebra $\mathcal{A}^{\infty}$ of analytic functions in
$\mathbb{D}$ which extend continuously to $\overline{\mathbb{D}}$, and so do
their derivatives of any order. Let us define

$S_{+}(E):=\Big{\\{}f\in L^{2}(E):\eqref{coefficientdecay}\text{ holds
}\Big{\\}}.$ (13)

It is not hard to see that a bilateral version of (11) cannot hold for all
possible sets $E$. A Fourier series with such bilateral rapid decay defines a
function of the class $C^{\infty}$, and a closed set $E$ which is nowhere
dense in $\mathbb{T}$ cannot support even a continuous function on
$\mathbb{T}$. Khrushchev found, by non-constructive methods, that $S_{+}(E)$
admits a non-zero function essentially if and only if $E$ contains a Beurling-
Carleson subset of positive Lebesgue measure. He showed also that if $E$
contains no Beurling-Carleson subsets of positive Lebesgue measure, then
$S_{+}(E)=\\{0\\}$.

In [12], a construction of a non-zero function $f\in S_{+}(E)$ has been
presented, whenever $E$ is a Beurling-Carleson set. From that construction it
follows easily that $S_{+}(E)$ is not only non-trivial, but even norm-dense in
$L^{2}(E)$, whenever $E$ is a Beurling-Carleson set. A natural follow-up is to
try to characterize the measurable subsets $E$ of $\mathbb{T}$ for which the
set $S_{+}(E)$ is norm-dense in $L^{2}(E)$. The investigation carried out in
the present paper leads us to the following conclusion.

######  Theorem B.

Let $E$ be a measurable subset of $\mathbb{T}$ of positive Lebesgue measure.
Then the norm-closure of the set $S_{+}(E)$ coincides with
$L^{2}(\text{core}(E))$. Hence the set $S_{+}(E)$ is dense in $L^{2}(E)$ if
and only if $|\text{res}(E)|=0$.

In fact, the condition (11) can be very significantly weakened, and theorem as
stated above still holds true even for this weaker condition. For instance, if
$|\text{res}(E)|>0$, then even functions $f$ living on $E$ for which the
Dirichlet integral of $C_{f}$ is finite, will not be dense in $L^{2}(E)$. As
usual, if $g$ is analytic in $\mathbb{D}$, then its Dirichlet integral is the
quantity

$\int_{\mathbb{D}}|g^{\prime}(z)|^{2}\,dA(z).$ (14)

We will establish this stronger versions in the proofs below.

Our proof of this stronger version, which is presented in Section 8, has the
following curious consequence. Assume that we have a function $f\in
L^{2}(\mathbb{T})$, and we have some information regarding where $f$ lives:

$f(\zeta)=0,\text{ for almost every }\zeta\in\mathbb{T}\setminus E,$ (15)

Assume further that we can cut the Fourier series of the function $f$ into two
pieces with semi-bounded spectrum

$f(\zeta)=\sum_{n<N}f_{n}\zeta^{n}+\sum_{n\geq
N}f_{n}\zeta^{n}:=f_{-}(\zeta)+f_{+}(\zeta)$

and measure that one of the pieces, say $f_{+}$, has finite logarithmic energy
(see [6]). Namely,

$\sum_{n\geq N}n|f_{n}|^{2}<\infty.$ (16)

In that case, we can improve our information on the localization of $f$ to

$f(\zeta)=0,\text{ for almost every
}\zeta\in\mathbb{T}\setminus\text{core}(E).$ (17)

###  1.4 Application II: approximations in $\mathcal{H}(b)$-spaces

Theorem B has a different interpretation in the context of the de Branges-
Rovnyak spaces $\mathcal{H}(b)$. The space $\mathcal{H}(b)$ can be defined as
the unique Hilbert space of analytic functions on $\mathbb{D}$ with a
reproducing kernel of the form

$k_{b}(z,\lambda)=\frac{1-\overline{b(\lambda)}b(z)}{1-\overline{\lambda}z},\quad
z,\lambda\in\mathbb{D}.$ (18)

Here $b:\mathbb{D}\to\mathbb{D}$ is an analytic self-map of the unit disk
$\mathbb{D}$. This class of spaces is important for its connections to the
model theory of general linear operators on Hilbert space, and also for its
various connections to complex function theory. One can consult the monographs
[14], [3] and [4] for more background.

It is in general hard to give explicit formulas for any inhabitants of a given
space $\mathcal{H}(b)$, except for the kernel functions in (18). Thus, of
special importance are results which explain what various functions enjoying
some additional regularity properties can be found in a given space
$\mathcal{H}(b)$, and to what extend these functions can be used to
approximate a general $\mathcal{H}(b)$-function. With this in mind, we present
a version of Theorem B which is specialized to the context of de Branges-
Rovnyak spaces. The set $E$ in the statement below is defined, up to a subset
of Lebesgue measure zero, by the usual non-tangential boundary values of the
analytic function $b$.

######  Theorem C.

Let $b:\mathbb{D}\to\mathbb{D}$ be analytic, and consider the measurable set

$E:=\\{\zeta\in\mathbb{T}:|b(\zeta)|<1\\}.$

A necessary condition for the functions with finite Dirichlet integral in (14)
to be norm-dense in $\mathcal{H}(b)$ is that $|\text{res}(E)|=0$.

The papers [11] and [12] deal with positive approximation results. It is
found, for instance, that if in addition to the condition $|\text{res}(E)|=0$,
the function $1-|b(\zeta)|$ is not too small on $E$, and the inner factor of
$b$ satisfies certain properties, then the set
$\mathcal{A}^{\infty}\cap\mathcal{H}(b)$ will be dense in $\mathcal{H}(b)$. In
total thus, three conditions are presented in [12] which imply this density
statement. By consideration of examples in [11], these three conditions are
presumed to be close to optimal. Theorem C confirms that one of these
conditions, the vanishing of the residual set above, is in fact necessary.

###  1.5 A comment on the weighted circle case

A more general weighted variant of our problem, in which the characteristic
function $1_{E}$ in (3) is replaced by a non-negative measurable function $w$
living on some carrier set $E$, is more difficult. In this setting, it is both
the structure of the carrier set $E$ and the local smallness of the weight $w$
that cause $L^{2}$-summands to appear in the Thomson decomposition of the
measure.

Significant efforts have been made to understand the conditions under which
the space $\mathcal{P}^{2}(\mu)$ contains $L^{2}(w\,d\textit{m})$, which of
course is the biggest possible non-analytic summand. The curious reader can
consult [10], in which a smallness criterion on the weight $w$ is shown to
imply this complete splitting of the measure into an analytic piece on
$\mathbb{D}$ and a full $L^{2}(w\,d\textit{m})$-piece on $\mathbb{T}$.
Interesting methods are employed in [10] which are different from ours, and
rely on estimates of certain composition operators. Our present work has a
different focus, and it is the identification of any $L^{2}$-summands inside
of $\mathcal{P}^{2}(\mu)$ that is our principal interest. In the weighted
context, we deduce from our results that $\mathcal{P}^{2}(\mu)$ contains an
$L^{2}$-summand if the carrier of the weight $w$ has a non-trivial residual.

######  Corollary D.

Let $\mu$ have the form

$d\mu=(1-|z|)^{\alpha}+w\,d\textit{m},$

where $\alpha>-1$ and $w$ is an integrable measurable non-negative weight on
$\mathbb{T}$. Consider the measurable set

$E:=\\{\zeta\in\mathbb{T}:w(\zeta)>0\\}$

and let $1_{\text{res}(E)}\cdot w$ be the restriction of the weight $w$ to the
residual set of $E$. Then $L^{2}(1_{\text{res}(E)}\cdot w\,d\textit{m})$ is
contained in $\mathcal{P}^{2}(\mu)$.

The corollary is a direct consequence of p2musplittheorem below, from which
Theorem A will also be deduced.

###  1.6 Outline of the paper

In Section 2 we recall some properties of the Hausdorff measures (or more
precisely, Hausdorff contents), and alter their definitions a bit to produce
Hausdorff-type measures which are more suitable to our purposes. The next
part, Section 3, deals with the definition and basic properties of the core-
residual decompositions. Here, the main result is mainCRestimate, which
connects the decomposition to the Hausdorff-type measures defined earlier. The
technical constructions of main objects occur in Sections 4 and 5, where we
specialize some ideas from [9]. The proof of the main result, Theorem A,
occurs in Sections 6 and 7. The last two Sections 8 and 9 deal with proofs of
Theorems B and C.

