# Weighted estimates for the Bergman projection on planar domains

A. Walton Green  A. Walton Green Department of Mathematics Washington
University in Saint Louis 1 Brookings Drive Saint Louis, MO 63130, USA
<EMAIL_ADDRESS>and  Nathan A. Wagner  Nathan A. Wagner Department of
Mathematics Brown University 151 Thayer Street Providence, RI, 02912 USA
<EMAIL_ADDRESS>
###### Abstract.

Let $\Omega\subset\mathbb{C}$ be a bounded, simply connected domain. We
generalize well-known weighted inequalities due to Békollè and Bonami on
$\mathbb{D}$ by giving a sufficient condition on the domain $\Omega$ such that
the Bergman projection $\Pi_{\Omega}$ is bounded on $L^{p}(\Omega,\sigma)$ for
$1<p<\infty$ and $\sigma$ belonging to the Muckenhoupt class
$\mathrm{B}_{p}(\Omega)$ whose basis consists of the images, under the Riemann
map, of Carleson boxes in $\mathbb{D}$. We also show that in the special case
of uniform domains, the same condition is also necessary for the full-range
weighted estimates, and moreover provide an alternative characterization of
the weight class that is intrinsic to $\Omega$. Our approach uses techniques
from both conformal mapping and dyadic harmonic analysis. We also apply these
methods to prove endpoint weak-type estimates and limited-range weighted
inequalities.

###### Key words and phrases:

Bergman projection, planar domains, weighted estimates, Riemann map, weak-type
estimates

###### 2020 Mathematics Subject Classification:

Primary: 30H20, 42B20. Secondary: 30C20

A. W. Green’s research partially supported by NSF grant DMS-2202813.

N. A. Wagner’s research partially supported by NSF grant DMS-2203272.

##  1\. Introduction

Let $\Omega$ be a bounded, simply connected domain in $\mathbb{C}$. A
fundamental object of study in complex analysis is the Bergman projection
$\Pi_{\Omega}$, which is the orthogonal projection from $L^{2}(\Omega)$ to the
closed subspace of holomorphic functions. Unweighted and weighted $L^{p}$
estimates for $\Pi_{\Omega}$ have close connections to duality of Bergman
spaces, conformal function theory, and complex partial differential equations.
They have been investigated by many authors, particularly in the case
$\Omega=\mathbb{D}.$ See references [bekolle1982, bekolle1986,
bekollebonami1978, borichev04, burbea83, burbea80, hedenmalm, lanzani-
stein-2004, PR, rtw2017] for some results pertaining to the regularity of the
Bergman projection.

For each interval $I\subset\mathbb{T}$, with center
$e^{\mathrm{i}\theta_{0}}$, the associated Carleson box is

(1.1)
$Q_{I}=\left\\{re^{\mathrm{i}\theta}\in\mathbb{D}:1-r\leq\ell(I),|\theta-\theta_{0}|\leq\frac{\ell(I)}{2}\right\\}.$

It is well-known that Carleson boxes form a Muckenhoupt basis, and moreover
the corresponding Muckenhoupt weight condition for $1<p<\infty$ (denoted by
$\sigma\in\mathrm{B}_{p}(\mathbb{D})$) characterizes the continuity of
$\Pi_{\mathbb{D}}$ on $L^{p}(\mathbb{D},\sigma)$ [bekollebonami1978,
bekolle1982]. More precisely, for $1\leq p<\infty$, we say a weight $\sigma$,
which is a locally integrable, almost everywhere positive function on
$\mathbb{D}$, belongs to the Békollè-Bonami class $\mathrm{B}_{p}(\mathbb{D})$
if the $\mathrm{B}_{p}(\mathbb{D})$ characteristic,

