# Restrictions of Békollé–Bonami weights and Bloch functions

Alberto Dayan  Fachrichtung Mathematik, Universität des Saarlandes, 66123
Saarbrücken, Germany<EMAIL_ADDRESS>,  Adrián Llinares  Department of
Mathematics and Mathematical Statistics, Umeå University, SE-90736 Umeå,
Sweden<EMAIL_ADDRESS>and  Karl-Mikael Perfekt  Department of
Mathematical Sciences, Norwegian University of Science and Technology, 7491
Trondheim, Norway<EMAIL_ADDRESS>
###### Abstract.

We characterize the restrictions of Békollé–Bonami weights of bounded
hyperbolic oscillation, to subsets of the unit disc, thus proving an analogue
of Wolff’s restriction theorem for Muckenhoupt weights. Sundberg proved a
discrete version of Wolff’s original theorem, by characterizing the trace of
$\operatorname*{BMO}$-functions onto interpolating sequences. We consider an
analogous question in our setting, by studying the trace of Bloch functions.
Through Makarov’s probabilistic approach to the Bloch space, our question can
be recast as a restriction problem for dyadic martingales with uniformly
bounded increments.

###### Key words and phrases:

Békollé–Bonami weights, Rubio de Francia factorization, Bloch space,
interpolating sequences

###### 2020 Mathematics Subject Classification:

46E30 (Primary); 30E05, 30H30, 42A61, 47B38 (Secondary)

##  1\. Introduction

For $1<p<\infty$, a positive measurable weight $w$ on the unit circle
$\mathbb{T}$ is said to be a Muckenhoupt $A_{p}$-weight if

$[w]_{A_{p}}:=\sup_{I\subset\mathbb{T}}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I}w\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I}w^{-\frac{1}{p-1}}\right)^{p-1}<\infty,$

where the supremum is taken over all subarcs $I$ of the unit circle, $|I|$
denotes the normalized arc length of $I$ and

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I}w:=\frac{1}{|I|}\int_{I}w(\theta)\frac{d\theta}{2\pi}$

denotes the average of $w$ on it. When $p=1$, we say that $w\in A_{1}$ if
there is a constant $C=[w]_{A_{1}}$ such that $\mathcal{M}w(\theta)\leq
Cw(\theta)$ for a.e. $\theta\in\mathbb{T}$, where $\mathcal{M}w$ is the
Hardy–Littlewood maximal function of $w$,

$\mathcal{M}w(\theta)=\sup_{I\ni\theta}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I}w.$

Muckenhoupt weights are extremely important, because, for $1<p<\infty$, it
holds that $w\in A_{p}$ if and only if the maximal function $\mathcal{M}$ is
bounded on $L^{p}(\mathbb{T},w\,d\theta)$, and, furthermore, if and only if
the Hilbert transform is bounded $L^{p}(\mathbb{T},w\,d\theta)$. In a famous
unpublished preprint, circa 1980, Wolff characterized the restrictions of
$A_{p}$-weights to subsets of the unit circle.

######  Theorem 1.1 (Wolff, as cited in [8]).

Let $w\geq 0$ be a weight on a measurable subset $\Omega\subset\mathbb{T}$,
and let $1\leq p<\infty$. Then the following are equivalent.

  1. (1)

There exists $W\in A_{p}$ such that $w(\theta)=W(\theta)$ for almost every
$\theta\in\Omega$.

  2. (2)

There exists $q>1$ such that $w^{q}\in A_{p,\Omega}$. That is, if $p>1$, it
holds that

$\sup_{I\subset\mathbb{T}}\left(\frac{1}{|I|}\int_{I\cap\Omega}w^{q}\right)\left(\frac{1}{|I|}\int_{I\cap\Omega}w^{-\frac{q}{p-1}}\right)^{p-1}<\infty,$

and similarly for $p=1$.

Following Békollé and Bonami [3], for $1<p<\infty$, the $B_{p}$-class is
defined as the family of all weights $w$ in the unit disc such $\mathbb{D}$
such that

(1.1) $[w]_{B_{p}}:=\sup_{I\subset\mathbb{T}}\leavevmode\nobreak\
\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}w\,dA\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}w^{-\frac{1}{p-1}}\,dA\right)^{p-1}<\infty,$

where $dA$ denotes normalized area measure on $\mathbb{D}$, and for an arc
$I$, $S(I)$ denotes the corresponding Carleson square, that is, the sector

$S(I):=\left\\{z\in\mathbb{D}\,:\,\frac{z}{|z|}\in I,1-|z|<|I|\right\\}.$

For $p=1$, we say that $w\in B_{1}$ if there is a constant $C=[w]_{B_{1}}$
such that $Mw(z)\leq Cw(z)$ for a.e. $z\in\mathbb{D}$, where $M$ now denotes
the associated maximal function,

$Mw(z):=\sup_{S(I)\ni z}\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)}w\,dA.$

As for Muckenhoupt weights, for $1<p<\infty$, we have that $w\in B_{p}$ if and
only if $M$ is bounded on $L^{p}(\mathbb{D},w\,dA)$, and, furthermore, if and
only if the Bergman projection is bounded on this weighted $L^{p}$-space.

However, the analogy stops at a certain point, since Békollé–Bonami weights
can be much more irregular than Muckenhoupt weights. In particular, they can
easily fail to satisfy the reverse Hölder and self-improvement properties that
$A_{p}$-weights enjoy (cf. Theorem 1.1 with $\Omega=\mathbb{T}$). Recently, it
was discovered that these desirable properties can be recovered [2] if we
consider weights $w\in B_{p}$ which are of _bounded hyperbolic oscillation_ ,
that is, weights for which there is a constant $L>0$ such that

(1.2) $\big{|}\log w(z)-\log w(\zeta)\big{|}\leq
L\big{(}1+\beta(z,\zeta)\big{)},\quad\textrm{a.e. }z,\zeta\in\mathbb{D},$

where $\beta$ denotes the hyperbolic metric on $\mathbb{D}$. Békollé–Bonami
weights satisfying (1.2), as well as their connection with the Bloch space,
were further explored in [13].

Under the regularity assumption (1.2), the main theorem of this article
establishes the analogue of the Wolff restriction theorem for quite general
subsets $\Omega\subset\mathbb{D}$. Namely, given a collection $\mathcal{J}$ of
subarcs $I\subset\mathbb{T}$, we will consider sets $\Omega$ of the type

(1.3) $\Omega=\bigcup_{I\in\mathcal{J}}T(I),$

where $T(I)$, as illustrated in Figure 1, is the top half of the Carleson
square $S(I)$,

$T(I):=\left\\{z\in\mathbb{D}\,:\,\frac{1}{2}<\dfrac{1-|z|}{|I|}\leq
1,\dfrac{z}{|z|}\in I\right\\}.$

$\frac{2-|I|}{2}$$1-|I|$$\bullet$0$\mathbb{T}$$T(I)$$S(I)$$\bullet$ Figure 1.
Example of $S(I)$, $T(I)$ and their centre.

######  Theorem 1.2.

Let $\Omega$ be a union (1.3) of top-halves of Carleson squares, and consider
a weight $w$ of bounded hyperbolic oscillation on $\Omega$, that is, a
measurable function $w\colon\Omega\to(0,\infty)$ satisfying (1.2) for points
$z,\zeta\in\Omega$. Then, for $1\leq p<\infty$, the following statements are
equivalent:

  1. (1)

There exists a weight $W\in B_{p}$ of bounded hyperbolic oscillation such that
$w(z)=W(z)$ for almost every $z\in\Omega$.

  2. (2)

There exists $q>1$ such that $w^{q}\in B_{p,\Omega}$. That is, if $p>1$, it
holds that

$\sup_{I\subset\mathbb{T}}\;\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w^{q}\>dA\left(\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w^{-\frac{q}{p-1}}\>dA\right)^{p-1}<\infty,$

and similarly for $p=1$.

To prove this theorem, we will in Section 2 first consider a dyadic setting in
which we are able to prove the analogous result by essentially following Rubio
de Francia’s approach to Muckenhoupt weights [8, Chapter IV]. In Section 3, we
will then prove our extension result by averaging over a continuous family of
dyadic problems. A noteworthy feature of the averaging argument is that we are
able to pass from quite special subsets $\Omega$ in the dyadic setting, unions
of dyadic top-halves, to quite general subsets in the continuous setting,
namely, unions of _any_ top-halves.

As a by-product of our techniques, in Corollary 3.10, we also prove that any
$w\in B_{p}$ of bounded hyperbolic oscillation, $p>1$, can be factored as
$w=w_{1}w_{2}^{1-p}$, where our novel contribution is to ensure that the
weights $w_{1},w_{2}\in B_{1}$ are also of bounded hyperbolic oscillation.
This type of factorization problem within a subclass of weights has previously
been raised by Borichev [4, Remark 2], where he considered a different class
of well-behaved Békollé–Bonami weights.

By taking logarithms in the Wolff restriction theorem, one obtains a
characterization of the restrictions of
$\operatorname*{BMO}(\mathbb{T})$-functions to $\Omega$ [8, Chapter IV,
Corollary 5.7]. The same is in principle true in our setting, when considering
the space

$\operatorname*{BMO}(\mathbb{D}):=\left\\{f\in
L^{1}_{\textrm{loc}}(\mathbb{D})\,:\,\sup_{D}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{D\cap\mathbb{D}}\left|f-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{D\cap\mathbb{D}}f\right|\,dA<\infty\right\\},$

where the supremum is taken over all discs $D$ centered at points in
$\mathbb{D}$. However, as shown in [13, Section 3], the logarithm of any
weight of bounded hyperbolic oscillation is automatically in
$\operatorname*{BMO}(\mathbb{D})$, and the restriction problem therefore boils
down to the extension of quasi-Lipschitz functions.

To obtain a more interesting restriction problem, we will in Section 4 instead
consider the space $\mathcal{B}$ of holomorphic functions in
$\operatorname*{BMO}(\mathbb{D})$. This space is precisely the Bloch space,
which is usually introduced as the space of holomorphic functions $b$ on the
unit disc such that

$\|b\|_{\mathcal{B}}:=|b(0)|+\sup_{z\in\mathbb{D}}|b^{\prime}(z)|(1-|z|^{2})<\infty.$

In other words, $\mathcal{B}$ is the space of holomorphic functions in
$\mathbb{D}$ which are Lipschitz with respect to the hyperbolic metric.
Further to this, the insight of Makarov [14] showed that Bloch functions are
in one-to-one correspondence with dyadic martingales with uniformly bounded
increments. Therefore, any restriction problem for the Bloch space can be
recast as a restriction problem for such martingales.

Before posing our precise problem for the Bloch space $\mathcal{B}$, we will
recall a result of Sundberg [19], which gave a discrete analogue of the
classical Wolff restriction theorem. To set the stage for Sundberg’s theorem,
we recall that a sequence $Z=(z_{n})_{n\in\mathbb{N}}$ in the unit disc is
interpolating for the space $H^{\infty}$ of bounded analytic functions on
$\mathbb{D}$ if, given any bounded sequence $(w_{n})_{n\in\mathbb{N}}$, there
exists a $\phi$ in $H^{\infty}$ such that $\phi(z_{n})=w_{n}$,
$n\in\mathbb{N}$. Let

$\rho(z,w):=\left|\frac{w-z}{1-\overline{w}z}\right|,\qquad z,w\in\mathbb{D},$

be the pseudo-hyperbolic distance on $\mathbb{D}$. Carleson [5] famously
showed that $Z$ is interpolating if and only if it is weakly separated,
$\inf_{n\neq j}\rho(z_{n},z_{j})>0$, and the measure

$\mu_{Z}:=\sum_{n\in\mathbb{N}}(1-|z_{n}|^{2})\delta_{z_{n}}$

satisfies the Carleson condition: $\mu_{Z}\big{(}S(I)\big{)}\leq C_{Z}|I|$,
where $C_{Z}>0$ is the Carleson constant for the measure and $I$ is any arc of
the unit circle. We remark that $Z$ satisfies this latter Carleson condition
if and only if

(1.4)
$\sup_{z\in\mathbb{D}}\sum_{n\in\mathbb{N}}\big{(}1-\rho^{2}(z,z_{n})\big{)}<\infty.$

Moreover, Carleson showed that the Carleson condition is equivalent to the
embedding condition

(1.5)
$\sum_{n\in\mathbb{N}}|g(z_{n})|^{2}(1-|z_{n}|^{2})\lesssim_{Z}\dfrac{1}{2\pi}\int_{0}^{2\pi}|g(e^{i\theta})|^{2}\,d\theta,\qquad
g\in L^{2}(\mathbb{T}),$

where $g(z)$ denotes the value at $z\in\mathbb{D}$ of the Poisson extension of
$g$ to the unit disc.

We now give a slightly modified version of the original statement of
Sundberg’s result, which can be read from its proof.

######  Theorem 1.3 (Sundberg [19]).

Given a sequence $(w_{n})_{n\in\mathbb{N}}$ in $\mathbb{C}$ and a sequence
$Z=(z_{n})_{n\in\mathbb{N}}$ in $\mathbb{D}$ satisfying the Carleson condition
(1.4), there exists a function $f$ in $\operatorname*{BMO}(\mathbb{T})$ such
that $(f(z_{n})-w_{n})_{n\in\mathbb{N}}\in\ell^{\infty}$ if and only if there
exists a function $\beta\colon\mathbb{D}\to\mathbb{C}$ and a number
$\lambda>0$ such that

(1.6)
$\sup_{z\in\mathbb{D}}\sum_{n\in\mathbb{N}}e^{\lambda|w_{n}-\beta(z)|}\big{(}1-\rho^{2}(z,z_{n})\big{)}<\infty.$

Moreover, if $Z$ additionally is weakly separated (and hence interpolating),
then, for every sequence $(w_{n})_{n\in\mathbb{N}}$ satisfying (1.6), there
exists a function $f$ in $\operatorname*{BMO}(\mathbb{T})$ with holomorphic
Poisson extension such that $f(z_{n})=w_{n}$, for every $n$.