###  1.7 Conventions and notation

We identify the circle $\mathbb{T}$ and the interval $[0,2\pi)$ by the usual
mapping. An arc $A$ of the circle is sometimes called for an interval, and the
notation $A=(a,b)$ means that $A$ is the shorter of the two circular arcs with
endpoints $a,b\in\mathbb{T}$. For subsets $A$ of $\mathbb{T}$, we use the
notation $|A|:=m(A)$ and a normalization so that $m(\mathbb{T})=1$. The symbol
$L^{2}(A)$ denotes the usual Lebesgue space of measurable functions living
only on $A$ and square-integrable with respect to the restriction of the
Lebesgue linear measure to $A$. That is, $L^{2}(A)=L^{2}(1_{A}\cdot m)$. For a
measure $\mu$, the notations $\mu_{\mathbb{D}}$ and $\mu_{\mathbb{T}}$ denote
the restrictions of the measure to $\mathbb{D}$ and $\mathbb{T}$,
respectively. We shall sometimes take complements of a closed set $A$ with
respect to an interval $I$. This complement is $I\setminus A$. Strictly
speaking this might not be open (considered as a subset of $\mathbb{T}$) if
$I$ is closed, since two of the complementary arcs might be only half-open. In
this case, being open means relatively open with respect to $I$.

##  2 Hausdorff-type measures

We begin our discussion by reviewing the notion of Hausdorff measures (or
Hausdorff contents) and their basic properties. We will introduce also some
useful variations of the classical definition.

###  2.1 Measure functions

Our Hausdorff-type measures will be defined using a measuring function $h$.
The functions which we consider in our discussions satisfy the following basic
assumptions:

  1. (i)

$h$ is defined for real $t\in[0,\delta)$, where $\delta$ is some positive
number, and it is increasing and continuous, with $h(0)=0$,

  2. (ii)

$h(t)/t$ is decreasing, with $\lim_{t\to 0^{+}}h(t)/t=+\infty$.

Any function satisfying the above requirements will be called a measure
function. We shall not make a fuss about the size of $\delta>0$ in $(i)$
above, so we assume that it is big enough so that all our computations are
valid. Any function $h$ satisfying $(i)$ and $(ii)$ can be modified or
extended to a function on $(0,\infty)$ with the same properties, in such a way
that the its values near $0$ remain unchanged. Such a modified function
induces equivalent Hausdorff-type measures.

An immediate consequence of $(ii)$ above is subadditivity:

$\displaystyle
h(t+s)=(t+s)\frac{h(t+s)}{t+s}=t\frac{h(t+s)}{t+s}+s\frac{h(t+s)}{t+s}$
$\displaystyle\leq t\frac{h(t)}{t}+s\frac{h(s)}{s}=h(s)+h(t).$

We will develop our most comprehensive results for a general measure function
$h$, but the main example we have in mind is

$h(t)=t\log(e/t),\quad t\in(0,1],$

which we extend by continuity to satisfy $h(0)=0$. The constant $e$ appearing
in the expression ensures that the function is increasing on $[0,1]$.

###  2.2 Hausdorff measures

For a measure function $h$ satisfying the above properties, we recall the
definition of the Hausdorff measure $\mathcal{M}_{h}$ of a subset $E$ of the
unit circle, given by the expression

$\mathcal{M}_{h}(E):=\inf_{L}\Big{\\{}\sum_{\ell\in
L}h(|\ell|):E\subset\cup_{\ell\in L},\,\ell\text{ open interval}\Big{\\}}.$
(19)

That is, $\mathcal{M}_{h}(E)$ is the infimum of the numbers $\sum_{\ell\in
L}h(|\ell|)$, taken over covers $L$ of $E$. By continuity of $h$, in fact it
matters not if we take the intervals $\ell$ to be open, closed, or half-open,
and sometimes it is convenient to assume that the intervals are one, or the
other.

We define also a measure where the covering families $L$ above are restricted
to consist of dyadic intervals. Let

$d_{n,j}=\\{e^{it}:t\in[2\pi 2^{-n}j,2\pi 2^{-n}(j+1)]\\},\quad
n\in\mathbb{N},\,0\leq j<2^{n}-1,$ (20)
$\mathcal{D}_{n}:=\\{d_{n,j}\\}_{j=1}^{2^{n-1}}$ (21)

and

$\mathcal{D}=\cup_{n\in\mathbb{N}}\mathcal{D}_{n}.$

The set $\mathcal{D}_{n}$ consists of the dyadic intervals of $n$th
generation, and $\mathcal{D}$ is the family of dyadic intervals covering
$\mathbb{T}$. We define

$\mathcal{M}_{h,d}(E):=\inf_{L}\Big{\\{}\sum_{d\in L}h(|d|):E\subset\cup_{d\in
L},\,d\in\mathcal{D}\Big{\\}}.$ (22)

The two quantities $\mathcal{M}_{h}(E)$ and $\mathcal{M}_{h,d}(E)$ are
comparable.

######  Proposition 2.1.

mhequiv For any set $E\subset\mathbb{T}$ we have

$\mathcal{M}_{h}(E)\leq\mathcal{M}_{h,d}(E)\leq 2\mathcal{M}_{h}(E).$

###### Proof.

The family of admissible covers $L$ is more restrictive in the definition of
$\mathcal{M}_{h,d}$, thus $\mathcal{M}_{h}(E)\leq\mathcal{M}_{h,d}(E).$ On the
other hand, if $L$ is any cover of $E$ consisting of any kind of intervals,
then each $\ell\in L$ can be covered by two dyadic intervals $d_{1},d_{2}$ of
equal length $|d_{1}|=|d_{2}|$ which satisfy $|d_{1}|<|\ell|\leq 2|d_{1}|$.
This implies, by the postulated properties of $h$, that

$h(|\ell|)\leq h(|d_{1}|)+h(|d_{2}|)\leq 2h(|\ell|).$

Summing over all $\ell\in L$ and passing to the infimum over all covers $L$,
we obtain $\mathcal{M}_{h,d}(E)\leq 2\mathcal{M}_{h}(E).$ ∎

The measure function $\mathcal{M}_{h,d}$ has the following crucial continuity
property. For a proof, we refer the reader to [6, page 301] or [2, page 9]

######  Proposition 2.2.

Mhdcont If $A_{n}$ is an increasing sequence of subsets of $\mathbb{T}$ for
which $A=\cup_{n}A_{n}$ holds, then

$\mathcal{M}_{h,d}(A)=\lim_{n\to\infty}\mathcal{M}_{h,d}(A_{n}).$

###  2.3 Small Hausdorff measures

We will also find it convenient to work with the following versions of
$\mathcal{M}_{h}$ and $\mathcal{M}_{h,d}$ which disregard sets of Lebesgue
measure zero. We define the quantity $\mathcal{M}^{0}_{h}(E)$ analogously to
(19), but the family $\\{\ell\\}_{\ell\in L}$ is only required to cover $E$ up
to a subset of Lebesgue measure zero. Hence

$\mathcal{M}^{0}_{h}(E):=\inf_{L}\Big{\\{}\sum_{\ell\in
L}h(|\ell|):|E\setminus(\cup_{\ell\in L}\ell)|=0,\,\ell\text{ open
interval}\Big{\\}}.$ (23)

and

$\mathcal{M}^{0}_{h,d}(E):=\inf_{L}\Big{\\{}\sum_{d\in
L}h(|\ell|):|E\setminus(\cup_{d\in L}d)|=0,\,d\in\mathcal{D}\Big{\\}}.$ (24)

A version of mhequiv obviously also holds for the measures
$\mathcal{M}^{0}_{h}$ and $\mathcal{M}^{0}_{h,d}$. Other simple relations
between our Hausdorff-type measures are the following.

######  Proposition 2.3.

mhinequalities For any measurable subset $E$ of $\mathbb{T}$ it holds that

$\mathcal{M}_{h}(E)\geq\mathcal{M}^{0}_{h}(E)\geq h(|E|)$

and

$\mathcal{M}_{h,d}(E)\geq\mathcal{M}^{0}_{h,d}(E)\geq h(|E|).$

###### Proof.

The ”small” versions $\mathcal{M}^{0}_{h}$ and $\mathcal{M}^{0}_{h,d}$ allow
for more general covers $L$, so the first inequalities in both lines are
clear. Moreover, if $|E\setminus(\cup_{\ell\in L}\ell)|=0$ then certainly
$|E|\leq\sum_{\ell\in L}|\ell|$, and by subadditivity of $h$ we have

$h(|E|)\leq\sum_{\ell\in L}h(|\ell|).$

By taking infimum over admissable families $L$ we obtain the right-most
inequalities above. ∎

##  3 Cores, residuals, and their properties

In this section we introduce our principal tool, which we call the core-
residual decomposition. We fix a measure function $h$ satisfying the
properties stated in Section 2.