(1.2)
$\left[\sigma\right]_{\mathrm{B}_{p}(\mathbb{D})}=\sup_{I}\left\langle\sigma\right\rangle_{Q_{I}}\left\langle\sigma^{-1}\right\rangle_{\frac{1}{p-1},Q_{I}}$

is finite, where we are using the following local average notation. For a
measurable function $f$, $0<p<\infty$, a weight $v$, and a measurable set $E$,

$\left\langle
f\right\rangle_{p,v,E}=\frac{\left\|f\right\|_{L^{p}(E,v)}}{\left(\int_{E}v\,dA\right)^{1/p}},\quad\left\|f\right\|_{L^{p}(E,v)}=\left(\int_{E}\left|f\right|^{p}v\,dA\right)^{1/p}.$

When $p=\infty$, we understand both quantities above to be the
$L^{\infty}(E)=L^{\infty}(E,v)$ norm. The measure $dA$ is the area measure in
$\mathbb{C}$ (normalized so that $A(\mathbb{D})=1$) and when $v$ is
identically $1$ or $p=1$ we omit those parameters in the notation.

Quantitative weighted estimates for $\Pi_{\mathbb{D}}$ in terms of the
characteristic (1.2) are also well-understood. For $1<p<\infty$,

(1.3)
$[\sigma]_{\mathrm{B}_{p}(\mathbb{D})}^{\frac{1}{2p}}\lesssim\left\|\Pi_{\mathbb{D}}\right\|_{\mathcal{L}(L^{p}(\mathbb{D},\sigma))}\lesssim[\sigma]_{\mathrm{B}_{p}(\mathbb{D})}^{\max\left\\{1,\frac{1}{p-1}\right\\}}.$

The lower inequality can be derived from [bekolle1982]*Proposition 2 together
with arguments in [rtw2017], and the upper was first established in [PR] and
was extended to the unit ball in $n$ dimensions in [rtw2017] 111The
quantitative upper estimate is not simply a matter of tracking the constants
in the qualitative case, but rather was delayed until the proof of the
$\mathrm{A}_{2}$ conjecture in harmonic analysis [hytonen12] and the ensuing
“sparse revolution” that took place in the early 2010’s [conde-
alonso2016pointwise, lerner-a2, lacey-a2, lerner2019intuitive] . A weak-type
analog of (1.3) also holds when $p=1$, see [bekolle1982], though the sharp
dependence on the $\mathrm{B}_{1}(\mathbb{D})$ characteristic is not known.

The following question is natural: what is the appropriate generalization of
(1.2) and (1.3) to general bounded simply connected planar domains $\Omega$?
One immediate obstacle is the apparent lack of canonical replacement for
Carleson boxes that respects the boundary geometry. A proposal was given by
Burbea in [burbea83, Theorem 2] which we slightly simplify here.

Let $\psi$ be a conformal map from $\mathbb{D}$ onto $\Omega$. For $1\leq
p<\infty$, we say a weight $\sigma$ on $\Omega$ belongs to the Békollè-Bonami
class $\mathrm{B}_{p}(\Omega)$ if the $\mathrm{B}_{p}(\Omega)$ characteristic,

(1.4)
$\left[\sigma\right]_{\mathrm{B}_{p}(\Omega)}=\sup_{I}\left\langle\sigma\right\rangle_{\psi(Q_{I})}\left\langle\sigma^{-1}\right\rangle_{\frac{1}{p-1},\psi(Q_{I})}$

is finite. For the domains we consider, the images $\\{\psi(Q_{I})\\}_{I}$ do
indeed form a Muckenhoupt basis (in the sense of [cmp2011, Definition 3.1])
and furthermore $\left[\sigma\right]_{\mathrm{B}_{p}(\Omega)}$ is, up to an
absolute constant, independent of the choice of conformal map (see Proposition
2.1 below).