Given an interpolating sequence $Z$, we therefore seek to characterize the
trace of the Bloch space $\mathcal{B}$ onto $Z$. The condition (1.6) is of
course directly related to the John–Nirenberg inequality for
$\operatorname*{BMO}(\mathbb{T})$. For the larger Bloch space, via the
identification of Makarov, we instead have the Azuma–Hoeffding inequality for
martingales at our disposal. In terms of Bloch functions, we may state this
inequality as the estimate

(1.7) $\sup_{r\in(0,1)}\int_{0}^{2\pi}e^{\frac{a\leavevmode\nobreak\
|b(re^{i\theta})|^{2}}{-\log(1-r^{2})}}\,\frac{d\theta}{2\pi}\leq
C_{a},\qquad\|b\|_{\mathcal{B}}\leq 1,\;0<a<1;$

see [10, Th. 3.3.1] for this precise statement.

The following theorem is the second main result of our paper.

######  Theorem 1.4.

Let $b$ be a Bloch function, and let $(z_{n})_{n\in\mathbb{N}}$ be a sequence
in $\mathbb{D}$ satisfying the Carleson condition (1.4). Then

(i):

There exists $\lambda>0$ such that

(1.8)
$\sup_{z\in\mathbb{D}}\sup_{r\in(0,1)}\sum_{\rho(z,z_{n})<r}e^{\frac{\lambda|b(z_{n})-b(z)|^{2}}{-\log(1-r^{2})}}\big{(}1-\rho^{2}(z,z_{n})\big{)}<\infty.$

(ii):

There exist $\lambda>0$ and $C>0$ such that, for all $z$ in $\mathbb{D}$, the
function

$\mathbb{N}\ni n\mapsto
e^{\frac{\lambda|b(z_{n})-b(z)|^{2}}{-\log(1-\rho^{2}(z,z_{n}))}}\big{(}1-\rho^{2}(z,z_{n})\big{)}$

has weak-$L^{1}$-norm less than $C$ with respect to the counting measure on
$\mathbb{N}$, that is,

(1.9)
$\\#\left\\{n\in\mathbb{N}\,\bigg{|}\,e^{\frac{\lambda|b(z_{n})-b(z)|^{2}}{-\log(1-\rho^{2}(z,z_{n}))}}\big{(}1-\rho^{2}(z,z_{n})\big{)}>t\right\\}\leq\frac{C}{t},\qquad
t>0.$

Furthermore,

(iii):

There exists a function $b\in\mathcal{B}$ and an interpolating sequence
$(z_{n})_{n\in\mathbb{N}}$, such that, for every $\lambda>0$, the strong
summability condition associated to (1.9) fails at $z=0$, that is,

(1.10)
$\sum_{n\in\mathbb{N}}e^{\frac{\lambda|b(z_{n})-b(0)|^{2}}{-\log(1-|z_{n}|^{2})}}(1-|z_{n}|^{2})=\infty.$

######  Remark 1.5.

Since Bloch functions are Lipschitz, we have that
$|b(z)-b(0)|\lesssim\log\frac{1}{1-|z|^{2}}.$ Therefore (1.10) also implies
that (1.6) fails.

Of course, our theorem begs the question.

######  Question.

Suppose additionally that $Z$ is weakly separated, and hence interpolating.
Given a sequence $(w_{n})$ in $\mathbb{C}$, are either of the conditions (1.8)
or (1.9) (with $w_{n}$ replacing $b(z_{n})$, and a general given function
$\beta(z)$ replacing $b(z)$) sufficient for the existence of a Bloch function
$b$ such that $b(z_{n})=w_{n}$?

We have not been able to determine the answer to this outstanding problem,
which seems equally as interesting when cast as a problem of martingales; see
Section 4 for details.

### Notation

Given two positive functions $f$ and $g$, $f(x)\lesssim g(x)$ denotes that
there exists a universal constant $C$ such that $f(x)\leq Cg(x)$ for all $x$.
If also $g(x)\lesssim f(x)$, we say that $f\simeq g$.

### Acknowledgments

We would like to express our gratitude to Artur Nicolau for helpful
discussions, especially concerning the sharpness of our second main theorem,
namely, Theorem 1.4 (iii).

The first author was partially supported by the grant 275113 of the research
Council of Norway through the Alain Bensoussan Fellowship Programme from
ERCIM, and by the Emmy Noether Program of the German Research Foundation (DFG
Grant 466012782). The work of the second author was funded by grant 275113 of
the Research Council of Norway through the Alain Bensoussan Fellowship
Programme from ERCIM, and his current research is funded by the postdoctoral
scholarship JCK22-0052 from The Kempe Foundations and is partially supported
by grant PID2019-106870GB-I00 from Ministerio de Ciencia e Innovación
(MICINN). The work of the third author was supported by grant 334466 of the
Research Council of Norway, “Fourier Methods and Multiplicative Analysis”.

##  2\. Dyadic Békollé–Bonami weights

###  2.1. Basic definitions and properties

We define a _dyadic grid_ of $\mathbb{T}$ as a collection of arcs
$\mathcal{D}$ with the properties that $\mathbb{T}\in\mathcal{D}$ and for any
arc $I\in\mathcal{D}$ we have that

  1. (1)

The subfamily $\\{J\in\mathcal{D}\>:\>|J|=|I|\\}$ is a partition of
$\mathbb{T}$.

  2. (2)

There exists $I_{1},I_{2}\in\mathcal{D}$ with the same length such that
$I=I_{1}\cup I_{2}$.

For technical simplicity, we will also assume that every $I\in\mathcal{D}$
contains its right, but not its left, endpoint.

Given a dyadic grid $\mathcal{D}$, a measurable subset
$\Omega\subset\mathbb{D}$ of positive area measure, and $p>1$, we say that an
integrable function $w:\Omega\rightarrow[0,+\infty)$ is a _restricted dyadic
Békollé–Bonami_ $B_{p,\mathcal{D},\Omega}$ weight if

$[w]_{B_{p,\mathcal{D},\Omega}}=\sup_{I\in\mathcal{D}}\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w\>dA\left(\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w^{-\frac{1}{p-1}}\>dA\right)^{p-1}<\infty.$

In the case that $\Omega=\mathbb{D}$, we will drop the term “restricted” and
simply write $B_{p,\mathcal{D}}$ and $[w]_{B_{p,\mathcal{D}}}$.

######  Remark 2.1.

Note that if $w\in B_{p,\mathcal{D},\Omega}$, then $w^{-\frac{1}{p-1}}\in
B_{\frac{p}{p-1},\mathcal{D},\Omega}$ and
$\left[w^{-\frac{1}{p-1}}\right]_{B_{\frac{p}{p-1}},\mathcal{D},\Omega}$
coincides with $\left[w\right]_{B_{p},\mathcal{D},\Omega}^{\frac{1}{p-1}}$.

When $p=1$, we introduce the class $B_{1,\mathcal{D},\Omega}$ (or
$B_{1,\mathcal{D}}$ when $\Omega=\mathbb{D}$) as the set of all integrable
weights $w$ for which there exists a constant $C=C(w)>0$ such that

(2.1) $M_{\mathcal{D},\Omega}w(z)\leq Cw(z),\quad\mbox{a.e. }z\in\Omega.$

Here $M_{\mathcal{D},\Omega}$ denotes the _(restricted) dyadic maximal
operator_ ,

(2.2) $M_{\mathcal{D},\Omega}f(z):=\sup_{I\in\mathcal{D},S(I)\cap\Omega\ni
z}\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}|f|\>dA.$

If $w\in B_{1,\mathcal{D},\Omega}$, the quantity
$[w]_{B_{1,\mathcal{D},\Omega}}$ will denote the infimum of all admissible $C$
in (2.1). We will use the notation
$\mathcal{D}_{\Omega}:=\\{I\in\mathcal{D}\>:\>A\big{(}S(I)\cap\Omega\big{)}>0\\}$,
and, given a measurable set $E$, we will write $w(E)$ to denote the quantity
$\int_{E}w\>dA$.

To begin, we observe that $B_{p,\mathcal{D},\Omega}$, $p>1$, can also be
characterized by means of the weak-type boundedness of the dyadic maximal
operator $M_{\mathcal{D},\Omega}$.

######  Theorem 2.2.

Let $w\in L^{1}(\Omega)\setminus\\{0\\}$ be a non-negative weight, and let
$p>1$. If $w\in B_{p,\mathcal{D},\Omega}$, then the weak-type inequality

(2.3)
$w\big{(}\\{z\in\Omega:M_{\mathcal{D},\Omega}f(z)>\lambda\\}\big{)}\leq\dfrac{[w]_{B_{p,\mathcal{D},\Omega}}}{\lambda^{p}}\|f\|_{L^{p}(\Omega,w)}^{p}$

holds for all $\lambda>0$ and $f\in L^{p}(\Omega,w)$. Conversely, if the
restricted dyadic maximal operator is of weak-type $(p,p)$ with respect to
$w$, then $w$ is a $B_{p,\mathcal{D},\Omega}$-weight.

###### Proof.

First, assume that $M_{\mathcal{D},\Omega}$ is of weak-type $(p,p)$. Take
$I\in\mathcal{D}_{\Omega}$. If $K\subset S(I)\cap\Omega$ is a measurable set
with positive area, we see that

(2.4) $w\big{(}S(I)\cap\Omega\big{)}\leq
w\left(\left\\{z\in\Omega:M_{\mathcal{D},\Omega}\chi_{K}(z)\geq\dfrac{A(K)}{A\big{(}S(I)\big{)}}\right\\}\right)\leq
C\left(\dfrac{A\big{(}S(I)\big{)}}{A(K)}\right)^{p}w(K).$

Choosing $I=\mathbb{T}$, we realize that $w$ cannot vanish in a set of
positive area.

Let $I\in\mathcal{D}_{\Omega}$ and take $n\geq 1$. Testing with the function

$f_{n}:=w^{-1/(p-1)}\chi_{S(I)\cap\Omega\cap\\{w^{-1/(p-1)}\leq n\\}}$

and the number $\lambda_{n}=\frac{1}{A(S(I))}\int_{S(I)\cap\Omega}f_{n}\>dA$
in (2.3), we see that

$\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w\>dA\left(\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega\cap\\{w^{-1/(p-1)}\leq
n\\}}w^{-\frac{1}{p-1}}\>dA\right)^{p-1}\leq C,\quad\forall n\geq 1.$

Therefore the monotone convergence theorem and the fact that $w>0$ almost
everywhere yield that

$\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w\>dA\left(\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w^{-\frac{1}{p-1}}\>dA\right)^{p-1}\leq
C,\quad\forall I\in\mathcal{D}_{\Omega},$

and hence $w$ is a restricted dyadic Békollé–Bonami weight.

Now, assume $w\in B_{p,\mathcal{D},\Omega}$, take $f\in L^{p}(\Omega,w)$, let
$\lambda>0$, and consider the set
$E_{\lambda}:=\\{z\in\Omega\>:\>M_{\mathcal{D},\Omega}f(z)>\lambda\\}$. If
$z\in E_{\lambda}$, then there exists $I\in\mathcal{D}_{\Omega}$ with $z\in
S(I)\cap\Omega$ and

$\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}|f|\>dA>\lambda,$

so that $S(I)\cap\Omega\subset E_{\lambda}$. Therefore, by the dyadic
structure, $E_{\lambda}=\cup_{I\in\mathcal{J}}\big{(}S(I)\cap\Omega\big{)}$
for a disjoint collection $\mathcal{J}\subset\mathcal{D}_{\Omega}$ satisfying
that

$\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}|f|\>dA>\lambda,\qquad
I\in\mathcal{J}.$

Therefore Hölder’s inequality implies that

$\displaystyle w\big{(}S(I)\cap\Omega\big{)}$ $\displaystyle\leq$
$\displaystyle\dfrac{\int_{S(I)\cap\Omega}|f|^{p}w\>dA}{\lambda^{p}}\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w\>dA\left(\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w^{-\frac{1}{p-1}}\>dA\right)^{p-1}$
$\displaystyle\leq$
$\displaystyle\dfrac{[w]_{B_{p,\mathcal{D},\Omega}}}{\lambda^{p}}\int_{S(I)\cap\Omega}|f|^{p}w\>dA$

for all $I\in\mathcal{J}$, and hence (2.3) holds. ∎

######  Remark 2.3.

When $\Omega=\mathbb{D}$, $w\in B_{p,\mathcal{D}}$ if and only if
$M_{\mathcal{D}}$ is _bounded_ on $L^{p}(\mathbb{D},w)$. Indeed, as in the
above proof we verify that the operator

$M_{\mathcal{D},w}f(z):=\sup_{z\in
S(I),I\in\mathcal{D}}\dfrac{1}{w\big{(}S(I)\big{)}}\int_{S(I)}|f|w\>dA$

is of weak-type $(1,1)$ with respect to the measure $w$, and thus
Marcinkiewicz’s theorem yields that

$\|M_{\mathcal{D},w}f\|_{L^{p}(w)}\leq
2\left(\dfrac{p}{p-1}\right)^{\frac{1}{p}}\|f\|_{L^{p}(w)}$

for all $p>1$. Now, following Lerner’s approach [12] one shows that if $w\in
B_{p,\mathcal{D}}$, then

$\displaystyle\|M_{\mathcal{D}}f\|_{L^{p}(w)}$ $\displaystyle\leq$
$\displaystyle[w]_{B_{p,\mathcal{D}}}^{\frac{1}{p-1}}\|M_{\mathcal{D},w}\|_{\frac{p}{p-1}}^{\frac{1}{p-1}}\left\|M_{\mathcal{D},w^{-\frac{1}{p-1}}}\right\|_{p}\|f\|_{L^{p}(w)}$
$\displaystyle\leq$ $\displaystyle
4\left(\dfrac{p^{2}}{p-1}\right)^{\frac{1}{p}}[w]_{B_{p,\mathcal{D}}}^{\frac{1}{p-1}}\|f\|_{L^{p}(w)}$

for all $f\in L^{p}(w)$, so $M_{\mathcal{D}}$ is bounded.