###  3.1 $h$-Carleson sets

The following definition is classical.

######  Definition 3.1.

A closed subset $A$ of $\mathbb{T}$ is called an $h$-Carleson set if the
system $L=\\{\ell_{n}\\}_{n}$ of disjoint open arcs which constitute the
complement $\mathbb{T}\setminus A$ satisfies

$\sum_{n}h(|\ell_{n}|)<\infty.$ (25)

For completeness, we spend a moment to recall the basic properties of the
$h$-Carleson sets which are needed in the following discussion. A simple first
observation which one can make is that if we start with a family of not
necessarily disjoint open arcs which satisfy (25), then the complement of
their union is an $h$-Carleson set.

######  Proposition 3.1.

constrhCarleson Let $L=\\{\ell\\}_{\ell\in L}$ be a family of open intervals
of $\mathbb{T}$ for which we have

$\sum_{\ell\in L}h(|\ell|)<\infty.$

Then $A=\mathbb{T}\setminus(\cup_{\ell\in L}\ell)$ is an $h$-Carleson set.

###### Proof.

The set $A$ is obviously closed. Its complement is a union

$\mathbb{T}\setminus A=\cup_{n}\alpha_{n}$

of disjoint open arcs $\alpha_{n}$, and each $\alpha_{n}$ is the union of
those arcs $\ell\in L$ which intersect it non-trivially. Thus
$|\alpha_{n}|\leq\sum_{|\ell\cap\alpha_{n}|>0}|\ell|,$ which by subadditivity
of $h$ implies

$h(|\alpha_{n}|)\leq\sum_{|\ell\cap\alpha_{n}|>0}|h(|\ell|).$

Whenever $\ell$ intersects $\alpha_{n}$ non-trivially, then $\ell$ is
contained in $\alpha_{n}$. Thus each $\ell\in L$ appears in the sum above only
for one distinct interval $\alpha_{n}$. Summing the above inequality over all
arcs $\alpha_{n}$ proves the claim. ∎

The class of $h$-Carleson sets is closed under taking finite unions and
intersections.

######  Proposition 3.2.

carlesonunionprop Let $A$ and $B$ be $h$-Carleson sets. Then their union
$A\cup B$ and their intersection $A\cap B$ are also $h$-Carleson sets.

###### Proof.

Let $\mathbb{T}\setminus A=\cup_{n}\alpha_{n}$, where $\alpha_{n}$ are
pairwise disjoint open arcs, and $\mathbb{T}\setminus B=\cup_{m}\beta_{m}$ be
the corresponding decomposition of $\mathbb{T}\setminus B$. The complement
$T\setminus(A\cap B)$ is a disjoint union of open arcs which are themselves
unions of arcs $\alpha_{n}$ and $\beta_{n}$. So the fact that $A\cap B$ is
$h$-Carleson follows from constrhCarleson.

For the case of the union, we have

$\mathbb{T}\setminus(A\cup B)=\cup_{n,m}\alpha_{n}\cap\beta_{m},$

and in order to verify that $A\cup B$ is an $h$-Carleson set, we need to show
that

$\sum_{n,m}h(|\alpha_{n}\cap\beta_{n}|)<\infty.$

Let us forget about the indices $(n,m)$ such that $\alpha_{n}\cap\beta_{m}$ is
void, they contribute nothing to the sum above. We partition the rest of the
indices $(n,m)$ into three families. The family $F_{1}$ consists of those
$(n,m)$ such that $\alpha_{n}\subset\beta_{m}$, the family $F_{2}$ consists of
those $(n,m)$ such that $\beta_{m}\subset\alpha_{n}$, and family $F_{3}$ of
those $(n,m)$ such that $\alpha_{n}\cap\beta_{m}$ is non-empty, but neither
$\alpha_{n}$ or $\beta_{m}$ is contained in the other. Our task is then to
show that

$\sum_{(n,m)\in F_{i}}h(|\alpha_{n}\cap\beta_{m}|)<\infty$

for $i=1,2,3$.

We start with $F_{1}$. If $(n,m)\in F_{1}$, then there is no other index
$m^{\prime}$ such that $(n,m^{\prime})\in F_{1}$, for that would imply
$\beta_{m}\cap\beta_{m^{\prime}}$ being non-trivial. It follows that each
integer $n$ appears at most once as a first coordinate of a pair in $F_{1}$.
Moreover, clearly $|\alpha_{n}\cap\beta_{m}|=|\alpha_{n}|$, and so

$\sum_{(n,m)\in F_{1}}h(|\alpha_{n}\cap\beta_{m}|)=\sum_{(n,m)\in
F_{1}}h(|\alpha_{n}|)\leq\sum_{n}h(|\alpha_{n}|)<\infty,$

by the assumption that $A$ is $h$-Carleson. The family $F_{2}$ is handled
symmetrically.

When it comes to family $F_{3}$, note that any integer $n$ appears as a first
coordinate in at most two pairs in $F_{3}$. Indeed, if $(n,m)\in F_{3}$, then
it follows that either the left or the right endpoint of $\alpha_{n}$ is
contained in $\beta_{m}$. Since $\alpha_{n}$ obviously has only two endpoints,
there can be at most two pairs in $F_{3}$ with first coordinate $n$. Thus

$\sum_{(n,m)\in F_{3}}h(|\alpha_{n}\cap\beta_{m}|)\leq\sum_{(n,m)\in
F_{3}}h(|\alpha_{n}|)\leq 2\sum_{n}h(|\alpha_{n}|)<\infty.$

By summing over all the three families we see that $A\cup B$ is an
$h$-Carleson set. ∎

###  3.2 Core-residual decomposition

With the definition and basic properties of $h$-Carleson sets in place, we can
introduce our decomposition of measurable sets. The decomposition is very
simple, but it has a crucial relation to the small Hausdorff-type measures.
The relation is stated in mainCRestimate.

Here and throughout the paper, if $|A\setminus B|=0$ holds between two
measurable subsets $A$ and $B$, we say that $A$ is almost contained in $B$.

######  Proposition 3.3.

Edecomp Let $E$ be any measurable subset of $\mathbb{T}$ and $h$ a measure
function. There exists a decomposition

$E=(F\cup\tilde{E})\setminus N,$

where $N$ is a set of Lebesgue measure zero, and the disjoint measurable
subsets $F$ and $\tilde{E}$ satisfy the following two properties:

  1. (i)

if $A$ is an $h$-Carleson set which is almost contained in $E$, then it is
almost contained in $\tilde{E}$,

  2. (ii)

$\tilde{E}$ is, up to a set of Lebesgue measure zero, a countable union of
$h$-Carleson sets almost contained in $E$.

The decomposition is unique, up to a difference of sets of Lebesgue measure
zero.

###### Proof.

We let the non-negative constant $C$ be the supremum of Lebesgue measures of
$h$-Carleson sets which are almost contained in $E$. If $C=0$ then we are done
by taking $E=F$. In the case $C>0$, let $\\{E_{n}\\}_{n}$ be a sequence of
$h$-Carleson sets almost contained in $E$ such that
$\lim_{n\to\infty}|E_{n}|=C$. By carlesonunionprop, we can assume that
$E_{n}\subset E_{n+1}$. Take $\tilde{E}:=\cup_{n}E_{n}$, and
$F=E\setminus\tilde{E}$. Clearly we have $|\tilde{E}|=C$, and property $(ii)$
holds.

Assume that $A$ is an $h$-Carleson set almost contained in $E$. Then $A\cup
E_{n}$ is also an $h$-Carleson almost contained in $E$, by carlesonunionprop.
We have

$C=\lim_{n\to\infty}|E_{n}|\leq\lim_{n\to\infty}|A\cup
E_{n}|=|A\cup\tilde{E}|.$

However, by definition of $C$ we have that $\lim_{n\to\infty}|A\cup E_{n}|\leq
C$, and so we infer that

$|A\cup\tilde{E}|=|\tilde{E}|.$

Thus $A$ is almost contained in $\tilde{E}$.