By changing variables, $[\sigma]_{\mathrm{B}_{p}(\Omega)}$ can be related to a
weighted characteristic on the unit disc. For weights $u,v$ on $\mathbb{D}$,
and $1\leq p<\infty$, we say $u$ belongs to the weighted Békollè-Bonami class
$\mathrm{B}_{p}(\mathbb{D},v)$ if

(1.5) $[u]_{\mathrm{B}_{p}(\mathbb{D},v)}=\sup_{I}\left\langle
u\right\rangle_{v,Q_{I}}\left\langle
u^{-1}\right\rangle_{\frac{1}{p-1},v,Q_{I}}<\infty.$

When $v$ is identically $1$, we remove it from the notation and recover (1.2).
A change of variable reveals

(1.6)
$\left\langle\sigma\right\rangle_{p,\psi(Q_{I})}=\left\langle\sigma\circ\psi\right\rangle_{p,|\psi^{\prime}|^{2},Q_{I}},$

hence, for $1\leq p<\infty$, $u=(\sigma\circ\psi)$, and
$v=|\psi^{\prime}|^{2}$,

(1.7) $[\sigma]_{\mathrm{B}_{p}(\Omega)}=[u]_{\mathrm{B}_{p}(\mathbb{D},v)}.$

On the other hand, the Bergman projection $\Pi_{\Omega}$ can be connected to
$\Pi_{\mathbb{D}}$ through the conformal map $\psi$. Consequently,

(1.8)
$\left\|\Pi_{\Omega}\right\|_{\mathcal{L}(L^{p}(\Omega,\sigma))}=\left\|\Pi_{\mathbb{D}}\right\|_{\mathcal{L}(L^{p}(\mathbb{D},(\sigma\circ\psi)|\psi|^{2-p}))};$

see, for example, [burbea83, Theorem 2] or [wagner22, Lemma 2.4]. So, in light
of (1.7), (1.8), and (1.3), we pose the following question. Under what
conditions on a weight $v$ does it hold that
$u\in\mathrm{B}_{p}(\mathbb{D},v)$ implies
$uv^{1-p/2}\in\mathrm{B}_{p}(\mathbb{D})$? A trivial sufficient condition is
that $v\in\mathrm{B}_{1}(\mathbb{D})$. Indeed, the definition of
$[v]_{\mathrm{B}_{1}(\mathbb{D})}$ implies

(1.9) $\left\langle uv^{1-p/2}\right\rangle_{Q_{I}}\left\langle
uv^{1-p/2}\right\rangle_{\frac{1}{p-1},Q_{I}}\leq[v]_{\mathrm{B}_{1}(\mathbb{D})}^{p}[u]_{\mathrm{B}_{p}(\mathbb{D},v)},\qquad
1\leq p<\infty.$

Thus, concerning the Bergman projection on $\Omega$, the following trivial
result holds, combining the above observations with the upper estimate in
(1.3).

######  Theorem A.

Let $\Omega\subsetneq\mathbb{C}$ be a bounded, simply connected domain with
$|\psi^{\prime}|^{2}\in\mathrm{B}_{1}(\mathbb{D})$, then for $1<p<\infty$,

(1.10)
$\|\Pi_{\Omega}\|_{\mathcal{L}(L^{p}(\Omega,\sigma))}\lesssim\left(\left[\left|\psi^{\prime}\right|^{2}\right]_{\mathrm{B}_{1}(\mathbb{D})}^{p}\left[\sigma\right]_{\mathrm{B}_{p}(\Omega)}\right)^{\max\\{1,\frac{1}{p-1}\\}}.$

Moreover, the implicit constant depends only on $p$.

In [bekolle1986], Békollè has shown that
$|\psi^{\prime}|^{2}\in\mathrm{B}_{1}(\mathbb{D})$ when $\Omega$ is convex.
Also, when the boundary of $\Omega$ is sufficiently smooth (e.g. Dini smooth,
which is slightly better than $C^{1}$), it is known that $\psi^{\prime}$
extends to a continuous, non-vanishing function on $\overline{\mathbb{D}}$
[pommerenke-book], so trivially
$|\psi^{\prime}|^{2}\in\mathrm{B}_{1}(\mathbb{D})$. A weaker version of
Theorem A in this latter special case of Dini smooth domains appears to have
been discovered by Burbea in [burbea80]. Beyond this, the authors are not
aware of geometric conditions on $\Omega$ which are necessary or sufficient
for $|\psi^{\prime}|^{2}\in\mathrm{B}_{1}(\mathbb{D})$.