Next we will prove a dyadic factorization theorem for
$B_{p,\mathcal{D},\Omega}$. A reader familiar with the theory of
$A_{p}$-weights might notice that this result is a consequence of Rubio de
Francia’s factorization techniques [17]. Nevertheless, we include a
constructive proof whose original idea is due to Coifman, Jones and Rubio de
Francia [6], as we will need to inspect the details of this proof when
considering classes of well-behaved dyadic weights in the next subsection.

######  Theorem 2.4.

Let $\mathcal{D}$ be a dyadic grid, $\Omega\subset\mathbb{D}$ a measurable set
with positive area and $p>1$. If $w\in L^{1}(\Omega)\setminus\\{0\\}$
satisfies that $M_{\mathcal{D},\Omega}$ is bounded on $L^{p}(\Omega,w)$ and
$L^{\frac{p}{p-1}}(\Omega,w^{-\frac{1}{p-1}})$, then there exist
$w_{1},w_{2}\in B_{1,\mathcal{D},\Omega}$ such that $w=w_{1}w_{2}^{1-p}$.

###### Proof.

Assume that $p\in(1,2]$ (if not, we can factor $w^{-\frac{1}{p-1}}\in
B_{\frac{p}{p-1},\mathcal{D},\Omega}$ instead). Consider the operator defined
as

(2.5)
$S(f):=M_{\mathcal{D},\Omega}(fw)w^{-1}+M_{\mathcal{D},\Omega}^{p-1}\left(|f|^{\frac{1}{p-1}}\right).$

It is plain that $M_{\mathcal{D},\Omega}$ is a positive and sublinear
operator. Since $1<p\leq 2$, it is not difficult to check that $S$ is also
positive and sublinear. Moreover, $S$ is bounded from
$L^{\frac{p}{p-1}}(\Omega,w)$ to itself. Indeed, if $f\in
L^{\frac{p}{p-1}}(\Omega,w)$ then $|f|^{\frac{1}{p-1}}\in L^{p}(\Omega,w)$ and
$fw\in L^{\frac{p}{p-1}}(\Omega,w^{-\frac{1}{p-1}})$, so we have that

$\left\|M_{\mathcal{D},\Omega}^{p-1}\left(|f|^{\frac{1}{p-1}}\right)\right\|_{L^{\frac{p}{p-1}}(\Omega,w)}\leq\|M_{\mathcal{D},\Omega}\|_{L^{p}(\Omega,w)}^{p-1}\|f\|_{L^{\frac{p}{p-1}}(\Omega,w)}$

and

$\left\|M_{\mathcal{D},\Omega}(fw)w^{-1}\right\|_{L^{\frac{p}{p-1}}(\Omega,w)}\leq\|M_{\mathcal{D},\Omega}\|_{L^{\frac{p}{p-1}}\left(\Omega,w^{-\frac{1}{p-1}}\right)}\|f\|_{L^{\frac{p}{p-1}}(\Omega,w)}.$

Observe also that, by definition, the operator $S$ satisfies

$M_{\mathcal{D},\Omega}^{p-1}\left(|f|^{\frac{1}{p-1}}\right)\leq
S(f)\quad\mbox{ and }\quad M_{\mathcal{D},\Omega}(fw)\leq S(f)w,$

for all $f\in L^{\frac{p}{p-1}}(\Omega,w)$.

Now, consider the function

(2.6) $f:=\sum_{k=0}^{\infty}\dfrac{S^{k}(u)}{(2\|S\|)^{k}}\in
L^{\frac{p}{p-1}}(\Omega,w),$

where $u$ is any positive $L^{\frac{p}{p-1}}(\Omega,w)$ function (for
instance, we might fix $u\equiv 1$). Since $S$ is positive and sublinear, an
immediate computation shows that

$S(f)\leq\sum_{k=1}^{\infty}\dfrac{S^{k+1}(u)}{(2\|S\|)^{k}}=2\|S\|\sum_{m=2}^{\infty}\dfrac{S^{m}(u)}{(2\|S\|)^{m}}\leq
2\|S\|f$

almost everywhere in $\Omega$. Therefore he have that

$M_{\mathcal{D},\Omega}\left(f^{\frac{1}{p-1}}\right)\leq(2\|S\|)^{\frac{1}{p-1}}f^{\frac{1}{p-1}}\quad\mbox{
and }\quad M_{\mathcal{D},\Omega}(fw)\leq 2\|S\|fw.$

In other words, we have that $w_{1}:=fw$ and $w_{2}:=f^{\frac{1}{p-1}}$ belong
to $B_{1,\mathcal{D},\Omega}$ and moreover

$\displaystyle[w_{1}]_{B_{1,\mathcal{D},\Omega}}$ $\displaystyle\leq$
$\displaystyle
2\left(\|M_{\mathcal{D},\Omega}\|_{L^{\frac{p}{p-1}}(\Omega,w^{-\frac{1}{p-1}})}+\|M_{\mathcal{D},\Omega}\|_{L^{p}(\Omega,w)}^{p-1}\right),$
$\displaystyle{[w_{2}]}_{B_{1,\mathcal{D},\Omega}}$ $\displaystyle\leq$
$\displaystyle
2^{\frac{1}{p-1}}\left(\|M_{\mathcal{D},\Omega}\|_{L^{\frac{p}{p-1}}(\Omega,w^{-\frac{1}{p-1}})}+\|M_{\mathcal{D},\Omega}\|_{L^{p}(\Omega,w)}^{p-1}\right)^{\frac{1}{p-1}}.$

Since $w=w_{1}w_{2}^{1-p}$, we have obtained the desired factorization. ∎

######  Remark 2.5.

The converse holds when $\Omega=\mathbb{D}$. By Remark 2.3 it is sufficient to
verify that $w_{1}w_{2}^{1-p}$ belongs to $B_{p,\mathcal{D}}$. But this is
straightforward consequence of the definition of $B_{1,\mathcal{D}}$. Indeed,

$\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)}w_{1}w_{2}^{1-p}\,dA\leq\dfrac{[w_{2}]_{B_{1,\mathcal{D}}}^{p-1}}{A\big{(}S(I)\big{)}}\int_{S(I)}w_{1}\,dA\left(\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)}w_{2}\,dA\right)^{1-p}$

and

$\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)}w_{2}w_{1}^{-\frac{1}{p-1}}\,dA\leq\dfrac{[w_{1}]_{B_{1,\mathcal{D}}}^{\frac{1}{p-1}}}{A\big{(}S(I)\big{)}}\int_{S(I)}w_{2}\,dA\left(\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)}w_{1}\,dA\right)^{-\frac{1}{p-1}}$

so it is clear that
$[w_{1}w_{2}^{1-p}]_{B_{p,\mathcal{D}}}\leq[w_{1}]_{B_{1,\mathcal{D}}}[w_{2}]_{B_{1,\mathcal{D}}}^{p-1}$.

We end this subsection with a lemma, whose proof is mostly based on Theorem
3.4 in Chapter II of [8], although the original idea goes back to [7].

######  Lemma 2.6.

Let $w$ be an integrable weight in $\mathbb{D}$ and $\mathcal{D}$ a dyadic
grid. Then, for every $\gamma\in(0,1)$ we have that
$(M_{\mathcal{D}}w)^{\gamma}\in B_{1,\mathcal{D}}$. Furthermore,
$[(M_{\mathcal{D}}w)^{\gamma}]_{B_{1,\mathcal{D}}}\leq\frac{2-\gamma}{1-\gamma}$.

###### Proof.

Take $z\in\mathbb{D}$ and let $I\in\mathcal{D}$ such that $z\in S(I)$.
Consider the weights $w_{1}:=w\chi_{S(I)}$ and $w_{2}:=w-w_{1}$. Since
$\gamma\in(0,1)$, we have that
$(M_{\mathcal{D}}w)^{\gamma}\leq(M_{\mathcal{D}}w_{1})^{\gamma}+(M_{\mathcal{D}}w_{2})^{\gamma}$
in $\mathbb{D}$.

On the one hand, we have that

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}(M_{\mathcal{D}}w_{1})^{\gamma}\>dA=\dfrac{\gamma}{A\big{(}S(I)\big{)}}\int_{0}^{\infty}t^{\gamma-1}A\big{(}\\{z\in
S(I):M_{\mathcal{D}}w_{1}(z)>t\\}\big{)}\>dt.$

Observe that

$\dfrac{\gamma}{A\big{(}S(I)\big{)}}\int_{0}^{\frac{w(S(I))}{A(S(I))}}t^{\gamma-1}A\big{(}\\{z\in
S(I):M_{\mathcal{D}}w_{1}(z)>t\\}\big{)}\>dt\leq\left(\dfrac{w\big{(}S(I)\big{)}}{A\big{(}S(I)\big{)}}\right)^{\gamma}$

and since $M_{\mathcal{D}}$ is of weak-type $(1,1)$ we deduce that

$\displaystyle\dfrac{\gamma}{A\big{(}S(I)\big{)}}\int_{\frac{w(S(I))}{A(S(I))}}^{\infty}t^{\gamma-1}A\big{(}\\{z\in
S(I):M_{\mathcal{D}}w_{1}(z)>t\\}\big{)}\>dt$ $\displaystyle\leq$
$\displaystyle\dfrac{\gamma\|w_{1}\|_{L^{1}(\mathbb{D})}}{A\big{(}S(I)\big{)}}\int_{\frac{w(S(I))}{A(S(I))}}^{\infty}t^{\gamma-2}\>dt$
$\displaystyle\leq$
$\displaystyle\dfrac{\gamma}{1-\gamma}\left(\dfrac{w\big{(}S(I)\big{)}}{A\big{(}S(I)\big{)}}\right)^{\gamma}$

and therefore we see that

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}(M_{\mathcal{D}}w_{1})^{\gamma}\>dA\leq\dfrac{1}{1-\gamma}(M_{\mathcal{D}}w)^{\gamma}(z),\quad\forall
z\in S(I).$

On the other hand, recall that $w_{2}$ is supported outside $S(I)$. Thus,
$M_{\mathcal{D}}w_{2}$ is constant in $S(I)$ and then

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}(M_{\mathcal{D}}w_{2})^{\gamma}\>dA=(M_{\mathcal{D}}w_{2})^{\gamma}(z)\leq(M_{\mathcal{D}}w)^{\gamma}(z)$

for all $z\in S(I)$.

Summing up, we have shown that the inequality

$M_{\mathcal{D}}(M_{\mathcal{D}}w)^{\gamma}\leq\dfrac{2-\gamma}{1-\gamma}(M_{\mathcal{D}}w)^{\gamma}$

holds in $\mathbb{D}$, so $(M_{\mathcal{D}}w)^{\gamma}\in B_{1,\mathcal{D}}$.
∎

###  2.2. Weights of bounded hyperbolic oscillation

For an arc $I\in\mathcal{D}$, we define its $\rho$-top half $T_{\rho}(I)$ as

$T_{\rho}(I):=\left\\{z\in\mathbb{D}\,:\,1-\rho<\dfrac{1-|z|}{|I|}\leq
1,\dfrac{z}{|z|}\in I\right\\}.$

In the case that $\rho=\frac{1}{2}$, we will simply write $T(I)$ to denote the
dyadic top-half $T_{1/2}(I)$.

We will say that a weight $w$ in $\mathbb{D}$ is _almost constant on dyadic
$3/4$-top halves_ if there exists a constant $C\geq 1$ such that

(2.7) $w(z)\leq Cw(\zeta)$

for almost every $z,\zeta\in T_{3/4}(I)$ and all $I\in\mathcal{D}$. In other
words, if $w$ is essentially constant in each union of $T(I)$ with the top
halves of the immediate successors of $I$ in the dyadic grid $\mathcal{D}$.

Note that in the continuous setting [11, Lemma 2.17], (2.7) holds for almost
every $z,\zeta\in T_{3/4}(I)$ and _every_ arc $I\subset\mathbb{T}$ if and only
if $w$ is of bounded hyperbolic oscillation, satisfying (1.2). Similarly, we
may understand weights which are almost constant on dyadic $3/4$-top halves by
means of the dyadic hyperbolic distance

(2.8)
$\beta_{\mathcal{D}}(z,\zeta):=\log_{2}\dfrac{|P_{\mathcal{D}}(\mathcal{I}_{z},\mathcal{I}_{\zeta})|}{\min\\{|\mathcal{I}_{z}|,|\mathcal{I}_{\zeta}|\\}},$

where $\mathcal{I}_{z}$ and $\mathcal{I}_{\zeta}$ are the unique arcs in
$\mathcal{D}$ such that $z\in T(\mathcal{I}_{z})$ and $\zeta\in
T(\mathcal{I}_{\zeta})$, and $P_{\mathcal{D}}(I,J)$ denotes the smallest arc
in $\mathcal{D}$ containing both $I$ and $J$. See Figure 2 for an example.
Observe that $\beta_{\mathcal{D}}$ is not a proper metric, but it is when
considering the quotient set corresponding to the equivalence relation arising
from the tilling $\big{(}T(I)\big{)}_{I\in\mathcal{D}}$. When this quotient
set is identified with a dyadic tree, $\beta_{\mathcal{D}}(z,\zeta)$ turns out
to be the number of nodes between the least common ancestor and the deeper of
the two points.