We establish the uniqueness of the decomposition. Let
$E=F_{1}\cup\tilde{E}_{1}$ be some other partition of $E$ where the above
stated properties hold and $\tilde{E}_{1}$ is (up to Lebesgue measure zero) a
union $\cup_{n}G_{n}$ of some increasing sequence of $h$-Carleson sets $G_{n}$
almost contained in $E$. It follows readily that the $h$-Carleson sets
$E_{n}\cup G_{n}$ are almost contained in the intersection
$\tilde{E}\cap\tilde{E}_{1}.$ But then

$|\tilde{E}_{1}|=\lim_{n}|G_{n}|\leq\lim_{n}|G_{n}\cup
E_{n}|\leq|\tilde{E}_{1}\cap\tilde{E}|,$

so we must have $\tilde{E}_{1}\subset\tilde{E}$ up to a set of Lebesgue
measure zero. By reversing the argument we see that $\tilde{E}=\tilde{E}_{1}$,
up to a set of Lebesgue measure zero, and so the uniqueness of the
decomposition is established. ∎

######  Definition 3.2.

Let $E$ be any measurable subset of $\mathbb{T}$, and consider any
decomposition of the form $E=(F\cup\tilde{E})\setminus N$ as in Edecomp. We
will say that $F$ and $\tilde{E}$ form the core-residual decomposition of $E$
with respect to $h$. We will use the notation

$F=\text{res}_{h}(E)$

and

$\tilde{E}=\text{core}_{h}(E),$

and call the two pieces for the residual and the core, respectively, of the
measurable subset $E$.

Note that $\text{res}_{h}(E)$ and $\text{core}_{h}(E)$ are defined, up to a
set of Lebesgue measure zero, for any measurable subset $E$ of $\mathbb{T}$.
The set $N$ with $|N|=0$ will play no role in our development, so we might
without loss of generality assume that $N$ is empty, by altering the set $E$
on a set of Lebesgue measure zero.

The core-residual decomposition does not change if we pass to a portion of $E$
contained in an interval, which is a fact that we will often implicitly use.

######  Proposition 3.4.

Let $E$ be a measurable subset of $\mathbb{T}$ and $I$ be an interval of
$\mathbb{T}$. Then

$\text{res}_{h}(I\cap E)=I\cap\text{res}_{h}(E)$

and

$\text{core}_{h}(I\cap E)=I\cap\text{core}_{h}(E).$

The above claim follows readily from the definition of the decomposition and
the intersection statement in carlesonunionprop. We skip the straight-forward
proof.

The next proposition is also quite simple, but it is our main one in this
section. It showcases a connection between the core-residual decompositions
and the Hausdorff-type measures introduced in Section 2. A prototype version
appears in [6, page 303]

######  Proposition 3.5.

mainCRestimate Let $E$ be any measurable subset of $\mathbb{T}$, and $I$ an
interval. Assume that $C$ is a measurable subset of $I$ which is almost
contained in $E$. Then

$\mathcal{M}^{0}_{h}\Big{(}I\setminus
C\Big{)}=\mathcal{M}^{0}_{h}\Big{(}I\setminus(C\cap\text{core}_{h}(E))\Big{)}.$

###### Proof.

We have the set inclusion $I\setminus C\subseteq
I\setminus(C\cap\text{core}_{h}(E))$, so the inequality

$\mathcal{M}^{0}_{h}\Big{(}I\setminus
C\Big{)}\leq\mathcal{M}^{0}_{h}\Big{(}I\setminus(C\cap\text{core}_{h}(E))\Big{)}$

is plain. To see the converse inequality, assume that $L$ is a family of open
intervals for which we have

$|(I\setminus C)\setminus(\cup_{\ell\in L}\ell)|=0$

and

$\sum_{\ell\in L}h(|\ell|)\leq\mathcal{M}^{0}_{h}(I\setminus C)+\epsilon,$
(26)

for some $\epsilon>0$. Let $A$ be the complement of $\cup_{\ell\in L}\ell$
with respect to $I$. Then $A$ is closed, and by constrhCarleson and (26), it
is an $h$-Carleson set. Moreover, clearly $A$ is almost contained in $C$,
which is almost contained in $E$ by hypothesis. By the definition of the core-
residual decomposition, it follows that $A$ is almost contained in
$C\cap\text{core}_{h}(E)$, and taking complements again, it follows that
$I\setminus(C\cap\text{core}_{h}(E))$ is almost contained in $\cup_{\ell\in
L}\ell$. Thus, the family $L$ is an admissible ”cover” of
$I\setminus(C\cap\text{core}_{h}(E))$ for the measure $\mathcal{M}^{0}_{h}$.
Consequently, we have

$\mathcal{M}^{0}_{h}(I\setminus(C\cap\text{core}_{h}(E))\leq\sum_{\ell\in
L}h(|\ell|)\leq\mathcal{M}^{0}_{h}(I\setminus C)+\epsilon.$

The proof of the proposition is complete by letting $\epsilon$ tend to zero. ∎

######  Remark 3.6.

To make some sense of the above result, take $C=E$ in the statement above.
Then the conclusion is that the $\mathcal{M}^{0}_{h}$-size of what remains
from the interval $I$ after removing $E$, is in fact the same as the
$\mathcal{M}^{0}_{h}$-size of what remains after removing only
$\text{core}_{h}(E)$. Thus, in a sense, the set $\text{res}_{h}(E)$ is
invisible for the Hausdorff-type measure $\mathcal{M}^{0}_{h}$.

##  4 A measure construction of Frostman

In the next result, we extend slightly the classical construction of the
Frostman measures (refer to [6, page 302] or [5, Appendix D] for details). The
properties $(ii)$ and $(iii)$ in the proposition below appear classically as
features of the Frostman measures, and we obtain a useful additional feature
$(i)$ through a simple averaging process.

######  Proposition 4.1.

frostmanmeasureprop Let $U=\cup_{n}\alpha_{n}$ be an open subset of
$\mathbb{T}$, where $\\{\alpha_{n}\\}_{n}$ is a disjoint family of open
intervals. There exists a positive absolutely continuous Borel measure $\nu$
which satisfies the following three properties:

  1. (i)

$\nu=\sum_{n=1}^{N}c_{n}1_{\alpha_{n}}$, where $N$ is some finite integer, and
$c_{n}$ are positive numbers,

  2. (ii)

$\nu(\Delta)\leq h(|\Delta|)$ for each interval $\Delta$ of $\mathbb{T}$,

  3. (iii)

$\nu(\mathbb{T})\geq\mathcal{M}_{h,d}(U)/24$.

###### Proof.

By Mhdcont there exist a positive integer $N$ and closed intervals
$\\{d^{\prime}_{n}\\}_{n=1}^{N}$ such that $d^{\prime}_{n}\subset\alpha_{n}$,
and if $D=\cup_{n=1}^{N}d^{\prime}_{n}$, then
$\mathcal{M}_{h,d}(D)\geq\mathcal{M}_{h,d}(U)/2$. The usual Frostman measure
construction (see, for instance, [6, page 302]) will produce a measure
$\tilde{\nu}$ which is of the form
$\tilde{\nu}=\sum_{n=1}^{N}\tilde{\nu}_{n}$, with $\tilde{\nu}_{n}$ supported
on a closed interval $d_{n}$ living inbetween $d^{\prime}_{n}$ and
$\alpha_{n}$, which satisfies $\tilde{\nu}(\Delta)\leq 4h(|\Delta|)$ for any
interval $\Delta\subset\mathbb{T}$, and
$\tilde{\nu}(D)\geq\mathcal{M}_{h,d}(D)$. Since $D\subset U$, we also have
$\tilde{\nu}(U)=\tilde{\nu}(D)\geq\mathcal{M}_{h,d}(D)\geq\mathcal{M}_{h,d}(U)/2$.
Now we replace the measure $\tilde{\nu}$ by its average over every
$\alpha_{n}$. That is, we take
$c_{n}=\tilde{\nu}_{n}(\alpha_{n})|\alpha_{n}|^{-1}$, and we define

$\nu:=\sum_{n=1}^{N}c_{n}1_{\alpha_{n}}.$ (27)

It is obvious that $\nu(U)=\tilde{\nu}(U)$, and even that
$\nu(\alpha_{n})=\tilde{\nu}(\alpha_{n})$. Moreover, we have not destroyed the
smoothness of the measure either, as we will show that $\nu(\Delta)\leq
12h(|\Delta|)$ for any interval $\Delta$. To this end, fix
$n\in\\{1,2,\ldots,N\\}$ and let $\delta$ be any positive number which is
smaller than the length of the interval $\alpha_{n}$. Let $a$ and $b$ be the
endpoints of $\alpha_{n}$. By Frostman’s construction, we have that
$\tilde{\nu}_{n}([t,t+\delta])\leq 4h(\delta)$ for any $t\in[a,b-\delta]$. It
follows then that

$\int_{a}^{b-\delta}\tilde{\nu}_{n}([t,t+\delta])dt\leq
4(b-\delta-a)h(\delta)\leq 4|\alpha_{n}|h(\delta).$

The left-hand side of this equation can be re-written using Fubini’s theorem:

$\displaystyle\int_{a}^{b-\delta}\tilde{\nu}_{n}([t,t+\delta])dt=\int_{a}^{b-\delta}\int_{\alpha_{n}}1_{t\leq
s\leq t+\delta}\,d\tilde{\nu}_{n}(s)dt$
$\displaystyle=\int_{\alpha_{n}}\int_{a}^{b-\delta}1_{t\leq s\leq
t+\delta}\,dt\,d\tilde{\nu}_{n}(s)=\delta\tilde{\nu}_{n}(\alpha_{n})=\delta
c_{n}|\alpha_{n}|.$

Let $\Delta$ be any subinterval of $\alpha_{n}$ of length $|\Delta|=\delta$.
Then, from the above computations and (27), we get

$\nu(\Delta)=\delta c_{n}\leq 4h(|\Delta|).$

Finally, let $\Delta$ be an arbitrary interval. Then $\Delta$ can be split up
into at most three contiguous intervals
$\Delta_{1}\cup\Delta_{2}\cup\Delta_{3}$, such that $\Delta_{1}$ and
$\Delta_{3}$ are partially contained in some intervals $\alpha_{n_{1}}$ and
$\alpha_{n_{2}}$, and $\Delta_{2}$ totally covers the intervals $\alpha_{n}$
which lie inbetween $\alpha_{n_{1}}$ and $\alpha_{n_{2}}$. For the interval
$\Delta_{2}$ we have $\nu(\Delta_{2})=\tilde{\nu}(\Delta_{2})\leq
4h(|\Delta_{2}|)\leq 4h(|\Delta|)$, and we have above verified that
$\nu(\Delta_{1})\leq 4h(|\Delta_{1}|)$ and $\nu(\Delta_{3})\leq
4h(|\Delta_{3}|)$. Thus $\nu(\Delta)\leq 12h(|\Delta|)$. Upon division of the
measure $\nu$ by 12, we finally obtain a measure satisfying $\nu(\Delta)\leq
h(|\Delta|)$ for all intervals $\Delta\subset\mathbb{T}$, and
$\nu(U)\geq\mathcal{M}_{h,d}(U)/24$.

∎

##  5 A real-variables construction

In this section we follow more or less the same path as Khrushchev does in [9,
Sections 4 and 10]. We utilize the core-residual decomposition to handle the
complications that arise due to our dealing with a completely general
measurable subset $E$.

Before going into the main construction, we single out the following technical
lemma which appears in a slightly more complicated version in Khrushchev’s
work in [9, Section 10]. We include a proof for the readers convenience.

######  Lemma 5.1.

khrushchevsetlemma Let $F$ be a measurable subset of an interval $I$. For any
$\epsilon>0$, there exists a closed subset $B$ of $I$ which is almost
contained in $F$, and for any interval $\ell$ which is complementary to $B$ in
$I$ it holds that

$|\ell\cap F|\leq\epsilon|\ell|.$

###### Proof.

Consider the set $X$ of points $x$ which are contained in $I\setminus F$ and
which are Lebesgue density points of $I\setminus F$, in the sense that for the
intervals $J(x,r)$ with center $x$ and length $r$, we have

$\lim_{r\to 0}\frac{|J(x,r)\cap F|}{r}=\lim_{r\to 0}\frac{|J(x,r)\cap
F|}{|J(x,r)|}=0.$ (28)

The set $X$ has full measure in $I\setminus F$. For each such $x$ we pick an
interval $J(x)=J(x,r)$ with $r=r(x)$, such that the interval which has five
times the length has small intersection with $F$:

$|J(x,5r)\cap F|\leq\epsilon|J(x,r)|.$

Then the set

$B:=I\setminus(\cup_{x\in X}J(x))$

satisfies the required properties. Since $|X|=|I\setminus F|$, certainly $B$
is almost contained in $F$. Let $\ell$ be any interval complementary to $B$.
Then $\ell$ is the union of those intervals $J(x)$ which are contained in it.
By the classical $5r$-covering theorem (see [13, page 23]) there exists a
countable subfamily $\\{J(x_{n})\\}_{n}$ of pairwise disjoint intervals which
are contained in $\ell$ and such that $\ell$ is contained in the union
$\cup_{n}J(x_{n},5r)$. Hence

$|\ell\cap F|\leq\sum_{n}|J(x_{n},5r)\cap
F|\leq\epsilon\sum_{n}|J(x_{n},r)|\leq\epsilon|\ell|,$

where the last inequality follows by the disjointness of $J(x_{n})$. ∎

The following construction is similar to one appearing in Khrushchev’s work
[9, Section 4], but additional technical difficulties arise because of the
presence of the set $\text{core}_{h}(E)$. It is here that mainCRestimate will
be used to obtain a crucial lower bound on a certain mass.

######  Proposition 5.2.

fnsequenceprop Let $E$ be a measurable subset of $\mathbb{T}$. There exists a
sequence of real-valued bounded measurable functions $\\{f_{n}\\}_{n}$ on
$\mathbb{T}$ which satisfy the following properties:

  1. (i)

$\int_{\Delta}f_{n}\,d\textit{m}\leq 2h(|\Delta|)$ for any interval $\Delta$
of $\mathbb{T}$,

  2. (ii)

$\int_{\mathbb{T}}f_{n}\,d\textit{m}=0$,

  3. (iii)

$f_{n}(x)=0$ for almost every $x\in\text{core}_{h}(E)$,

  4. (iv)

$f_{n}(x)\leq 0$ for almost every $x\in\text{res}_{h}(E)$,

  5. (v)

$\lim_{n\to\infty}f_{n}(x)=-\infty$ uniformly for $x\in\text{res}_{h}(E)$, up
to a set of Lebesgue measure zero.

###### Proof.

Fix a positive integer $n$ and recall the definitions (20) and (21) of the
$n$th generation dyadic intervals. We will show how to define $f_{n}$ piece by
piece on the interior $I$ of any dyadic interval in $\mathcal{D}_{n}$. If
$|I\cap\text{res}_{h}(E)|=0$, then we simply set $f_{n}\equiv 0$ on $I$.

In the case $|I\cap\text{res}_{h}(E)|>0$, we apply khrushchevsetlemma with
data $F=I\cap E$ and $\epsilon=1/2$. The lemma produces a closed set
$B:=H_{I}$ contained in $I$ and almost contained in $E$, with the property
that if $\ell$ is an interval complementary to $H_{I}$ in $I$, then $|\ell\cap
E|\leq|\ell|/2$. We next apply frostmanmeasureprop to the open set
$U=I\setminus H_{I}$. Let

$\tilde{\nu}_{I}=\sum_{n=1}^{N}c_{n}1_{\alpha_{n}}$

be the measure produced by that proposition, where $\alpha_{n}$ are open arcs
complementary to $H_{I}$ and contained in $I$. This measure satisfies

$\tilde{\nu}_{I}(\Delta)\leq h(|\Delta|)$ (29)

for any interval $\Delta$ of $\mathbb{T}$, and

$\tilde{\nu}_{I}(I)\geq\mathcal{M}_{h,d}(I\setminus H_{I})/24.$ (30)

We produce a new smaller measure $\nu_{I}$ by restricting $\tilde{\nu}_{I}$ to
the set $I\setminus E$. Thus, we set $\beta_{n}=\alpha_{n}\cap(I\setminus E)$
and let

$\nu_{I}:=\sum_{n=1}^{N}c_{n}1_{\beta_{n}}.$

Note that

$\displaystyle\tilde{\nu}_{I}(I)=\sum_{n=1}^{N}c_{n}|\alpha_{n}|=\nu_{I}(I)+\sum_{n=1}^{N}c_{n}|\alpha_{n}\cap
E|$
$\displaystyle\leq\nu_{I}(I)+\Big{(}\sum_{n=1}^{N}c_{n}|\alpha_{n}|\Big{)}/2=\nu_{I}(I)+\tilde{\nu}_{I}(I)/2$

where we used that $|\alpha_{n}\cap E|\leq|\alpha_{n}|/2$, a consequence of
khrushchevsetlemma. Hence we obtain that

$\nu_{I}(I)\geq\tilde{\nu}_{I}(I)/2.$

Together with (30), this gives

$\nu_{I}(I)\geq\mathcal{M}_{h,d}(I\setminus H_{I})/48,$

and clearly

$\mathcal{M}_{h,d}(I\setminus H_{I})/48\geq\mathcal{M}^{0}_{h}(I\setminus
H_{I})/48.$

Further, using mainCRestimate and mhinequalities we obtain

$\displaystyle\mathcal{M}^{0}_{h}(I\setminus
H_{I})=\mathcal{M}^{0}_{h}(I\setminus(H_{I}\cap\text{core}_{h}(E)))$
$\displaystyle\geq\mathcal{M}^{0}_{h}(I\setminus\text{core}_{h}(E))\geq
h(|I\setminus\text{core}_{h}(E)|).$