Here, our goal is to go beyond the simple observations which to lead to
Theorem A in two different, but related directions.

###  1.1. Sharpened conditions on $\psi$ for weighted estimates

First we will show that the assumption
$|\psi^{\prime}|^{2}\in\mathrm{B}_{1}(\mathbb{D})$ can be weakened while
maintaining the full range of weighted estimates. In fact, in Theorem E below,
we give sufficient conditions on domains $\Omega$, in terms of
$|\psi^{\prime}|$, for which full or limited range weighted estimates hold for
$\Pi_{\Omega}$ with respect to the weight classes defined by (1.4). We have
the following special case of Theorem E for the full range of $p$:

######  Theorem B.

Let $\Omega\subsetneq\mathbb{C}$ be a bounded, simply connected domain. If

(1.11) $|\psi^{\prime}|^{2}\in\bigcap_{p>1}\mathrm{B}_{p}(\mathbb{D}),$

then for each $\sigma\in\mathrm{B}_{p}(\Omega)$,
$\Pi_{\Omega}:L^{p}(\Omega,\sigma)\to L^{p}(\Omega,\sigma)$.

Theorem B sharpens Theorem A in a non-trivial way. It is indeed the case that
there exist domains $\Omega$ for which (1.11) is satisfied, but
$|\psi^{\prime}|^{2}\not\in\mathrm{B}_{1}(\mathbb{D})$. See Section 2.2.1
below for an example. The main obstacle to obtaining the sufficiency in
Theorem B is that $\mathrm{B}_{p}(\mathbb{D})$ weights do not in general
satisfy a reverse Hölder inequality. If they did, then the strategy
implemented by Johnson and Neugenbauer [JN] concerning homeomorphisms which
preserve the $\mathrm{A}_{p}(\mathbb{R}^{n})$ classes could be easily
repeated. In particular, a parallel theorem for the Szegő projection can be
established by directly adapting the arguments in the proof of [JN, Theorem
2.7], so we do not focus on it in this paper. Nonetheless, we get around this
difficulty for the Bergman projection by using the special form of the
homeomorphism $\psi$, namely its conformality, as well as more modern tools
from dyadic harmonic analysis.

In the case when $\Omega$ is a uniform domain, the condition (1.11) is sharp
for weighted estimates (see Theorem G below), and furthermore we can replace
the $\mathrm{B}_{p}(\Omega)$ characteristic by the following equivalent one
which is intrinsic to $\Omega$,

(1.12)
$\left[\sigma\right]_{\mathrm{D}_{p}(\Omega)}=\sup_{D}\left\langle\sigma\right\rangle_{D\cap\Omega}\left\langle\sigma^{-1}\right\rangle_{\frac{1}{p-1},D\cap\Omega}$

where the supremum is taken over all Euclidean disks $D$ centered on the
boundary $\partial\Omega$.

######  Theorem C.

Let $\Omega$ be a simply connected, bounded uniform domain. Then,

(1.13) $|\psi^{\prime}|^{2}\in\bigcap_{p>1}\mathrm{B}_{p}(\mathbb{D}),$

if and only if for each $\sigma\in\mathrm{D}_{p}(\Omega)$,
$\Pi_{\Omega}:L^{p}(\Omega,\sigma)\to L^{p}(\Omega,\sigma)$.