$\bullet$$z$$\bullet$$\zeta$$P_{\mathcal{D}}(\mathcal{I}_{z},\mathcal{I}_{\zeta})$$\mathbb{T}$
Figure 2. Example of $P_{\mathcal{D}}(\mathcal{I}_{z},\mathcal{I}_{\zeta})$.

######  Lemma 2.7.

Let $f$ be a positive function in $\mathbb{D}$. Then, the following statements
are equivalent:

  1. (1)

$f$ is almost constant on dyadic $3/4$-top halves.

  2. (2)

There exists a constant $L$ such that

$|\log f(z)-\log f(\zeta)|\leq L\big{(}1+\beta_{\mathcal{D}}(z,\zeta)\big{)}$

for every $z,\zeta\in\mathbb{D}$.

###### Proof.

First, assume that (1) holds, and without loss of generality suppose that the
top halves of $\mathcal{I}_{z}$ and $\mathcal{I}_{\zeta}$ are centered at $z$
and $\zeta$ (see Figure 1). In the associated dyadic tree, there are up to
$2\beta_{\mathcal{D}}(z,\zeta)$ nodes separating $z$ and $\zeta$, and then the
triangle inequality yields that

$|\log f(z)-\log f(\zeta)|\leq 2C\beta_{\mathcal{D}}(z,\zeta)$

since each node is included in the $3/4$-top half of its predecessor.

It is trivial that $\beta_{\mathcal{D}}(z,\zeta)\leq 2$ for all $z,\zeta\in
T_{3/4}(I)$ and every $I\in\mathcal{D}$, and hence $f$ is almost constant in
top halves if $\log f$ is quasi-Lipschitz with respect to
$\beta_{\mathcal{D}}$. ∎

######  Remark 2.8.

In fact, it follows from the above argument that we could replace $T_{3/4}(I)$
by $T_{\rho}(I)$ for any $\frac{1}{2}<\rho<1$. In contrast to the continuous
case, it is not true for $0<\rho\leq\frac{1}{2}$. Indeed, the function
$f(z):=\exp\big{(}\beta_{\mathcal{D}}^{2}(z,0)\big{)}$ is constant in every
top half $T(I)$, $I\in\mathcal{D}$, but does not satisfy property (2).

The proof of Lemma 2.7 is still valid if we replace $\mathbb{D}$ by some
domain $\Omega$ whose corresponding subgraph in the dyadic tree is connected.
However, this equivalence is not true for a general $\Omega$. Therefore, in
the restricted dyadic setting, we need to distinguish between the two
properties. Given a weight $w:\Omega\rightarrow[0,+\infty]$, we will say that
$w$ is _almost constant on dyadic $3/4$-top halves_ if (2.7) holds for almost
every $z,\zeta\in T_{3/4}(I)\cap\Omega$ and all $I\in\mathcal{D}$. In general,
we will use the notation $C_{w}$ to denote the infimum among all admissible
$C$ in the inequality (2.7). Furthermore, we say that $w$ is of _dyadic
bounded hyperbolic oscillation_ (or $\mathcal{D}$\- bounded hyperbolic
oscillation, whenever it is necessary to specify the dyadic grid considered)
if

(2.9) $|\log w(z)-\log w(\zeta)|\leq
L_{w}\big{(}1+\beta_{\mathcal{D}}(z,\zeta)\big{)}$

for almost every $z$ and $\zeta$ in $\Omega$. If $w$ is of dyadic bounded
hyperbolic oscillation, $L_{w}$ will denote the smallest possible constant in
(2.9). It is worth noticing that any weight on $\Omega$ that is of dyadic
bounded hyperbolic oscillation is almost constant on dyadic $3/4$-top halves,
while the converse holds only for certain $\Omega$ (Lemma 2.7 considers the
case $\Omega=\mathbb{D}$).

######  Lemma 2.9.

Let $f$ be a positive and integrable function in $\Omega$. Then
$M_{\mathcal{D},\Omega}f$ is almost constant on dyadic $3/4$-top halves with
constant $C_{M_{\mathcal{D},\Omega}f}\leq 4$.

###### Proof.

By construction, $M_{\mathcal{D},\Omega}$ is constant in dyadic top halves and
“monotonic” in the sense that if $J$ is an immediate successor of $I$ in
$\mathcal{D}$, then we have that $M_{\mathcal{D},\Omega}f(z)\leq
M_{\mathcal{D},\Omega}f(\zeta)$ for all $z\in T(I)$ and every $\zeta\in T(J)$.
Thus, we only need to check that

$M_{\mathcal{D},\Omega}f(z)\leq 4M_{\mathcal{D},\Omega}f(\zeta)$

in the case that $\zeta\in T(I)$ and $z\in T(J)$. It is plain that this holds,
since

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(J)}f\,dA\leq\dfrac{A\big{(}S(I)\big{)}}{A\big{(}S(J)\big{)}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}f\,dA\leq
4M_{\mathcal{D},\Omega}f(\zeta).\qed$

Suppose that $w$ is almost constant on dyadic $3/4$-top halves. In light of
the above lemma, observe that for any positive $u$ the operator $S(u)$
introduced in (2.5) is almost constant on $3/4$-top halves as well, with
constant less than or equal to $4C_{w}$. Hence, the function defined in (2.6)
verifies that $C_{f}\leq 4C_{w}$, and therefore Rubio de Francia’s
factorization result also holds within this subclass of weights.

######  Theorem 2.10.

Let $\mathcal{D}$ be a dyadic grid, $\Omega\subset\mathbb{D}$ a measurable set
with positive area, and $p\in(1,2]$. If $w\in L^{1}(\Omega)\setminus\\{0\\}$
satisfies that $M_{\mathcal{D},\Omega}$ is bounded on $L^{p}(\Omega,w)$ and
$L^{\frac{p}{p-1}}(\Omega,w^{-\frac{1}{p-1}})$, then there exist
$B_{1,\mathcal{D},\Omega}$ weights $w_{1}$ and $w_{2}$ such that
$w=w_{1}w_{2}^{1-p}$ and

$[w_{1}]_{B_{1,\mathcal{D},\Omega}},[w_{2}]^{p-1}_{B_{1,\mathcal{D},\Omega}}\leq
2\left(\|M_{\mathcal{D},\Omega}\|_{L^{\frac{p}{p-1}}\left(\Omega,w^{-\frac{1}{p-1}}\right)}+\|M_{\mathcal{D},\Omega}\|_{L^{p}(\Omega,w)}^{p-1}\right).$

Moreover, if $w$ is almost constant on $3/4$-top halves, then we can choose
$w_{1}$ and $w_{2}$ to satisfy the same property, with constants
$C_{w_{1}}\leq 4C_{w}^{2}$ and $C_{w_{2}}\leq(4C_{w})^{\frac{1}{p-1}}$.

As explained in the introduction, $B_{p}$-weights of bounded hyperbolic
oscillation do satisfy the reverse Hölder property. This continues to hold
true in the dyadic setting, as is seen by following the proof [2] verbatim.

######  Theorem 2.11 ([2]).

Let $\mathcal{D}$ be a dyadic grid. If $w\in B_{p,\mathcal{D}}$, $1\leq
p<\infty$, is of dyadic bounded hyperbolic oscillation, then there exists
$r>1$ such that

(2.10) $\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}w^{r}\,dA\right)^{\frac{1}{r}}\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}w\,dA$

for all $I\in\mathcal{D}$.

###### Proof.

One follows exactly the proof of the implication
$\textrm{(3)}\implies\textrm{(4)}$ in the proof of Theorem 1.6 in [2]. Note
that property (3) from [2] is ensured by the inequality (2.4) in the present
paper. ∎

######  Remark 2.12.

The reverse Hölder inequality leads to the _self-improvement property_.
Namely, if $w\in B_{p,\mathcal{D}}$, $1\leq p<\infty$, then there is $q>1$
such that $w^{q}\in B_{p,\mathcal{D}}$. Indeed, if $p>1$, since both $w$ and
$w^{-\frac{1}{p-1}}$ are dyadic Békollé–Bonami weights, there exist
$r_{1},r_{2}>1$ such that

$\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}w^{r_{1}}\,dA\right)^{\frac{1}{r_{1}}}\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}w\,dA\quad\mbox{ and
}\quad\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}w^{-\frac{r_{2}}{p-1}}\,dA\right)^{\frac{1}{r_{2}}}\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}w^{-\frac{1}{p-1}}\,dA$

for all $I\in\mathcal{D}$. Choosing $q:=\min\\{r_{1},r_{2}\\}$ we have that
$[w^{q}]_{B_{p,\mathcal{D}}}\lesssim[w]_{B_{p,\mathcal{D}}}^{q}$ and hence the
self-improvement property holds. A similar argument applies when $p=1$.

Finally, we are ready to show that the Wolff restriction theorem can be
extended to dyadic Békollé–Bonami weights of dyadic bounded hyperbolic
oscillation. In doing so, we must introduce some restrictions on the domain
$\Omega$. For the purposes of proving Theorem 1.2, it turns out that it is
sufficient to consider $\Omega$ to be a union of dyadic top halves. These sets
are advantageous, because even if their associated subgraphs are disconnected,
restricted dyadic Békollé–Bonami weights which are almost constant on 3/4-top
halves must be of dyadic bounded hyperbolic oscillation.

######  Lemma 2.13.

Let $\mathcal{D}$ be a dyadic grid of $\mathbb{T}$ and let $\Omega$ be the
union of top halves of a subcollection of intervals in $\mathcal{D}$. Consider
$1\leq p<\infty$. If $w\in B_{p,\mathcal{D},\Omega}$ is almost constant on
dyadic 3/4-top halves, then $w$ is of dyadic bounded hyperbolic oscillation.

###### Proof.

First, assume that $p=1$. Considering $w$ as a weight defined on $\mathbb{D}$
(setting $w(z):=0$ for all $z\in\mathbb{D}\setminus\Omega$), we realize that
for almost every $z\in\Omega$, we have that

$w(z)\leq C_{w}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T_{3/4}(\mathcal{I}_{z})\cap\Omega}w\>dA\leq
4C_{w}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(\mathcal{I}_{z})}w\>dA\leq
4C_{w}M_{\mathcal{D}}w(z),$

where $\mathcal{I}_{z}$ is the unique arc in $\mathcal{D}$ with $z\in
T(\mathcal{I}_{z})$. Here we used that $w$ is almost constant on 3/4-top
halves and that $T(\mathcal{I}_{z})\subset\Omega$. Since $w\in
B_{1,\mathcal{D},\Omega}$, we furthermore have that

$M_{\mathcal{D}}w(z)=M_{\mathcal{D},\Omega}w(z)\leq[w]_{B_{1,\mathcal{D},\Omega}}w(z)$

for almost every $z\in\Omega$. Thus, we have that

$w(z)=k(z)M_{\mathcal{D}}w(z),\quad\mbox{a.e. }z\in\Omega,$

where $\log k$ is bounded from above and below in $\Omega$. Since
$M_{\mathcal{D}}w$ is of dyadic bounded hyperbolic oscillation in $\mathbb{D}$
by Lemmas 2.7 and 2.9, this shows that $w$ is of dyadic bounded hyperbolic
oscillation in $\Omega$.

If $1<p$, we can apply Jensen’s inequality to see that $w^{\delta}\in
B_{\delta p+1-\delta,\mathcal{D},\Omega}$ and $w^{-\frac{\delta}{p-1}}\in
B_{\delta\frac{p}{p-1}+1-\delta,\mathcal{D},\Omega}$ for every
$\delta\in(0,1)$. Thus we may use Marcinkiewicz’s interpolation theorem to see
that $M_{\mathcal{D},\Omega}$ is bounded on $L^{p}(\Omega,w^{\delta})$ and
$L^{\frac{p}{p-1}}(\Omega,w^{-\frac{\delta}{p-1}})$. Therefore Theorem 2.10
applies, yielding that $w^{\delta}=w_{1}w_{2}^{1-p}$, for some $w_{1},w_{2}\in
B_{1,\mathcal{D},\Omega}$ which are almost constant on 3/4-top halves.
According to the previous discussion, the weights $w_{1}$ and $w_{2}$ are both
of bounded hyperbolic oscillation, and therefore so is $w$. ∎

######  Theorem 2.14.

Let $\mathcal{D}$ be a dyadic grid and $1<p<\infty$. Let $\Omega$ be the union
of top halves of a subcollection of intervals in $\mathcal{D}$. Consider a
weight $w$ of dyadic bounded hyperbolic oscillation in $\Omega$. If $w^{q}\in
B_{q,\mathcal{D},\Omega}$ for some $q>1$, then there exists $W\in
B_{p,\mathcal{D}}$ of dyadic bounded hyperbolic oscillation such that
$W(z)=w(z)$ for almost every $z\in\Omega$. Furthermore, for the Békollé–Bonami
and oscillation constants, we have that

$[W]_{B_{p,\mathcal{D}}}\leq M_{1}\quad\mbox{ and }\quad L_{W}\leq M_{2},$

where $M_{1}$ and $M_{2}$ only depend on $q$, $p$, $L_{w}$ and
$[w^{q}]_{B_{p,\mathcal{D},\Omega}}$.