All in all, the string of inequalities gives us a lower estimate for the mass
that $\nu_{I}$ places on $I$:

$\nu_{I}(I)\geq\frac{h(|I\setminus\text{core}_{h}(E)|)}{48}.$ (31)

We define $f_{n}$ on the interior of the interval $I$ by the formula

$f_{n}|I=\sum_{n=1}^{N}c_{n}1_{\beta_{n}}-\frac{\nu_{I}(I)}{|I\cap\text{res}_{h}(E)|}1_{I\cap\text{res}_{h}(E)}.$
(32)

For definiteness let us put $f_{n}$ to equal zero on the endpoints of $I$. The
functions $f_{n}$ are clearly bounded. Our construction ensures that

$\int_{I}f_{n}dm=0$ (33)

for any $I$ which is a dyadic arc of $n$th generation. Consequently,

$\int_{\mathbb{T}}f_{n}d\textit{m}=0$

also holds, so $(ii)$ above is satisfied. The properties $(iii)$ and $(iv)$
also hold. If $\Delta$ is an interval and it is contained in a $n$th
generation dyadic interval $I$, then

$\int_{\Delta}f_{n}d\textit{m}=\nu_{I}(\Delta)-\sum_{n=1}^{N}c_{n}|\beta_{n}\cap\Delta|\leq\tilde{\nu}_{I}(\Delta)\leq
h(|\Delta|).$

If $\Delta$ is an interval, but it is not contained in any single such
interval $I$, then we can split $\Delta$ up into contiguous intervals
$\Delta_{1}\cup\Delta_{2}\cup\Delta_{3}$, such that $\Delta_{1}$ and
$\Delta_{3}$ are contained in some dyadic intervals
$I_{1},I_{2}\in\mathcal{D}_{n}$, and $\Delta_{2}$ is a union of the dyadic
intervals in $\mathcal{D}_{n}$ which lie inbetween $I_{1}$ and $I_{2}$. Then,
using (33), we can estimate

$\int_{\Delta}f_{n}d\textit{m}=\int_{\Delta_{1}}f_{n}d\textit{m}+\int_{\Delta_{3}}f_{n}d\textit{m}\leq
2h(|\Delta|).$

So $(i)$ also holds. Finally, consider $x\in\text{res}_{h}(E)$. Using (31),
the obvious inequality

$|I\setminus\text{core}_{h}(E)|\geq|I\cap\text{res}_{h}(E)|$

and the properties of $h$ postulated in Section 2, we have

$\displaystyle-
f_{n}(x)=\frac{\nu_{I}(I)}{|I\cap\text{res}_{h}(E)|}\geq\frac{h(|I\setminus\text{core}_{h}(E)|}{48|I\cap\text{res}_{h}(E)|}\geq\frac{h(|I\cap\text{res}_{h}(E)|)}{48|I\cap\text{res}_{h}(E)|}\geq\frac{h(|I|)}{48|I|}.$

But $|I|=2\pi 2^{-n}$, and consequently by the assumption that $\lim_{t\to
0^{+}}h(t)/t=\infty$ we conclude that

$\lim_{n\to\infty}-f_{n}(x)=\infty$

for $x\in\text{res}_{h}(E)$. This settles $(v)$, and the proof is complete. ∎

##  6 A sequences which splits $\mathcal{P}^{2}(\mu)$

We will now show that for a large family of measures $\mu$, the condition that
$|\text{res}_{h}(E)|>0$ for the carrier set of $E$ of $\mu_{\mathbb{T}}$
implies that $\mathcal{P}^{2}(\mu)$ contains an $L^{2}$-summand. We start by
recalling some well-known results which will be used in the proof.

For a real-valued measure $\nu$ living on $\mathbb{T}$ we define its Herglotz
integral by the formula

$H_{\nu}(z):=\int_{\mathbb{T}}\frac{\zeta+z}{\zeta-z}d\nu(\zeta),\quad
z\in\mathbb{D}.$

The function $H_{\nu}$ is analytic in $\mathbb{D}$, and its real part
$\operatorname{Re}(H_{\nu})$ is the usual Poisson integral $P_{\nu}$ of $\nu$.
It is well-known that if $d\nu=fd\textit{m}$ is absolutely continuous, then
$|H_{\nu}(z)|=\exp(|f(z)|)$ for almost every $z\in\mathbb{T}$, in the sense of
the usual non-tangential boundary values.

The following estimate for the growth of $P_{\nu}$ in terms of smoothness of
the measure $\nu$ (or more precisely, smoothness of its positive part) is also
well-known (see, for instance, [6, page 297]).

######  Lemma 6.1.

poissonintGrowth Let $h$ be a measure function satisfying the properties
stated in Section 2. There exists an absolute constant $C>0$ such that if
$\nu(\Delta)\leq h(|\Delta|)$ for any interval $\Delta$ in $\mathbb{T}$, then

$P_{\nu}(z)=\operatorname{Re}(H_{\nu}(z))\leq C\frac{h(1-|z|)}{1-|z|},\quad
z\in\mathbb{D}.$ (34)

We will also need the following approximation result which is a direct
consequence of the classical Beurling-Wiener theorem. For a proof, see [7],
for instance.

######  Lemma 6.2.

beurling-wiener Let $E$ be a measurable subset of $\mathbb{T}$ which satisfies
$|E|<|\mathbb{T}|$. If a function $g\in L^{2}(E)$ is non-zero almost
everywhere on a set $F\subset E$, then the norm-closure of the analytic
polynomial multiples of $g$ in $L^{2}(E)$ contains the space $L^{2}(F)$.

We now state the main result of this section, and the most general result of
the paper.

######  Proposition 6.3.

p2musplittheorem Let $\mu=\mu_{\mathbb{D}}+\mu_{\mathbb{T}}$ be a positive
finite Borel measure on the closed disk $\overline{\mathbb{D}}$, where
$d\mu_{\mathbb{T}}=w\,d\textit{m}$ is absolutely continuous, and
$\mu_{\mathbb{D}}$ is such that for all bounded analytic functions $p$, the
estimate

$\int_{\mathbb{D}}|p(z)|^{2}d\mu_{\mathbb{D}}(z)\leq
c_{1}\sup_{z\in\mathbb{D}}|p(z)|^{2}\exp\Bigg{(}-c_{2}\frac{h(1-|z|)}{1-|z|}\Bigg{)}$
(35)

holds for some measure function $h$ and some positive constants $c_{1},c_{2}$.
Let $E$ be a carrier set of $w$, i.e., a measurable subset of $\mathbb{T}$
such that $w=w1_{E}$ almost everywhere, and
$1_{\text{res}_{h}(E)}\cdot\mu_{\mathbb{T}}$ be the restriction of
$\mu_{\mathbb{T}}$ to the set $\text{res}_{h}(E)$. If $|\text{res}_{h}(E)|>0$,
then the space $L^{2}(1_{\text{res}_{h}(E)}\cdot\mu_{\mathbb{T}})$ is
contained in $\mathcal{P}^{2}(\mu)$.

Before going into the proof, we make two remarks.

######  Remark 6.4.

Firstly, note that in the case $h(t)=t\log(e/t)$, the inequality (35) is
satisfied for some small $c_{2}>0$ for any measure $\mu_{\mathbb{D}}$ of the
form

$d\mu_{\mathbb{D}}=(1-|z|)^{\alpha}dA,\quad\alpha>-1.$ (36)

Indeed, in that case the exponential expression in (35) equals a constant
multiple of $(1-|z|)^{c_{2}}$, and the estimate holds for $\alpha-c_{2}>-1$,
since

$\displaystyle\int_{\mathbb{D}}|p(z)|^{2}(1-|z|)^{\alpha}dA(z)\leq\sup_{z\in\mathbb{D}}|p(z)|^{2}(1-|z|)^{c_{2}}\int_{\mathbb{D}}(1-|z|)^{\alpha-
c_{2}}dA(z)$ $\displaystyle\leq
c_{1}\,\sup_{z\in\mathbb{D}}|p(z)|^{2}(1-|z|)^{c_{2}},$

where

$c_{1}=\int_{\mathbb{D}}(1-|z|)^{\alpha-c_{2}}dA(z)<\infty.$

######  Remark 6.5.

Secondly, in fact $L^{2}(\text{res}_{h}(E))$ is the biggest $L^{2}$-summand
contained in $\mathcal{P}^{2}(\mu)$ in the case when $w$ is the characteristic
function of $E$, $h(t)=t\log(e/t)$ and $\mu_{\mathbb{D}}$ is of the form (36).
We will elaborate on this in the next section, thus proving the first of the
main theorems stated in the introduction.