We postpone the precise definition of uniform domain until Section 2, but the
class includes all Lipschitz domains and furthermore, the boundary can be
quite irregular, allowing for Hausdorff dimension arbitrarily close to $2$.
Furthermore, the condition (1.13) is satisfied whenever $\Omega$ is
asymptotically conformal (see Section 2.2.2 below) therefore the full range of
weighted estimates hold on these domains.

We remark in passing that our results should be extendable to unbounded graph
domains. One only needs the existence of a conformal map that maps the upper-
half space to $\Omega$, and has a continuous extension to the Riemann sphere
that maps the real axis to $\partial\Omega\setminus\\{\infty\\}$ and fixes the
point at infinity (such a conformal map is guaranteed to exist for graph
domains, see [lanzani-stein-2004, Proposition 2.2]). In this case, one
proceeds by using the upper-half plane rather than the unit disk as the model
domain. The Békollè-Bonami regions in the upper-half space are again Carleson
tents, see [PR]. We have not checked all the details, but we believe that the
proofs should be completely analogous.

###  1.2. Weighted weak-type estimates

Our second extension of Theorem A is to obtain the weighted weak-type $(1,1)$
analogue of (1.10). The main difference is that when establishing the analogue
of (1.8) for the $L^{p,\infty}$ spaces, the conformal map appears partly as a
multiplier, and partly as a measure (for the $L^{p}$ norm there is no
distinction between a multiplier and a measure). The argument is similar to
the one outlined in [bekolle1986, Lemma 3.1], but we will need a
generalization of the result there to weighted spaces and we include all the
arguments for completeness. Weak-type estimates in which some or all of the
weight is treated as a multiplier, which now appear to be called mixed weak-
type inequalities, begin with Muckenhoupt and Wheeden [muckenhoupt1977]. There
was limited development of these types of estimates [sawyer-weak, bonami82]
until their systematic study by Cruz-Uribe, Martell, and Pérez [dcu-martell].
[dcu-martell] and subsequent modern developments are motivated by
interpolation problems with a change of measure, but as noted in Proposition
5.2 below, and in [bonami82, bekolle1986], they also arise in establishing
weak-type estimates through a change of variable. The following theorem is
proved in Section 5 below.

######  Theorem D.

Let $\Omega\subsetneq\mathbb{C}$ be a simply connected domain with
$\left|\psi^{\prime}\right|^{2}\in\mathrm{B}_{1}(\mathbb{D})$, then

(1.14) $\|\Pi_{\Omega}\|_{L^{1}(\Omega,\sigma)\to
L^{1,\infty}(\Omega,\sigma)}\lesssim[\sigma]_{\mathrm{B}_{1}(\Omega)}^{3},$

where the implicit constant depends polynomially on
$\left[\left|\psi^{\prime}\right|^{2}\right]_{\mathrm{B}_{1}(\mathbb{D})}$.

###  1.3. Organization

This paper is organized as follows. In Section 2, we collect useful
preliminaries regarding dyadic structures on $\mathbb{D}$, uniform domains,
properties of conformal maps, and weight classes. In Section 3, we introduce
and prove a general version of Theorem B, namely Theorem E. In Section 4, we
focus on the category of uniform domains and prove a converse to Theorem E
which implies Theorem C. Finally, in Section 5 we prove Theorem D.

##  2\. Preliminaries: Dyadic structures, conformal maps, and uniform domains

In this section we collect some preliminaries concerning dyadic harmonic
analysis, conformal maps, and uniform domains.

###  2.1. Dyadic harmonic analysis in $\mathbb{D}$

Given a Carleson box $Q_{I}$, defined by (1.1), let $T_{I}$ denote the
corresponding “top half”

$T_{I}=\left\\{re^{\mathrm{i}\theta}\in\mathbb{D}:\frac{\ell(I)}{2}\leq
1-r\leq\ell(I),|\theta-\theta_{0}|\leq\frac{\ell(I)}{2}\right\\},$

where $\ell(I)$ denotes the arclength of the interval $I$, normalized so that
$\ell(\mathbb{T})=1$. $T_{I}$ coincides with $T_{I,\rho}$ from [APR] with
$\rho=\frac{1}{2}$.