Conversely, if $w$ coincides almost everywhere in $\Omega$ with some $W\in
B_{p,\mathcal{D}}$ of dyadic bounded hyperbolic oscillation, then there exists
$q>1$ such that $w^{q}\in B_{p,\mathcal{D},\Omega}$.

###### Proof.

We will only consider the case $1<p\leq 2$, since we may otherwise consider
$w^{-\frac{1}{p-1}}$. First, assume that $w^{q}\in B_{p,\mathcal{D},\Omega}$.
Using Jensen’s inequality we see that $w^{\delta q}\in B_{\delta
p+1-\delta,\mathcal{D},\Omega}$ for all $\delta\in(0,1)$ and, moreover,
$[w^{\delta q}]_{B_{\delta
p+1-\delta,\mathcal{D},\Omega}}\leq[w^{q}]_{B_{p,\mathcal{D},\delta}}^{\delta}$.
Since $\delta p+1-\delta<p$, Marcinkiewicz’s interpolation theorem yields that

$\|M_{\mathcal{D},\Omega}\|_{L^{p}(\Omega,w^{\delta q})}\leq
2\left(\dfrac{p}{(p-1)(1-\delta)}\right)^{\frac{1}{p}}[w^{q}]_{B_{p,\mathcal{D},\Omega}}^{\frac{\delta}{\delta
p+1-\delta}}.$

Similarly, $w^{-\frac{\delta q}{p-1}}\in
B_{\delta\frac{p}{p-1}+1-\delta,\mathcal{D},\Omega}$ and hence

$\displaystyle\|M_{\mathcal{D},\Omega}\|_{L^{\frac{p}{p-1}}\left(\Omega,w^{-\frac{\delta
q}{p-1}}\right)}$ $\displaystyle\leq$ $\displaystyle
2\left(\dfrac{p}{1-\delta}\right)^{\frac{p-1}{p}}[w^{-\frac{q}{p-1}}]_{B_{\frac{p}{p-1},\mathcal{D},\Omega}}^{\frac{\delta}{\delta
p/(p-1)+1-\delta}}$ $\displaystyle=$ $\displaystyle
2\left(\dfrac{p}{1-\delta}\right)^{\frac{p-1}{p}}[w^{q}]_{B_{p,\mathcal{D},\Omega}}^{\frac{\delta}{p+\delta-1}}.$

Because $w$ is of bounded hyperbolic oscillation, it is clear that $w^{\delta
q}$ is almost constant on dyadic $3/4$-top halves and, moreover, $C_{w^{\delta
q}}=C_{w}^{\delta q}\leq e^{3\delta qL_{w}}$. Then Theorem 2.10 yields that
there exist $w_{1},w_{2}\in B_{1,\mathcal{D},\Omega}$ such that $w^{\delta
q}=w_{1}w_{2}^{1-p}$ and furthermore

$w_{1}(z)\leq 4C_{w}^{2\delta q}w_{1}(\zeta)\quad\mbox{ and }\quad
w_{2}(z)\leq 4^{\frac{1}{p-1}}C_{w}^{\frac{\delta q}{p-1}}w_{2}(\zeta)$

for almost every $z$ and $\zeta$ in $T_{3/4}(I)\cap\Omega$ and all
$I\in\mathcal{D}$. Also, we have that

$[w_{1}]_{B_{1,\mathcal{D},\Omega}},[w_{2}]_{B_{1,\mathcal{D},\Omega}}^{p-1}\leq
2\left(\|M_{\mathcal{D},\Omega}\|_{L^{p/(p-1)}(\Omega,w^{-\delta
q/(p-1)})}+\|M_{\mathcal{D},\Omega}\|_{L^{p}(\Omega,w^{\delta
q})}^{p-1}\right).$

In summary, the constants $C_{w_{1}}$, $C_{w_{2}}$,
$[w_{1}]_{B_{1,\mathcal{D},\Omega}}$ and $[w_{2}]_{B_{1,\mathcal{D},\Omega}}$
are dominated by some quantities depending only on $L_{w}$,
$[w^{q}]_{B_{p,\mathcal{D},\Omega}}$, $p$, $q$ and $\delta$.

Now, considering $w_{1}$ as a weight defined on $\mathbb{D}$ (setting
$w_{1}(z):=0$ for all $z\in\mathbb{D}\setminus\Omega$), and arguing as in
Lemma 2.13, we see that

$M_{\mathcal{D}}w_{1}(z)=M_{\mathcal{D},\Omega}w_{1}(z)\leq[w_{1}]_{B_{1,\mathcal{D},\Omega}}w_{1}(z)$

for almost every $z\in\Omega$. Repeating this observation with $w_{2}$ in
place of $w_{1}$, we deduce that there exists a function $k$ in $\Omega$, with
$\frac{(4C_{w_{2}})^{1-p}}{[w_{1}]_{B_{1,\mathcal{D},\Omega}}}\leq k(z)\leq
4C_{w_{1}}[w_{2}]_{B_{1,\mathcal{D},\Omega}}^{p-1}$, such that

$w(z)=\big{(}k(z)M_{\mathcal{D}}w_{1}(z)\big{)}^{\frac{1}{\delta
q}}\big{(}M_{\mathcal{D}}w_{2}(z)\big{)}^{\frac{1-p}{\delta q}}=:W(z)$

for almost every $z\in\Omega$. Finally, fixing $\delta>\frac{1}{q}$ (for
instance $\delta=\frac{q+1}{2q}$) and setting $k(z):=1$ for
$z\in\mathbb{D}\setminus\Omega$, Lemma 2.6 yields that $W\in
B_{p,\mathcal{D}}$ and, moreover,

$\displaystyle[W]_{B_{p,\mathcal{D}}}$ $\displaystyle\leq$
$\displaystyle[(kM_{\mathcal{D}}w_{1})^{\frac{1}{\delta
q}}]_{B_{1,\mathcal{D}}}[(M_{\mathcal{D}}w_{2})^{\frac{1}{\delta q}}]^{p-1}$
$\displaystyle\leq$
$\displaystyle\big{(}4^{p}C_{w_{1}}C_{w_{2}}^{p-1}[w_{1}]_{B_{1,\mathcal{D},\Omega}}[w_{2}]_{B_{1,\mathcal{D},\Omega}}^{p-1}\big{)}^{\frac{1}{\delta
q}}\left(\frac{2-\frac{1}{\delta q}}{1-\frac{1}{\delta q}}\right)^{p}.$

In addition to this, since both $M_{\mathcal{D}}w_{1}$ and
$M_{\mathcal{D}}w_{2}$ are almost constant in $3/4$-top halves we realize that

$C_{W}\leq\big{(}4^{p}C_{w_{1}}C_{w_{2}}^{p-1}[w_{1}]_{B_{1,\mathcal{D},\Omega}}[w_{2}]_{B_{1,\mathcal{D},\Omega}}^{p-1}\big{)}^{\frac{1}{\delta
q}}$

and therefore $W$ that is of dyadic bounded hyperbolic oscillation, in light
of Lemma 2.7.

The converse result follows the from reverse Hölder property for dyadic
Békollé–Bonami weights of dyadic bounded hyperbolic oscillation. ∎

The proof also works equally well for $p=1$.

######  Theorem 2.15.

Let $\mathcal{D}$ be a dyadic grid and let $\Omega$ be the union of top halves
of a subcollection of $\mathcal{D}$. If $w$ is a weight of dyadic bounded
hyperbolic oscillation such that $w^{q}\in B_{1,\mathcal{D},\Omega}$ for some
$q>1$, then there exists $W\in B_{1,\mathcal{D}}$ such that $W(z)=w(z)$ for
almost every $z\in\Omega$ and

$[W]_{B_{1,\mathcal{D}}}\leq\frac{2q-1}{q-1}[w^{q}]_{B_{1,\mathcal{D},\Omega}}^{\frac{1}{q}}e^{3L_{w}}\quad\mbox{
and }\quad
L_{W}\leq\dfrac{\log(64[w^{q}]_{B_{1,\mathcal{D},\Omega}})}{q}+3L_{w}.$

Conversely, if $w$ coincides almost everywhere in $\Omega$ with some $W\in
B_{1,\mathcal{D}}$ of dyadic bounded hyperbolic oscillation, then there exists
$q>1$ such that $w^{q}\in B_{1,\mathcal{D},\Omega}$.

##  3\. The average of dyadic extensions

In the continuous setting, for a non-negative weight $w$ on a measurable
subset $\Omega\subset\mathbb{D}$, we say that $w\in B_{p,\Omega}$ if

$[w]_{B_{p,\Omega}}:=\sup_{I\subset\mathbb{T}}\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w\>dA\left(\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w^{-\frac{1}{p-1}}\>dA\right)^{p-1}<\infty.$

When $p=1$, we define $B_{1,\Omega}$ as the space of all weights $w$ in
$\Omega$ such that

$M_{\Omega}w(z):=\sup_{S(I)\ni
z}\dfrac{1}{A\big{(}S(I)\big{)}}\int_{S(I)\cap\Omega}w\,dA\leq Cw(z)$

for almost every $z\in\Omega$. In this section we will push the extension
results of the previous section, Theorems 2.14 and 2.15, from the dyadic to
the continuous setting by forming geometric averages of dyadic weights. Note
that averaging techniques have a successful history in weighted theory, for
instance, see [9] and [16].

Given a parameter $\theta\in[0,1]$, let $\mathcal{D}(\theta)$ be the dyadic
grid consisting of all arcs of the form

$\left\\{e^{it}:\dfrac{\pi k}{2^{n-1}}\leq
t-\theta<\dfrac{\pi(k+1)}{2^{n-1}}\right\\},\qquad n\geq
0,\;k\in\\{0,\ldots,2^{n}-1\\}.$

For the sake of brevity, we will denote by $\beta_{\theta}$ the associated
metric $\beta_{\mathcal{D}(\theta)}$ defined in (2.8). Let $\beta$ denote the
hyperbolic metric on $\mathbb{D}$,

(3.1)
$\beta(z,\zeta)=\dfrac{1}{2}\log\dfrac{1+|\rho(z,\zeta)|^{2}}{1-|\rho(z,\zeta)|^{2}},\quad
z,\zeta\in\mathbb{D}.$

We wish to relate this continuous metric to the the average of dyadic
hyperbolic metrics. For $z\in\mathbb{D}$, let $I_{z}$ denote the arc centered
at $z/|z|$ of length $1-|z|$, and, for $z,\zeta\in\mathbb{D}$, let
$P(z,\zeta)$ be the smallest arc (not necessarily dyadic) that contains both
$I_{z}$ and $I_{\zeta}$. See Figure 3 for an example. Similarly, we denote by
$\mathcal{I}^{\theta}_{z}$ the interval in $\mathcal{D}(\theta)$ such that
$z\in T(\mathcal{I}^{\theta}_{z})$, and by $P_{\theta}(z,\zeta)$ the smallest
interval in $\mathcal{D}(\theta)$ containing $\mathcal{I}^{\theta}_{z}$ and
$\mathcal{I}^{\theta}_{\zeta}$. Further, for any arc $I\subset\mathbb{T}$,
$P_{\theta}(I)$ will denote the smallest interval in $\mathcal{D}(\theta)$
containing $I$. With these definitions, we clearly have that

(3.2) $P_{\theta}(z,\zeta)\subseteq
P_{\theta}\big{(}P(z,\zeta)\big{)},\qquad\theta\in[0,1].$

$\bullet$$z$$\bullet$$\zeta$$P(z,\zeta)$$\mathbb{T}$ Figure 3. Example of
$P(z,\zeta)$.

######  Lemma 3.1.

There exists an $M>0$ such that, for all $z$ and $\zeta$ in $\mathbb{D}$,

$\log\frac{|P(z,\zeta)|}{\min\\{|I_{z}|,|I_{\zeta}|\\}}\leq
M\big{(}1+\beta(z,\zeta)\big{)}.$

###### Proof.

We will use the estimate

(3.3)
$|1-z\overline{\zeta}|\simeq\max\\{1-|z|^{2},1-|\zeta|^{2},|z^{*}-\zeta^{*}|\\},\qquad
z,\zeta\in\overline{\mathbb{D}}\setminus\\{0\\},$

where $z^{*}:=z/|z|$. When $\zeta=1$ the above estimate is an exercise in
plane geometry. For general points
$z,\zeta\in\overline{\mathbb{D}}\setminus\\{0\\}$ it follows from the previous
case and the estimate

$\max\\{1-|z|^{2},1-|\zeta|^{2}\\}\leq 1-|z\zeta|\leq
2\max\\{1-|z|^{2},1-|\zeta|^{2}\\}.$

Since

$1-\rho^{2}(z,\zeta)=\frac{(1-|z|^{2})(1-|\zeta|^{2})}{|1-\overline{\zeta}z|^{2}},\qquad
z,\zeta\in\mathbb{D},$

we have, thanks to (3.1) and (3.3), that

(3.4)
$\beta(z,\zeta)\simeq\log\frac{\max\\{(1-|z|^{2}),(1-|\zeta|^{2}),|z^{*}-\zeta^{*}|\\}}{\sqrt{(1-|z|^{2})(1-|\zeta|^{2})}}.$

To conclude we consider three different cases.