###### Proof of p2musplittheorem.

We start by noting that if $g$ is a bounded analytic function in $\mathbb{D}$,
then the function in $L^{2}(\mu)$ which coincides with $g$ in $\mathbb{D}$ and
(almost everywhere) with the usual non-tangential boundary values of $g$ on
$\mu_{\mathbb{T}}$, is actually contained in $\mathcal{P}^{2}(\mu)$. This
follows easily by dilating $g_{r}(z):=g(rz)$ and approximating the dilated
functions uniformly in $\overline{\mathbb{D}}$ by analytic polynomials.

Let $\\{f_{n}\\}_{n}$ be the sequence of real-valued bounded functions
constructed in fnsequenceprop. By property $(v)$ in that proposition, there
exists an increasing sequence of integers $\\{n_{k}\\}_{k}$ such that

$\exp(f_{n_{k}}(\zeta)/k)\leq 1/k,\text{ for almost every
}\zeta\in\text{res}_{h}(E).$ (37)

We construct the bounded analytic functions

$g_{k}(z):=\exp\big{(}H_{f_{n_{k}}}(z)/k\big{)},\quad z\in\mathbb{D}$ (38)

By property $(i)$ in fnsequenceprop and poissonintGrowth, for sufficiently
large $k$ the functions $g_{k}$ satisfy a growth estimate

$|g_{k}(z)|^{2}\leq\exp\Big{(}\frac{4C}{k}\frac{h(1-|z|)}{1-|z|}\Big{)}\leq\exp\Big{(}c_{2}\frac{h(1-|z|)}{1-|z|}\Big{)},\quad
z\in\mathbb{D}.$ (39)

The functions $g_{k}$ are bounded analytic functions in $\mathbb{D}$, so they
are contained in $\mathcal{P}^{2}(\mu)$ in the sense explained at the
beginning of the proof. In terms of boundary values, the functions $g_{k}(z)$
converge uniformly almost everywhere to 0 on $\text{res}_{h}(E)$, and by
property $(iii)$ in fnsequenceprop, we have $|g_{k}(z)|=1$ almost everywhere
on $\text{core}_{h}(E)$, thus $|g_{k}(z)|\leq 1$ almost everywhere on
$\mu_{\mathbb{T}}$. Keeping in mind the hypothesis (35) stated in the lemma,
we conclude that $\\{g_{k}\\}_{k}$ is a norm-bounded sequence in
$\mathcal{P}^{2}(\mu)$. The estimate (39) shows also that the family
$\\{g_{k}\\}_{k}$ is pointwise bounded uniformly on compact subsets of
$\mathbb{D}$, hence we can extract a subsequence of $\\{g_{k}\\}_{k}$ which
converges uniformly on compacts to some analytic function and also converges
weakly to a function $g\in\mathcal{P}^{2}(\mu)$. The function $g$ vanishes
almost everywhere on $\text{res}_{h}(E)$, by the uniform convergence of the
functions $g_{k}$ to zero on $\text{res}_{h}(E)$. By property $(ii)$ of
fnsequenceprop, the functions $g_{k}$ satisfy $g_{k}(0)=1$, and so $g(0)=1$.
Next, the inequality in (39) shows that

$|g(z)|=\lim_{k\to\infty}|g_{k}(z)|\leq\lim_{k\to\infty}\exp\Big{(}\frac{4C}{k}\frac{h(1-|z|)}{1-|z|}\Big{)}=1.$

By the maximum modulus principle for analytic functions it follows now that
$g(z)=1$, for all $z\in\mathbb{D}$.

Consider now $1-g\in\mathcal{P}^{2}(\mu)$. This function lives only on
$\mu_{\mathbb{T}}$, and it equals to $1$ almost everywhere on
$\text{res}_{h}(E)$. It follows that the entire subspace $\mathcal{M}\subseteq
L^{2}(\mu_{\mathbb{T}})$, which is the norm-closure of analytic polynomial
multiples of $1-g$, is contained in $\mathcal{P}^{2}(\mu)$. The linear mapping
$T$ defined by $f\mapsto f\sqrt{w}$ is a unitary map between
$L^{2}(\mu_{\mathbb{T}})$ and $L^{2}(E)$, and it preserves subspaces invariant
under multiplication by analytic polynomials. If $|\text{res}_{h}(E)|>0$, then
certainly $|E|<|\mathbb{T}|$, so beurling-wiener applies to tell us that
$T\mathcal{M}$ equals $L^{2}(F)$, for some measurable subset $F$ which
contains $\text{res}_{h}(E)$. By applying the inverse mapping $T^{-1}$ to
$L^{2}(F)$ we get that $L^{2}(1_{\text{res}_{h}(E)}\cdot\mu_{\mathbb{T}})$ is
contained in $\mathcal{P}^{2}(\mu)$. ∎

##  7 Proof of Theorem A

We specialize to the case

$h(t)=t\log(e/t).$ (40)

For this choice, the $h$-Carleson sets are also known as Beurling-Carleson
sets, or simply Carleson sets, as was noted in the introduction. We specialize
also to the measures $\mu$ supported on the closed disk
$\overline{\mathbb{D}}$ which are of the form

$d\mu=d\mu_{\mathbb{D}}+d\mu_{\mathbb{T}}=(1-|z|)^{\alpha}dA+1_{E}d\textit{m},$
(41)

where $\alpha>-1$ and $E$ is any measurable subset of $\mathbb{T}$.

The lemma below is a direct consequence of, for instance, [12, Corollary 5.3].
It is also more or less known as a folklore theorem, and is a consequence of
Khrushchev’s work [9, Section 3] in slightly greater generality than we shall
present it here. However, we have not found a precise reference for these more
general statements, and so we present only what is necessary to complete the
proof of our main theorem. The interested reader is invited to analyze
Khrushchev’s proof in [9] to extract a more general statement which involves
other measure functions $h$.

######  Lemma 7.1.

p2munonsplitBC If $E$ is a Beurling-Carleson set, then the space
$\mathcal{P}^{2}(\mu)$ with measure $\mu$ of the form (41) will contain no
characteristic functions of any subset of $\overline{\mathbb{D}}$, except the
trivial ones: the zero function and the characteristic function of the carrier
set, i.e., $1\in\mathcal{P}^{2}(\mu)$.

We are now ready to prove our our main result, Theorem A of Section 1. We
restate it for convenience.

######  Theorem 7.2.

theoremArestated Let $\mu$ be of the form (41). Assume that for $h$ of the
form (40) we have $|\text{res}_{h}(E)|>0$. Let

$d\mu_{1}:=d\mu-1_{\text{res}_{h}(E)}d\textit{m}=d\mu_{\mathbb{D}}+1_{\text{core}_{h}(E)}d\textit{m}.$

Then

$\mathcal{P}^{2}(\mu)=L^{2}(\text{res}_{h}(E))\oplus\mathcal{P}^{2}(\mu_{1})$
(42)

where $\mathcal{P}^{2}(\mu_{1})$ contains no non-trivial characteristic
functions.

###### Proof.

By p2musplittheorem and the remarks appearing after its statement, the space
$\mathcal{P}^{2}(\mu)$ contains $L^{2}(\text{res}_{h}(E))$. The orthogonal
complement of this space in $\mathcal{P}^{2}(\mu)$ is obviously
$\mathcal{P}^{2}(\mu_{1})$, the point of contention is only if
$\mathcal{P}^{2}(\mu_{1})$ contains a non-trivial characteristic function or
not. It is easy to see that if $\mathcal{P}^{2}(\mu_{1})$ contains a non-
trivial characteristic function, then it will contain the characteristic
function $1_{G}$ of some measurable subset $G$ of $\text{core}_{h}(E)$, since
the functions in $\mathcal{P}^{2}(\mu)$ are analytic in $\mathbb{D}$.

Let $\text{core}_{h}(E)=\cup_{n}{E_{n}}$, where the $E_{n}$ are Beurling-
Carleson sets. Fix a sequence of analytic polynomials $\\{p_{n}\\}_{n}$ which
converges to $1_{G}$ in the norm of $\mathcal{P}^{2}(\mu_{1})$. Then this
sequence converges also in the norm induced by the smaller measures
$d\mu_{n}:=d\mu_{\mathbb{D}}+1_{E_{n}}d\textit{m}$, to the characteristic
function $1_{G\cap E_{n}}$. By p2munonsplitBC the sequence must converge to
$0$, i.e., $|G\cap E_{n}|=0$. Since this is true for all $n$,
$|G\cap\text{core}_{h}(E)|=0$, and so $|G|=0$. ∎

We have thus completed the proof of Theorem A of the introduction.