It is well-known that any Carleson box $Q_{I}$ can be well-approximated by
$Q_{J}$ where $J$ belongs to one of two dyadic systems of intervals; say
$\mathcal{D}_{1},\mathcal{D}_{2}.$ By well-approximated, we mean that given an
arbitrary interval $I$ on $\mathbb{T}$, there exists intervals
$J,J^{\prime}\in\mathcal{D}_{1}\cup\mathcal{D}_{2}$ satisfying
$J^{\prime}\subset I\subset J$ and $|Q_{I}|\sim|Q_{J}|\sim|Q_{J^{\prime}}|.$
For example, one can take

(2.1) $\begin{gathered}\mathcal{D}_{1}=\left\\{\left[\frac{2\pi
j}{2^{k}},\frac{2\pi(j+1)}{2^{k}}\right):j,k\in\mathbb{N},0\leq
j<2^{k}\right\\},\\\ \mathcal{D}_{2}=\left\\{\left[\frac{2\pi
j}{2^{k}}+\frac{2\pi}{3},\frac{2\pi(j+1)}{2^{k}}+\frac{2\pi}{3}\right):j,k\in\mathbb{N},0\leq
j<2^{k}\right\\}.\end{gathered}$

Therefore, when computing the Békollè-Bonami characteristic of $u$, it
suffices to compute averages over Carleson boxes $Q_{I}$ where $I$ belongs to
$\mathcal{D}_{1}$ or $\mathcal{D}_{2}.$ The precise form of $\mathcal{D}_{j}$
is not important, but rather we note a few essential properties of any dyadic
grid $\mathcal{D}$:

  * (i)

For each $k\in\mathbb{N}$, $\\{I\in\mathcal{D}:\ell(I)=2^{-k}\\}$ forms a
partition of $\mathbb{T}$.

  * (ii)

$\\{T_{I}:I\in\mathcal{D}\\}$ forms a partition of $\mathbb{D}$.

  * (iii)

Any two $I,J\in\mathcal{D}$ are either disjoint, or one is contained in the
other.

  * (iv)

For each $I\in\mathcal{D}$ and $k\geq 1$, we can subdivide $I$ into
subintervals $\\{I_{j}^{k}\\}_{j=1}^{2^{k}}$ with side length
$\ell(I_{j}^{k})=2^{-k}\ell(I)$ which we call the $k$-th generation of $I$.
For $J=I_{j}^{1}$, we say $I$ is the dyadic parent of $J$ and analogously,
$Q_{I}$ the dyadic parent of $Q_{J}$.

Throughout $u$, $v$, and $w$ will be generic weights on $\mathbb{D}$. We say a
weight $u$ is doubling (with respect to $\mathcal{D}$) if there exists
$c_{u}>0$ such that for all intervals $I\subset J$ in $\mathbb{T}$ (resp. in
$\mathcal{D}$) with $\ell(J)=2\ell(I)$,

(2.2) $c_{u}^{-1}\int_{Q_{J}}u\,dA\leq\int_{Q_{I}}u\,dA\leq
c_{u}\int_{T_{I}}u\,dA.$

If only the first inequality in (2.2) holds, then we say $u$ is weakly
doubling. It is not difficult to show that any weight
$u\in\mathrm{B}_{p}(\mathbb{D})$ is doubling, $1\leq p<\infty.$

For a weight $v$ on $\mathbb{D}$, define the maximal operator

(2.3) $M_{v}f(z):=\sup_{I:Q_{I}\ni z}\left\langle
f\right\rangle_{v,Q_{I}},\quad f\in L^{1}(\mathbb{D},\sigma),z\in\mathbb{D}.$