Case (i):

$z$ belongs to $S(I_{\zeta})$. In this case, the maximum of (3.3) is
comparable to $(1-|\zeta|^{2})$, and by construction $|P(z,\zeta)|\leq
2|I_{\zeta}|=2(1-|\zeta|^{2})$. Therefore (3.4) becomes

$\beta(z,\zeta)\gtrsim\log\frac{\sqrt{1-|\zeta|^{2}}}{\sqrt{1-|z|^{2}}}=\frac{1}{2}\Big{(}\log\frac{1}{2}+\log\frac{|P(z,\zeta)|}{|I_{z}|}\Big{)}$

and this concludes the proof, since if $z$ belongs to $S(I_{\zeta})$, then
$|I_{z}|\leq|I_{\zeta}|$.

Case (ii):

$\zeta$ belongs to $S(I_{z})$. This case is treated exactly as the previous
one, with $z$ and $\zeta$ exchanged.

Case (iii):

Neither $z\in S(I_{\zeta})$, nor $\zeta\in S(I_{z})$. In this case, the
maximum in (3.3) is comparable to $|z^{*}-\zeta^{*}|$, and

$|P(z,\zeta)|\lesssim|z^{*}-\zeta^{*}|+\frac{1}{2}(1-|z|^{2})+\frac{1}{2}(1-|\zeta|^{2})\leq
2|z^{*}-\zeta^{*}|.$

Hence (3.4) becomes

$\beta(z,\zeta)\gtrsim\log\frac{1/2|P(z,\zeta)|}{\sqrt{(1-|z|^{2})(1-|\zeta|^{2})}}\geq\log\frac{1/2|P(z,\zeta)|}{\min\\{|I_{z}|,|I_{\zeta}|\\}}.\qed$

######  Remark 3.2.

In general, $\beta(z,\zeta)\lesssim\big{(}1+\beta_{\theta}(z,\zeta)\big{)}$
for all $\theta\in[0,1]$ and $z,\zeta\in\mathbb{D}$. However, the two
quantities are not comparable. For instance, it is enough to test with
$z_{k}=(1-2^{-k})e^{i(\theta+2^{-k-2})}$ and
$\zeta_{k}=(1-2^{-k})e^{i(\theta-2^{-k-2})}$ to realize that
$\beta_{\theta}(z_{k},\zeta_{k})=k+1$ while
$\big{(}\beta(z_{k},\zeta_{k})\big{)}_{k\geq 1}$ is bounded because both
$z_{k}$ and $\zeta_{k}$ belong to
$T\big{(}\\{e^{it}:|t-\theta|<2^{-k-1}\\}\big{)}$ for every $k$.

Thanks to Lemma 3.1 we can estimate the (continuous) hyperbolic distance
$\beta$ in terms of a quantity that is reminiscent of the dyadic hyperbolic
metric defined in (2.8). The aim of the next results is to show that for
separated points, $\beta_{\theta}$ and $\beta$ are actually comparable on
average, cf. [15, Sec. 4]

######  Lemma 3.3.

Let $I$ be a subarc of $\mathbb{T}$, and let $N\in\mathbb{N}$ be the number
such that $2^{-N}\leq|I|<2^{-N+1}$. Then

$\left|\left\\{\theta\in[0,1]\,:\,|P_{\theta}(I)|=2^{k-N}\right\\}\right|\lesssim
2^{-k},\qquad 2\leq k\leq N.$

###### Proof.

The center of $P_{\theta}(I)$ is always contained in $I$. Hence, if
$|P_{\theta}(I)|=2^{k-N}$, there exists an interval $J$ in
$\mathcal{D}(\theta)$ of that length whose center is contained in $I$. So we
only need to note that the Lebesgue measure of the set of $\theta$ for which
such an interval $J$ exists is $2^{N-k}\cdot|I|\lesssim 2^{-k}$. ∎

######  Theorem 3.4.

There exists a constant $\widetilde{M}$ such that, for all $z$ and $\zeta$ in
$\mathbb{D}$, we have that

$\int_{0}^{1}\beta_{\theta}(z,\zeta)\,d\theta\leq\widetilde{M}\big{(}1+\beta(z,\zeta)\big{)}.$

###### Proof.

By (3.2), we have that

$\int_{0}^{1}\beta_{\theta}(z,\zeta)\,d\theta=\int_{0}^{1}\log_{2}\frac{|P_{\theta}(z,\zeta)|}{\min\\{|\mathcal{I}^{\theta}_{z}|,|\mathcal{I}^{\theta}_{\zeta}|\\}}\,d\theta\leq\int_{0}^{1}\log_{2}\frac{\big{|}P_{\theta}\big{(}P(z,\zeta)\big{)}\big{|}}{\min\\{|I_{z}|,|I_{\zeta}|\\}}.$

Hence we divide and multiply the argument of the logarithm by $|P(z,\zeta)|$
and we obtain, in view of Lemma 3.1 and Lemma 3.3, that

$\begin{split}\int_{0}^{1}\beta_{\theta}(z,\zeta)\,d\theta&\lesssim\big{(}1+\beta(z,\zeta)\big{)}+\int_{0}^{1}\log_{2}\frac{|P_{\theta}(P(z,\zeta))|}{|P(z,\zeta)|}\,d\theta\\\
&=\big{(}1+\beta(z,w)\big{)}+\sum_{k=0}^{\infty}k\cdot|\\{\theta\in[0,1]\,:\,|P_{\theta}(P(z,w))|=2^{k}|P(z,w)|\\}|\\\
&\lesssim\big{(}1+\beta(z_{1},z_{2})\big{)}+\sum_{k=0}^{\infty}k2^{-k}.\qed\end{split}$

######  Remark 3.5.

Theorem 3.4 implies that, given a family of functions $f_{\theta}$ with the
property that

$|f_{\theta}(z)-f_{\theta}(\zeta)|\leq
L_{\theta}\big{(}1+\beta_{\theta}(z,\zeta)\big{)},\quad\forall
z,w\in\mathbb{D},$

and $\sup_{\theta}L_{\theta}<\infty$, the average

$f(z):=\int_{0}^{1}f_{\theta}(z)\,d\theta,\quad z\in\mathbb{D},$

is quasi-Lipschitz with respect the hyperbolic metric $\beta$,

$|f(z)-f(\zeta)|\lesssim\big{(}1+\beta(z,\zeta)\big{)}.$

Our goal is to show that, given a family of dyadic weights
$(W_{\theta})_{\theta\in[0,1]}$, their geometric average

$W(z):=\exp\left(\int_{0}^{1}\log W_{\theta}(z)\,d\theta\right)$

is a $B_{p}$ weight, as long as the quantities
$([W_{\theta}]_{B_{p,\mathcal{D}(\theta)}})_{\theta\in[0,1]}$ are uniformly
bounded. Moreover, we will also prove that $W$ is of bounded hyperbolic
oscillation if the weights $W_{\theta}$ are uniformly of dyadic bounded
hyperbolic oscillation. To do this, the following version of Minkowski’s
inequality will come in handy.

######  Lemma 3.6.

Let $\mu_{1}$ and $\mu_{2}$ be two probability measures on $X_{1}$ and
$X_{2}$. Then

(3.5)
$\int_{X_{1}}e^{\int_{X_{2}}\log|f(x_{1},x_{2})|\,d\mu_{2}(x_{2})}\,d\mu_{1}(x_{1})\leq
e^{\int_{X_{2}}\log\left(\int_{X_{1}}|f(x_{1},x_{2})|\,d\mu_{1}(x_{1})\right)\,d\mu_{2}(x_{2})}$

for every measurable function $f:X_{1}\times X_{2}\rightarrow\mathbb{R}$.

###### Proof.

We start from the classical version of Minkowski’s inequality,

$\left(\int_{X_{1}}\left(\int_{X_{2}}|F(x_{1},x_{2})|\,d\mu_{2}(x_{2})\right)^{P}\,d\mu_{1}(x_{1})\right)^{\frac{1}{P}}\leq\int_{X_{2}}\left(\int_{X_{1}}|F(x_{1},x_{2})|^{P}\,d\mu_{1}(x_{1})\right)^{\frac{1}{P}}\,d\mu_{2}(x_{2}),$

which holds for all $1\leq P<\infty$ and every measurable function
$F:X_{1}\times X_{2}\rightarrow\mathbb{R}$. If $0<q<1$, choosing
$P=\frac{1}{q}$ and $F=|f|^{q}$ gives that

$\int_{X_{1}}\left(\int_{X_{2}}|f(x_{1},x_{2})|^{q}\,d\mu_{2}(x_{2})\right)^{\frac{1}{q}}\,d\mu_{1}(x_{1})\leq\left(\int_{X_{2}}\left(\int_{X_{1}}|f(x_{1},x_{2})|\,d\mu_{1}(x_{1})\right)^{q}\,d\mu_{2}(x_{2})\right)^{\frac{1}{q}}.$

Letting $q\rightarrow 0^{+}$, we therefore see that (3.5) holds. Indeed, for
any probability measure $\mu$ on $X$ and measurable function $g$, it is an
exercise with Jensen’s inequality to show that
$\left(\int_{X}|g|^{q}\,d\mu\right)^{1/q}$ converges to the geometric mean of
$|f|$ as $q\to 0^{+}$. ∎

######  Lemma 3.7.

Let $(W_{\theta})_{\theta\in[0,1)}$ be a measurable family of weights such
that

$\sup_{\theta\in[0,1)}[W_{\theta}]_{B_{1},\mathcal{D}(\theta)}<\infty.$

Then

$W(z):=\exp\left(\int_{0}^{1}\log W_{\theta}(z)\,d\theta\right)$

is a $B_{1}$ weight. Moreover, if each $W_{\theta}$ is of
$\mathcal{D}(\theta)$-bounded hyperbolic oscillation and the constant
$L_{W_{\theta}}$ in (2.9) is uniformly bounded in $\theta$, then $W$ is of
bounded hyperbolic oscillation.

###### Proof.

Let $z\in\mathbb{D}$, and let $I$ be an arc such that $z\in S(I)$. Then, (3.5)
yields that

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}W(\zeta)\,dA(\zeta)$ $\displaystyle\leq$
$\displaystyle\exp\left(\int_{0}^{1}\log\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(I)}W_{\theta}(\zeta)\,dA(\zeta)\right)\,d\theta\right)$
$\displaystyle\leq$
$\displaystyle\exp\left(\int_{0}^{1}\log\left(\dfrac{A\big{(}S(P_{\theta}(I))\big{)}}{A\big{(}S(I)\big{)}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S(P_{\theta}(I))}W_{\theta}(\zeta)\,dA(\zeta)\right)\,d\theta\right)$
$\displaystyle\leq$
$\displaystyle\exp\left(\int_{0}^{1}\log\left(\dfrac{A\big{(}S(P_{\theta}(I))\big{)}}{A\big{(}S(I)\big{)}}[W_{\theta}]_{B_{1,\mathcal{D}(\theta)}}W_{\theta}(z)\right)\,d\theta\right).$

Thus an application of Lemma 3.3 shows that
$[W]_{B_{1}}\lesssim\sup_{\theta\in[0,1)}[W^{(\theta)}]_{B_{1},\mathcal{D}(\theta)}$.

The second part of the theorem follows immediately from Theorem 3.4, cf.
Remark 3.5. ∎

Now, we are ready to prove our main extension theorem, Theorem 1.2. We split
the proof into two parts, first considering the case $p=1$.

######  Theorem 3.8 (Extension of $B_{1,\Omega}$ weights).

Let $\Omega$ be the union of top halves of Carleson squares, that is,
$\Omega=\bigcup_{z\in C}T(I_{z})$, for some set $C\subseteq\mathbb{D}$.
Consider a weight $w$ on $\Omega$ of bounded hyperbolic oscillation. Then, the
following statements are equivalent:

  1. (1)

There exists $q>1$ such that $w^{q}\in B_{1,\Omega}$.

  2. (2)

There exists $W\in B_{1}$ of bounded hyperbolic oscillation such that
$w(z)=W(z)$ for almost every $z\in\Omega$.

###### Proof.

That the second statement implies the first is a direct consequence of the
reverse Hölder property proved in [2], cf. Remark 2.12.

Therefore, assume there exists $q>1$ such that $w^{q}\in B_{1,\Omega}$. Take
$\theta\in[0,1]$ and consider the following sub-collection of
$\mathcal{D}(\theta)$,

$\mathcal{G}_{\theta}:=\left\\{I\in\mathcal{D}(\theta)\>:\>A\big{(}T(I)\cap\Omega\big{)}\geq\dfrac{A\big{(}T(I)\big{)}}{18}\right\\}.$

Since $\Omega$ is the union of top-halves, $\mathcal{G}_{\theta}$ is non-
empty. Indeed, given $T(I_{z})\subset\Omega$, let $J\in\mathcal{D}(\theta)$ be
such that $|J|\geq|I_{z}|>\frac{|J|}{2}$ and $|I_{z}\cap
J|\geq\frac{|I_{z}|}{2}$. If neither $J$ nor its children belong to
$\mathcal{G}_{\theta}$, we would have that
$A\big{(}T_{3/4}(J)\cap\Omega\big{)}<\frac{A(T_{3/4}(J))}{18}$, but, on the
other hand,

$\displaystyle\dfrac{A\big{(}T_{3/4}(J)\cap\Omega\big{)}}{A\big{(}T_{3/4}(J)\big{)}}$
$\displaystyle\geq$ $\displaystyle\dfrac{A\big{(}T_{3/4}(J)\cap
T(I_{z})\big{)}}{A\big{(}T_{3/4}(J)\big{)}}=\dfrac{|I_{z}\cap
J|\left(\left(1-\frac{|I_{z}|}{2}\right)^{2}-(1-|I_{z}|)^{2}\right)}{|J|\left(\left(1-\frac{|J|}{4}\right)^{2}-(1-|J|)^{2}\right)}$
$\displaystyle\geq$
$\displaystyle\dfrac{1}{4}\dfrac{1-\frac{3|J|}{4}}{3-\frac{15|J|}{8}}\geq\dfrac{1}{18}.$

Thus $\mathcal{G}_{\theta}$ cannot be empty. Furthermore, the argument shows
that every arc $I_{\zeta}$, $\zeta\in C$, intersects at least one arc of
$\mathcal{G}_{\theta}$.