##  8 Proof of Theorem B

Theorem B stated in the introduction will follow easily from previous results
found in [12], and from the next lemma.

######  Lemma 8.1.

lemmaForTheoremB Let $E$ be a measurable subset of $\mathbb{T}$, and $f\in
L^{2}(E)$ be such that the Cauchy transform $C_{f}$ has a finite Dirichlet
integral:

$\int_{\mathbb{D}}|C^{\prime}_{f}|^{2}dA<\infty.$

Then $f$ vanishes almost everywhere on $\text{res}(E)$.

###### Proof.

A simple computation shows that if $g$ is an analytic function in
$\mathbb{D}$, and $\\{g_{n}\\}_{n}$ is the sequence of its Taylor
coefficients, then we have

$\int_{\mathbb{D}}|g|^{2}dA=\sum_{n=0}^{\infty}\frac{|g_{n}|^{2}}{n+1}.$ (43)

Thus, if $f(\zeta)=\sum_{n\in\mathbb{Z}}f_{n}\zeta^{n}$ is the Fourier
expansion of $f$, then the finiteness of the Dirichlet integral of $C_{f}$ is
equivalent to

$\sum_{n=0}^{\infty}(n+1)|f_{n}|^{2}<\infty.$ (44)

Let $p(z)=\sum_{n=0}^{N}p_{n}z^{n}$ be any analytic polynomial. We can use
Cauchy-Schwarz inequality to estimate

$\displaystyle\Big{|}\sum_{n=0}^{N}f_{n}\overline{p_{n}}\Big{|}\leq\Big{(}\sum_{n=0}^{\infty}(n+1)|f_{n}|^{2}\Big{)}^{1/2}\Big{(}\sum_{n=0}^{N}\frac{|p_{n}|^{2}}{n+1}\Big{)}^{1/2}$
$\displaystyle\lesssim\Big{(}\int_{\mathbb{D}}|C^{\prime}_{f}|^{2}dA\Big{)}^{1/2}\Big{(}\int_{\mathbb{D}}|p|^{2}dA\Big{)}^{1/2}$
(45)

Now consider $h\in L^{2}(\text{res}(E))$. By Parseval’s identity, and the fact
that $f$ lives only on the set $E$, we have

$\displaystyle\int_{E}f\overline{h}\,d\textit{m}=\int_{E}f\overline{(h-p)}\,d\textit{m}+\int_{E}f\overline{p}\,d\textit{m}$
$\displaystyle=\int_{E}f\overline{(h-p)}+\sum_{n=0}^{N}f_{n}\overline{p_{n}}.$
(46)

By theoremArestated for the case $\alpha=0$, for any $\epsilon>0$ we can find
an analytic polynomial $p$ which satisfies

$\int_{\mathbb{D}}|p|^{2}dA<\epsilon$ (47)

and

$\int_{E}|p_{n}-h|^{2}d\textit{m}<\epsilon.$ (48)

Consequently, a combination of (45), (46), (47) and (48) shows that

$\int_{E}f\overline{h}\,d\textit{m}=0,$

for any $h\in L^{2}(\text{res}(E))$. Thus $f$ must vanish almost everywhere on
$\text{res}(E)$. ∎

###### Proof of Theorem B.

A set of functions in $S_{+}(E)$ which live only on $\text{core}(E)$ and are
norm-dense in $L^{2}(\text{core}(E))$ has been constructed in [12].
Conversely, lemmaForTheoremB shows that if the Cauchy transform $C_{f}$ of any
function $f\in L^{2}(E)$ has a finite Dirichlet integral, then $f$ lives on
$\text{core}(E)$. The theorem is thus proved. ∎

##  9 Proof of Theorem C

Let $D$ be the space of analytic functions in $\mathbb{D}$ with a finite
Dirichlet integral in (14). We shall give two proofs of Theorem C stated in
the introduction.

Firstly, the validity of the claim in Theorem C follows immediately from
p2musplittheorem and the main result in [11] which characterizes the norm-
density of $D\cap\mathcal{H}(b)$ in $\mathcal{H}(b)$ in terms of certain
operator-theoretic properties of the multiplication operator
$M_{z}:\mathcal{P}^{2}(\mu)\to\mathcal{P}^{2}(\mu)$ for the measure

$d\mu=dA+(1-|b|^{2})d\textit{m},$ (49)

the operator $M_{z}$ being one of multiplication by the coordinate function z:

$M_{z}f(z)=zf(z).$

A necessary condition for density of $D\cap\mathcal{H}(b)$ is, according to
[11], that $M_{z}:\mathcal{P}^{2}(\mu)\to\mathcal{P}^{2}(\mu)$ admits non non-
trivial invariant subspace on which it acts as an isometry. Since the set

$E:=\\{\zeta\in\mathbb{T}:|b(\zeta)|<1\\}$ (50)

is the carrier set on $\mathbb{T}$ of the weight $1-|b|^{2}$, p2musplittheorem
implies that if $|\text{res}(E)|>0$, then $\mathcal{P}^{2}(\mu)$ admits such a
subspace. So in that case $D\cap\mathcal{H}(b)$ cannot be dense in
$\mathcal{H}(b)$.

We can however give a proof of Theorem C which utilizes Theorem B and only the
classical theory of $\mathcal{H}(b)$-spaces, passing around the abstract
formulation in [11]. Since we have already seen one proof, we give only a
sketch of the next argument. For references to the facts about
$\mathcal{H}(b)$s-spaces used below, one can consult [3] and [4].

###### Sketch of an alternative proof of Theorem C.

Let $\Delta(\zeta):=\sqrt{1-|b(\zeta)|^{2}},\zeta\in\mathbb{T}$. It is well-
known that for each $f\in L^{2}(E)$, with $E$ given by (50), the Cauchy
transform $C_{f\Delta}$ of the function $f\Delta$ (see (12)) is contained in
$\mathcal{H}(b)$. The linear manifold of all such functions is usually denoted
by $\mathcal{H}(\overline{b})$ in the literature. If $b$ is a so-called
extreme point of the unit ball of $H^{\infty}$ (which is the only case of
interest for us), then it is known that the transform $f\mapsto C_{f\Delta}$
is one-to-one on $L^{2}(E)$, and we have

$\|C_{f\Delta}\|_{\mathcal{H}(b)}\leq\|C_{f\Delta}\|_{\mathcal{H}(\overline{b})}:=\|f\|_{L^{2}(E)}.$
(51)

If $T_{\overline{b}}$ is the usual Toeplitz operator with the co-analytic
symbol $\overline{b}$, then
$T_{\overline{b}}:\mathcal{H}(b)\to\mathcal{H}(\overline{b})$ is a
contraction, in particular it is a bounded operator. It is also known that the
space $D$ is invariant for any Toeplitz operator with a bounded co-analytic
symbol.

Assume that $D\cap\mathcal{H}(b)$ is dense in $\mathcal{H}(b)$. Our task is
then to show that the set $E$ given by (50) is residual-free. Let
$\\{g_{n}\\}_{n}$ be a sequence in $D\cap\mathcal{H}(b)$ which tends to the
function $C_{1_{E}\Delta}\in\mathcal{H}(b)$ in the norm of $\mathcal{H}(b)$.
By the invariance of $D$ for the operator $T_{\overline{b}}$, it follows that
the sequence of $D$-functions $T_{\overline{b}}g_{n}$ tends to
$T_{\overline{b}}C_{\Delta}=C_{\overline{b}\Delta}$ in the norm of
$\mathcal{H}(\overline{b})$. Since the functions $T_{\overline{b}}g_{n}$ are
in $\mathcal{H}(\overline{b})$, they are Cauchy transforms of the form
$T_{\overline{b}}g_{n}=C_{f_{n}\Delta}$, for some $f_{n}\in L^{2}(E)$. By the
norm-equality in (51) we have that $f_{n}\to\overline{b}$ in $L^{2}(E)$, and
so we can assume that $f_{n}\to\overline{b}$ pointwise almost everywhere on
$E$ by passing to a subsequence. By Theorem B and the fact that
$C_{f_{n}\Delta}=T_{\overline{b}}g_{n}\in D$ we must have $f_{n}\equiv 0$ on
$\text{res}(E)$. But $\overline{b}\neq 0$ almost everywhere on $E$ (since $b$
is analytic), so we see that $\text{res}(E)$ must have Lebesgue measure zero,
as claimed in the statement of Theorem C. ∎

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Bartosz Malman,
KTH Royal Institute of Technology,
Stockholm, Sweden
<EMAIL_ADDRESS>