Note when $v\equiv 1$, $M_{v}$ corresponds to the ordinary maximal function in
the Bergman setting, and in this case we simply write $M_{v}=M.$ When the
supremum is restricted only to $I\in\mathcal{D}$, we use the notation
$M^{\mathcal{D}}_{v}$. Standard considerations show that for any $v$ and any
dyadic grid $\mathcal{D}$, $M^{\mathcal{D}}_{v}:L^{p}(\mathbb{D},v)\to
L^{p}(\mathbb{D},v)$ with norm $p^{\prime}=\frac{p}{p-1}$
[lerner2019intuitive]*Theorem 15.1. By the approximation property of
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$,

(2.4) $M_{v}f(z)\lesssim
M_{v}^{\mathcal{D}_{1}}f(z)+M_{v}^{\mathcal{D}_{2}}f(z),\quad f\in
L^{1}(\mathbb{D},v),z\in\mathbb{D}$

holds whenever $v$ is doubling.

We also outline the following sparse estimate in the Bergman setting which we
will use, but is quite standard by now. Let

(2.5) $\mathcal{A}_{\mathcal{D},v}f=\sum_{I\in\mathcal{D}}\left\langle
f\right\rangle_{v,Q_{I}}\chi_{Q_{I}}.$

Then,

$\displaystyle\left|\left\langle\mathcal{A}_{\mathcal{D},v}f,g\,wv\right\rangle\right|$
$\displaystyle=\sum_{I\in\mathcal{D}}\left(\int_{Q_{I}}v\,dA\right)\left\langle
f\right\rangle_{v,Q_{I}}\left\langle gw\right\rangle_{v,Q_{I}}$
$\displaystyle\leq
c_{v}\sum_{I\in\mathcal{D}}\left(\int_{T_{I}}v\,dA\right)\left\langle
w\right\rangle_{v,Q_{I}}\left\langle w^{-1}\right\rangle_{v,Q_{I}}\left\langle
fw\right\rangle_{w^{-1}v,Q_{I}}\left\langle g\right\rangle_{wv,Q_{I}}$
$\displaystyle\leq
c_{v}\left[w\right]_{\mathrm{B}_{2}(v)}\int_{\mathbb{D}}M_{w^{-1}v}(fw)w^{-\frac{1}{2}}M_{wv}(g)w^{\frac{1}{2}}v\,dA.$
$\displaystyle\leq
4c_{v}\left[w\right]_{\mathrm{B}_{2}(v)}\left\|f\right\|_{L^{2}(wv)}\left\|g\right\|_{L^{2}(wv)}.$

This, together with a slight modification for $p\neq 2$ (see, e.g. [moen2012])
gives

(2.6) $\left\|A_{\mathcal{D},v}\right\|_{L^{p}(\mathbb{D},wv)}\leq
pp^{\prime}c_{v}\left[w\right]_{\mathrm{B}_{p}(v)}^{\max\left\\{1,\frac{1}{p-1}\right\\}}.$

###  2.2. Conformal estimates

We first verify that the class $\mathrm{B}_{p}(\Omega)$ is independent of the
choice of conformal map, and that the collection
$\\{\psi(Q_{I})\\}_{I\subseteq\mathbb{T}}$ forms a Muckenhoupt basis, which
means the maximal function

(2.7) $M_{\psi}f(z)=\sup_{I:z\in\psi(Q_{I})}\left\langle
f\right\rangle_{\psi(Q_{I})}$

is bounded on $L^{p}(\Omega,\sigma)$ when $\sigma\in\mathrm{B}_{p}(\Omega)$.

######  Proposition 2.1.

Suppose there exists a conformal map $\psi_{0}$ from $\mathbb{D}$ onto
$\Omega$ such that $\left|\psi_{0}^{\prime}\right|^{2}$ is a doubling weight
(2.2). Then for any other such conformal map $\psi$,

  * i.

$\left|\psi^{\prime}\right|^{2}$ is a weakly doubling measure,

  * ii.