Let $\Omega_{\theta}:=\bigcup_{I\in\mathcal{G}_{\theta}}T(I)$, and define

(3.6)
$w_{\theta}(z):=\sum_{I\in\mathcal{G}_{\theta}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(I)\cap\Omega}w\,dA\right)\leavevmode\nobreak\
\chi_{T(I)}(z),\qquad z\in\Omega_{\theta}.$

Then $w_{\theta}$ is a weight of dyadic bounded hyperbolic oscillation on
$\Omega_{\theta}$, uniformly in $\theta$, since $\log w$ is quasi-Lipschitz in
$\Omega$ with respect to the (continuous) hyperbolic metric $\beta$.

Moreover, $w_{\theta}^{q}\in B_{1,\mathcal{D}(\theta),\Omega_{\theta}}$; it is
equivalent to see that there exists a constant $K$ such that

$\dfrac{1}{A\big{(}S(J)\big{)}}\int_{S(J)\cap\Omega_{\theta}}w_{\theta}^{q}\,dA\leq
K\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(I)\cap\Omega}w\,dA\right)^{q}$

for all $J\in\mathcal{D}(\theta)$ containing $I\in\mathcal{G}_{\theta}$. Using
the hypothesis that $w^{q}$ is a $B_{1,\Omega}$ weight together with Hölder’s
inequality, we realize that

$\displaystyle\dfrac{1}{A\big{(}S(J)\big{)}}\int_{S(J)\cap\Omega_{\theta}}w_{\theta}^{q}\,dA$
$\displaystyle=\dfrac{1}{A\big{(}S(J)\big{)}}\int_{S(J)\cap\Omega_{\theta}}\left(\sum_{J^{\prime}\in\mathcal{G}_{\theta},J^{\prime}\subset
J}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(J^{\prime})\cap\Omega}w\,dA\chi_{T(J^{\prime})}\right)^{q}\,dA$
$\displaystyle=\dfrac{1}{A\big{(}S(J)\big{)}}\sum_{J^{\prime}\in\mathcal{G}_{\theta},J^{\prime}\subset
J}A\big{(}T(J^{\prime})\big{)}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(J^{\prime})\cap\Omega}w\,dA\right)^{q}$ (3.7)
$\displaystyle\leq\dfrac{1}{A\big{(}S(J)\big{)}}\sum_{J^{\prime}\in\mathcal{G}_{\theta},J^{\prime}\subset
J}A\big{(}T(J^{\prime})\big{)}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(J^{\prime})\cap\Omega}w^{q}\,dA$
$\displaystyle\leq\dfrac{18}{A\big{(}S(J)\big{)}}\int_{S(J)\cap\Omega}w^{q}\,dA\leq
18[w^{q}]_{B_{1,\Omega}}w^{q}(z),$

for almost every $z\in T(I)\cap\Omega$. Hence, since
$I\in\mathcal{G}_{\theta}$ and $w^{q}$ is of bounded hyperbolic oscillation,
we have that

$\dfrac{1}{A\big{(}S(J)\big{)}}\int_{S(J)\cap\Omega}w_{\theta}^{q}\,dA\leq
18C_{w}^{q}[w^{q}]_{B_{1,\Omega}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(I)\cap\Omega}w\,dA\right)^{q}.$

Thus $[w_{\theta}^{q}]_{B_{1,\mathcal{D}(\theta),\Omega_{\theta}}}\lesssim
C_{w}^{q}[w^{q}]_{B_{1,\Omega}}$.

Since $\Omega_{\theta}$ is a union of $\mathcal{D}(\theta)$-dyadic top halves,
Theorem 2.15 gives us the existence of a weight $W_{\theta}\in
B_{1,\mathcal{D}(\theta)}$ of $\mathcal{D}(\theta)$-bounded hyperbolic
oscillation, with constants depending only on $C_{w}$, $q$ and
$[w^{q}]_{B_{1,\Omega}}$, such that $W_{\theta}(z)=w_{\theta}(z)$ for almost
every $z\in\Omega_{\theta}$. By Lemma 3.7 we thus have that the function

$W(z):=\exp\left(\int_{0}^{1}\log W_{\theta}(z)\,d\theta\right)$

is a $B_{1}$ weight of bounded hyperbolic oscillation.

To finish the proof we will show that there exists a positive function $F$ on
$\mathbb{D}$, such that $\log F$ is bounded, and

$w(z)=F(z)W(z),\qquad z\in\Omega.$

This yields the desired extension of $w$.

If $\zeta\in C$, let $z_{1},\ldots,z_{4}$ be the four corners of
$T(I_{\zeta})$, and let $z_{5}$ be the midpoint of the arc that joins the two
outside corners (the corners of greater modulus). Since
$T(I_{\zeta})\subset\Omega$, for every $\theta\in[0,1]$, at least one of the
points $z_{1},\ldots,z_{5}$ has to belong to $\Omega_{\theta}$; call it
$z_{\theta}$. Indeed, as remarked earlier, there is $J\in\mathcal{G}_{\theta}$
which intersects $I_{\zeta}$, and then $T(J)$ must contain at least one of the
points (see Figure 4 for some examples). For the avoidance of doubt, we
provide a proof of this claim. If $|I_{\zeta}|$ is dyadic, then $T(I_{\zeta})$
is covered by at most two dyadic top halves, each of which contains an inner
corner point. One of these top-halves coincides with $T(J)$, and we are done.
Otherwise, let $k$ be the unique non-negative integer such that
$2^{-k}>|I_{\zeta}|>2^{-k-1}$, and, for the sake of contradiction, assume that
$T(J)\cap\\{z_{1},\ldots,z_{5}\\}=\emptyset$. Since $T(J)\cap T(I_{\zeta})$ is
non-empty, the generation of $J$ is either $k$ or $k+1$, and we therefore have
two possible cases:

  * •

If $|J|=2^{-k}$, then, since $T(J)$ does not contain either of the two inner
corners of $T(I_{\zeta})$, it must be that $|J|<|I_{\zeta}|<2^{-k}$. This is
impossible.

  * •

If $|J|=2^{-k-1}$, we may split $I_{\zeta}$ into two arcs of the same length.
Since $T(J)$ does not contain $z_{5}$ or the outer corners of $T(I_{\zeta})$,
$J$ must be completely contained in one of the halves of $I_{\zeta}$, but then
$|J|<\frac{|I_{\zeta}|}{2}<2^{-k-1}$, which is absurd.

In either case we have reached a contradiction, and the existence of
$z_{\theta}\in\Omega_{\theta}$ is proven.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

Figure 4. The top-half $T(I)$ can always be covered by at most 5 dyadic top
halves.

In the case that $z_{j}\not\in\Omega$ for some $j$, we may define

$w(z_{j}):=\liminf_{r\rightarrow 0^{+}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\\{|z-z_{j}|<r\\}\cap\Omega}w\,dA.$

For any $z\in T(I_{\zeta})$, note then that

$\displaystyle|\log w(z)-\log w(z_{\theta})|$ $\displaystyle\leq$
$\displaystyle L_{w},$ $\displaystyle|\log W_{\theta}(z_{\theta})-\log
w(z_{\theta})|$ $\displaystyle=$ $\displaystyle|\log
w_{\theta}(z_{\theta})-\log w(z_{\theta})|\lesssim L_{w},$
$\displaystyle\int_{0}^{1}|\log W_{\theta}(z)-\log
W_{\theta}(z_{\theta})|\,d\theta$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{5}\int_{0}^{1}|\log W_{\theta}(z)-\log
W_{\theta}(z_{j})|\,d\theta$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{5}\int_{0}^{1}L_{W_{\theta}}\big{(}1+\beta_{\theta}(z,z_{j})\big{)}\,d\theta\leq
K.$

These constants are independent of $\zeta$, and thus $|\log w-\log W|$ is
uniformly bounded on $\Omega$. Defining $F$ accordingly on $\Omega$, and
setting $F(z):=1$ for $z\in\mathbb{D}\setminus\Omega$, we obtain the desired
extension of $w$. ∎

The proof for the case $p>1$ is similar, but additionally relies on
factorization.

######  Theorem 3.9 (Extension of $B_{p,\Omega}$ weights).

Let $\Omega$ be the union of top halves of Carleson squares, that is,
$\Omega=\bigcup_{z\in C}T(I_{z})$, for some set $C\subseteq\mathbb{D}$.
Consider a weight $w$ on $\Omega$ of bounded hyperbolic oscillation. Then, for
$1<p<\infty$, the following statements are equivalent:

  1. (1)

There exists $q>1$ such that $w^{q}\in B_{p,\Omega}$.

  2. (2)

There exists $W\in B_{p}$ of bounded hyperbolic oscillation such that
$w(z)=W(z)$ for almost every $z\in\Omega$.

###### Proof.

By possibly replacing $w$ with $w^{-\frac{1}{p-1}}$, we may suppose that
$1<p\leq 2$. Proceeding like in the previous proof, we have that the weights
introduced in (3.6) are of uniformly bounded hyperbolic oscillation. Moreover,
arguing as in (3), replacing Hölder’s inequality with Jensen’s inequality when
estimating the averages of $w_{\theta}^{-q/(p-1)}$, we obtain

$[w_{\theta}^{q}]_{B_{p,\mathcal{D}(\theta),\Omega_{\theta}}}\leq
18^{p}[w^{q}]_{B_{p,\Omega}}.$

Thus, Theorem 2.14 yields that every $w_{\theta}$ may be extended to a weight
$W_{\theta}\in B_{p,\mathcal{D}(\theta)}$ of dyadic bounded hyperbolic
oscillation, with uniform bounds. We now apply dyadic factorization, Theorem
2.10, with $\Omega=\mathbb{D}$, in which case we know that
$M_{\mathcal{D}(\theta)}$ is $L^{p}(W_{\theta})$-bounded if and only if
$W_{\theta}\in B_{p,\mathcal{D}(\theta)}$. Thus, taking also Lemma 2.7 into
account, we are able to write $W_{\theta}=W_{1,\theta}W_{2,\theta}^{1-p}$, for
some $W_{1,\theta},W_{2,\theta}\in B_{1,\mathcal{D}(\theta)}$ of
$\mathcal{D}(\theta)$-bounded hyperbolic oscillation, with uniform estimates.
Their geometric average $W$ satisfies that

(3.8) $W=\left(\exp\int_{0}^{1}\log
W_{1,\theta}\,d\theta\right)\left(\exp\int_{0}^{1}\log
W_{2,\theta}\,d\theta\right)^{1-p}.$

By Lemma 3.7, $W\in B_{p}$ and is of bounded hyperbolic oscillation. The same
discussion as in Theorem 3.8 shows that $W$ coincides with $w$ in $\Omega$, up
to a positive function $F$ such that $\log F$ is bounded. This completes the
proof. ∎

From the proof of Theorem 3.9 we also deduce that $B_{p}$ weights of bounded
hyperbolic oscillation can be factorized in terms of $B_{1}$ weights of
bounded hyperbolic oscillation.

######  Corollary 3.10.

Let $p>1$ and let $w$ be a weight on $\mathbb{D}$ of bounded hyperbolic
oscillation. Then $w\in B_{p}$ if and only if there exist $w_{1},w_{2}\in
B_{1}$ of bounded hyperbolic oscillation such that $w=w_{1}w_{2}^{1-p}$.

##  4\. Restrictions of Bloch functions and martingales

We first recall the probabilistic framework, due to Makarov [14], connecting
Bloch functions and martingales. To this end, in this section, let
$\mathcal{D}_{k}$ be the set of dyadic sub-intervals of $[0,1]$ of length
$2^{-k}$,

$\mathcal{D}_{k}:=\\{[j/2^{-k},(j+1)/2^{-k})\,:\,j=0,\dots,2^{k}-1\\},$

and let $\mathcal{D}=\bigcup_{k=0}^{\infty}\mathcal{D}_{k}$ be the collection
of all dyadic intervals. A dyadic martingale is a sequence of random variables
$M=(M_{n})_{n\in\mathbb{N}}$, defined on $[0,1]$, such that: i) $M_{k}$ is
constant on each interval $I\in\mathcal{D}_{k}$ – denote the value by $M_{I}$,
and ii) whenever $I=I_{1}\cup I_{2}$, $I\in\mathcal{D}_{k}$,
$I_{j}\in\mathcal{D}_{k+1}$ for $j=1,2$, it holds that

$M_{I}=\frac{1}{2}\left(M_{I_{1}}+M_{I_{2}}\right).$

Depending on the context, we will refer to $M$ either as the sequence
$(M_{n})_{n\in\mathbb{N}}$ or the family $(M_{I})_{I\in\mathcal{D}}$. When
identifying $\mathcal{D}$ with a dyadic tree, one can view a dyadic martingale
as a function on the tree nodes, whose value at each node is the average of
its values at that node’s children.

Makarov first observed that given a Bloch function $b$ and a dyadic interval
$I$, the limit

(4.1) $b_{I}:=\lim_{r\to 1}\frac{1}{|I|}\int_{I}b(re^{2\pi i\theta})\,d\theta$

exists, and the collection $(b_{I})_{I\in\mathcal{D}}$ defines a dyadic
martingale on $[0,1]$. Moreover, there exists a universal constant $C$ such
that

(4.2) $|b_{I}-b(z)|<C\|b\|_{\mathcal{B}},\qquad z\in T(I).$

Therefore, since $b$ is Lipschitz with respect to the hyperbolic metric of
$\mathbb{D}$, the martingale $(b_{I})_{I\in\mathcal{D}}$ has uniformly bounded
increments. Of course, this observation extends beyond the scope of the dyadic
structure. That is,

(4.3) $|b_{I}-b_{J}|<C\|b\|_{\mathcal{B}},$

for every pair of adjacent dyadic intervals $I$ and $J$ of the same length.
Moreover, it turns out that (4.3) characterizes those dyadic martingales which
arise from Bloch functions.

######  Lemma 4.1 ([14, Lemma 2.1]).

A real dyadic martingale $(S_{I})_{I\in\mathcal{D}}$ satisfies (4.3) if and
only if there exists a Bloch function $b$ such that
$\operatorname{Re}b_{I}=S_{I}$ for all $I\in\mathcal{D}$.

A real dyadic martingale satisfying (4.3) is hence known as a _Bloch
martingale_.

######  Example 4.2 (The random walk and Kahane’s martingale).

For any $x$ in $[0,1]$, let $(\varepsilon_{n}(x))_{n\in\mathbb{N}}$ be its
dyadic expansion, so that

$x=\sum_{n=1}^{\infty}\varepsilon_{n}(x)2^{-n},\qquad\varepsilon_{n}(x)\in\\{0,1\\}.$

For any $n\in\mathbb{N}$, let $Y_{n}\colon[0,1]\to\\{-1,1\\}$ be the random
variable defined as

$Y_{n}(x):=2\varepsilon_{n}(x)-1.$

Then the sequence $(S_{n})_{n\in\mathbb{N}}$ defined as

$S_{n}:=\sum_{k=1}^{n}Y_{k}$

is a dyadic martingale, which corresponds to the _random walk_ on the dyadic
tree. It is a dyadic martingale with bounded increments, but it is not a Bloch
martingale. Indeed, if $I_{k}=[1/2-2^{-k},1/2)$ and $J_{k}=[1/2,1/2+2^{-k})$,
then for all $k\geq 2$ we have $S_{I_{k}}=k-2=-S_{J_{k}}$, so that (4.3)
fails.

$\bullet$-4,-2,-2,0,-2,0,0,2,-2,0,0,2,0,2,2,4$\bullet$-3,-1,-1,1,-1,1,1,3$\bullet$-2,0,0,2$\bullet$-1,
1$\bullet$0

Figure 5. The first generations of the random walk.

On the other hand, one can obtain a Bloch martingale by switching to $4$-adic
expansions. Let $x=\sum_{n=1}^{\infty}\nu_{n}(x)4^{-n}$ be the $4$-adic
expansion of a point $x$ in $[0,1]$, $\nu_{n}(x)\in\\{0,1,2,3\\}$, and
consider the sequence of random variables

$W_{n}(x):=\begin{cases}1\quad&\text{if}\quad\nu_{n}(x)\in\\{0,3\\}\\\
-1\quad&\text{if}\quad\nu_{n}(x)\in\\{1,2\\}\end{cases}.$

Then _Kahane’s martingale_ $K$, $K_{n}:=\sum_{k=1}^{n}W_{k}$ is a Bloch
martingale [14, Prop. 2.2], which can be described as follows: $K_{[0,1]}=0$,
and if $I$ is in $\mathcal{D}_{k}$, then

  * •

if $k$ is odd, then $K_{I}=K_{I^{\prime}}$ where $I^{\prime}$ is the unique
interval in $\mathcal{D}_{k-1}$ containing $I$;

  * •

if $k$ is even, then at the left-most and the right-most sub-intervals of $I$
in $\mathcal{D}_{k+2}$ the martingale $K$ assumes the value $K_{I}+1$, whereas
at the other two sub-intervals of $I$ in $\mathcal{D}_{k+2}$, the martingale
$K$ assumes the value $K_{I}-1$.

In particular, the jumps of $K$ between adjacent intervals, as in (4.3), never
exceed $2$.

$\bullet$2,0,0,2,0,-2,-2,0,0,-2,-2,0,2,0,0,2$\bullet$1,1,-1,-1,-1,-1,1,1$\bullet$1,-1,-1,1$\bullet$0,0$\bullet$0

Figure 6. The first generations of the Kahane martingale.

Probabilistic tools turn out to be extremely valuable in the study of the mean
growth and boundary behaviour of Bloch functions $b$. Makarov showed in [14,
Corollary 3.2] that there exists a universal constant $C$ such that

(4.4) $\limsup_{r\to 1^{-}}\frac{\operatorname{Re}b(re^{2\pi
i\theta})}{\sqrt{\log\frac{1}{1-r}\log\log\log\frac{1}{1-r}}}\leq
C\|b\|_{\mathcal{B}},\qquad\textrm{a.e. }\theta\in[0,1].$

Moreover, the inequality (1.7) can for sufficiently small $a$ be derived from
the Azuma–Hoeffding concentration inequality for martingales with bounded
increments. We single out the following formulation of the Azuma inequality,
as stated in [18, Lemma 4.7].

######  Lemma 4.3.

Let $b$ be a Bloch function of norm one, $\|b\|_{\mathcal{B}}=1$, and for any
$I\in\mathcal{D}$, let $l_{I}(\varepsilon,k)$ be the number of dyadic
intervals $J$ contained in $I$, of length $2^{-k}|I|$, and such that
$|b_{J}-b_{I}|>\varepsilon k$. Then, there exist universal constants
$\gamma<1$ and $C>0$ such that

$l_{I}(\varepsilon,k)\leq C2^{k}\leavevmode\nobreak\
e^{-\gamma\varepsilon^{2}k}\leavevmode\nobreak\ |I|.$

We now have all the necessary tools to prove Theorem 1.4.

###### Proof of Theorem 1.4.

(i):

By conformal invariance, it suffices to prove (1.8) for $z=0$, uniformly for
all Bloch functions of norm less than or equal to $1$. To do so, observe that
for $r\in(0,1)$ and $\lambda>0$, the function

$g(z):=e^{\frac{\lambda|b(z)-b(0)|^{2}}{-\log(1-r^{2})}},\qquad|z|\leq r,$

is subharmonic. Therefore, if $G_{r}$ is the Poisson integral of the
restriction of $g$ to $\\{|z|=r\\}$, we have that $G(z)\geq g(z)$ for all
$|z|\leq r$, and since $Z=(z_{n})_{n\in\mathbb{N}}$ satisfies the Carleson
embedding condition (1.5),

$\displaystyle\sum_{|z_{n}|<r}e^{\frac{\lambda|b(z_{n})-b(0)|^{2}}{-\log(1-r^{2})}}(1-|z_{n}|^{2})$
$\displaystyle=\int_{|z|<r}g(z)\,d\mu_{Z}(z)$
$\displaystyle\leq\int_{|z|<r}G(z)\,d\mu_{Z}(z)\lesssim\int_{0}^{2\pi}|g(re^{i\theta})|^{2}\,d\theta.$

The conclusion then follows from (1.7) upon choosing $\lambda$ sufficiently
small.

(ii):

Again, by conformal invariance, it suffices to consider (1.9) for $z=0$, as
long as the estimates we prove are uniform in Bloch functions $b$ such that
$\|b\|_{\mathcal{B}}\leq 1$. Without loss of generality we may also assume
that $b(0)=0$. We are going to use the dyadic decomposition of the unit disc.
Since $Z$ satisfies the Carleson condition, $Z$ is a finite union of weakly
separated sequences[1, Proposition 9.11]. By possibly decomposing each weakly
separated subsequence into further finite unions, we may assume that each
dyadic top-half contains at most one point of the sequence. Let
$G_{k}=\\{T(I)\,:\,I\in\mathcal{D}_{k}\\}$ denote the top halves of generation
$k$.

Since $Z$ satisfies the Carleson condition,
$M=\sum_{n}(1-|z_{n}|^{2})<\infty$, and thus

$\\#\\{n\,:\,1-|z_{n}|^{2}\geq t\\}\leq\frac{M}{t}.$

Therefore, it is sufficient to estimate

$\\#\left\\{n\,:\,t>(1-|z_{n}|^{2})\,,\,e^{\frac{\lambda|b(z_{n})|^{2}}{-\log(1-|z_{n}|^{2})}}(1-|z_{n}|^{2})>t\right\\}\\\
=\sum_{k\geq\log_{2}\frac{1}{t}}\\#\left\\{z_{n}\in G_{k}\,:\,|b(z_{n})|>k\log
2\sqrt{\frac{1}{\lambda}\left(1+\frac{\log\leavevmode\nobreak\ t}{k\log
2}\right)}\right\\}\\\
\lesssim\sum_{k\geq\log_{2}\frac{1}{t}}2^{k}\leavevmode\nobreak\
e^{-\frac{\gamma k\log^{2}2}{\lambda}\left(1+\frac{\log\leavevmode\nobreak\
t}{k\log 2}\right)}=t^{-\frac{\gamma}{\lambda}\leavevmode\nobreak\ \log
2}\sum_{k\geq\log_{2}\frac{1}{t}}\,2^{k\left(1-\frac{\gamma}{\lambda}\leavevmode\nobreak\
\log 2\right)}\simeq\frac{1}{t},$

where we used Lemma 4.3 and assumed that $\lambda$ was sufficiently small,
$\lambda<\gamma\log 2$.

(iii):

We show that the sum in (1.10) diverges at $z=0$ for the Bloch function $K$
corresponding to the Kahane martingale and a suitably chosen interpolating
sequence $(z_{n})_{n\in\mathbb{N}}$. It is important for the counterexample
that $K$ satisfies the converse inequality to (4.4), see [14, Corollary 3.5].
Namely, there exists a constant $c>0$ such that

(4.5) $\limsup_{r\to 1^{-}}\frac{\operatorname{Re}K(re^{2\pi
i\theta})}{\sqrt{\log\frac{1}{1-r}\log\log\log\frac{1}{1-r}}}\geq
c,\qquad\textrm{a.e. }\theta\in[0,1].$

Let $(s_{j})_{j\in\mathbb{N}}$ be an increasing sequence of positive numbers
such that $s_{j}/j\to\infty$ as $j\to\infty$. We construct the desired
interpolating sequence recursively. We start the process by selecting points
$z_{1}^{1},z_{2}^{1},\dots,z^{1}_{l_{0}}$ for which the corresponding Carleson
squares $(S(I_{z_{i}^{1}}))_{i=1}^{l_{0}}$ are pairwise disjoint. Because of
(4.5), we may choose the points such that

$\frac{|K(z^{1}_{i})|^{2}}{\log\frac{1}{1-|z^{1}_{i}|^{2}}}\geq\frac{c}{2}\log\log\log\frac{1}{1-|z^{1}_{i}|}\geq
s_{1},\qquad i=1,\dots,l_{0},$

and

$\frac{1}{4}\leq\sum_{i=1}^{l_{0}}(1-|z^{1}_{i}|^{2})\leq\frac{1}{2}.$

In this construction, we will refer to the points
$\Lambda_{1}=\\{z_{1}^{1},z_{2}^{1},\dots,z^{1}_{l_{0}}\\}$ as the points of
the first generation.

Recursively, given a point $w\in\Lambda_{j}$ of generation $j$, we choose its
successors in generation $j+1$, $z_{1,w}^{j+1},\dots,z_{l_{w},w}^{j+1}$, so
that $I_{z_{i,w}^{j+1}}\subset I_{w}$, $I_{z_{i,w}^{j+1}}\cap
I_{z_{k,w}^{j+1}}=\emptyset$ if $i\neq k$, and such that

(4.6)
$\frac{|K(z^{j+1}_{i,w})|^{2}}{\log\frac{1}{1-|z^{j+1}_{i,w}|^{2}}}\geq\frac{c}{2}\log\log\log\frac{1}{1-|z^{j+1}_{i,w}|}\geq
s_{j+1},\qquad i=1,\dots,l_{w},$

and

(4.7)
$\frac{1}{4}(1-|w|^{2})\leq\sum_{i=1}^{l_{w}}(1-|z^{j+1}_{i,w}|^{2})\leq\frac{1}{2}(1-|w|^{2}).$

Let $\Lambda_{j+1}$ be the set of all successors to points $w\in\Lambda_{j}$.

We consider the sequence
$Z=(z_{n})_{n\in\mathbb{N}}:=\bigcup_{j=1}^{\infty}\Lambda_{j}.$ Since the
Carleson squares associated with points of the same generation are disjoint,
$Z$ is weakly separated, and the upper bound in (4.7) implies that $\mu_{Z}$
satisfies the Carleson measure condition. Thus $Z$ is interpolating. On the
other hand, the lower bound in (4.7) yields

$\sum_{z\in\Lambda_{j}}(1-|z|^{2})\geq 4^{-j},\qquad j\geq 1,$

and therefore

$\sum_{n\in\mathbb{N}}e^{\frac{\lambda|K(z_{n})-K(0)|^{2}}{-\log(1-|z_{n}|^{2})}}(1-|z_{n}|^{2})\geq\sum_{j=1}^{\infty}e^{\lambda
s_{j}}\sum_{z\in\Lambda_{j}}(1-|z|^{2})=\infty,$

for every $\lambda>0$.

∎

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