$\displaystyle\sup_{\psi}\sup_{I}\left\langle\sigma\right\rangle_{\psi(Q_{I})}\left\langle\sigma^{-1}\right\rangle_{\frac{1}{p-1},\psi(Q_{I})}\lesssim\inf_{\psi}\sup_{I}\left\langle\sigma\right\rangle_{\psi(Q_{I})}\left\langle\sigma^{-1}\right\rangle_{\frac{1}{p-1},\psi(Q_{I})}$.

If in addition,
$\left|\psi_{0}^{\prime}\right|^{2}\in\cup_{p\in[1,\infty)}\mathrm{B}_{p}(\mathbb{D})$,
then

  * iii.

$|\psi^{\prime}|^{2}$ is doubling,

  * iv.

$\\{\psi(Q_{I})\\}_{I}$ forms a Muckenhoupt basis.

###### Proof.

Since the claims are obvious for $\psi$ which is a rotation of $\psi_{0}$, it
is enough to consider $\psi=\psi_{0}\circ\tau_{\lambda}$ where
$\tau_{\lambda}(z)=\frac{\lambda-z}{1-\bar{\lambda}z}$ is the automorphism of
$\mathbb{D}$ that exchanges the points $0$ and $\lambda$. First, notice that
$\left|\tau_{\lambda}^{\prime}\right|^{2}$ itself is a doubling weight with
doubling constant independent of $\lambda$. We also rely on the following
almost invariance property of $Q_{I}$ under $\tau_{\lambda}$: there exists
$C>0$ such that for each $\lambda\in\mathbb{D}$ and $I\subset\mathbb{T}$,
there exists $I_{\lambda},{I}_{\lambda}^{*}\subset\mathbb{T}$ satisfying

(2.8) $Q_{I_{\lambda}}\subset\tau_{\lambda}(Q_{I})\subset
Q_{I_{\lambda}^{*}},\qquad
C^{-1}\leq\frac{\ell(I_{\lambda}^{*})}{\ell(I_{\lambda})}\leq C.$

This geometric property can be seen, for example, by considering the Möbius-
invariant family of regions of $\mathbb{D}$ bounded by a geodesic in the
hyperbolic metric and an arc in $\mathbb{T}$ (see, e.g. [luecking1985,
p.321]), as such regions well-approximate Carleson tents.

Now, let $I\subset J$ with $\ell(J)=2\ell(I)$. Since $\tau_{\lambda}$ is
doubling, $|Q_{J_{\lambda}^{*}}|\sim|Q_{I_{\lambda}}|$, so

$|\psi(Q_{J})|\leq|\psi_{0}(Q_{J_{\lambda}^{*}}))|\lesssim|\psi_{0}(Q_{I_{\lambda}})|\leq|\psi(Q_{I})|,$

which establishes i. To prove ii., let $\psi_{1},\psi_{2}$ be conformal maps.
Then, $\psi_{1}=\psi_{2}\circ\tau_{\lambda}$ for some $\lambda\in\mathbb{D}$.
Therefore, by (2.8) and the doubling property of $\psi_{2}$,

$\left\langle\sigma\right\rangle_{\psi_{1}(Q_{I})}\leq\frac{\int_{\psi_{2}(Q_{I_{\lambda}^{*}})}\sigma}{|\psi_{2}(Q_{I_{\lambda}^{*}})|}\lesssim\left\langle\sigma\right\rangle_{\psi_{2}(Q_{\hat{I}_{\lambda}})},$

and the same for $\sigma^{\frac{-1}{p-1}}$, which establishes ii. The proof of
iii. is easier than i., simply noting that
$|\tau_{\lambda}(Q_{J})|\sim|\tau_{\lambda}(Q_{I})|\sim|\tau_{\lambda}(T_{I})|$.

Now, to prove iv., let $f\in L^{1}(\Omega)$ be nonnegative,
$I\subset\mathbb{T}$, and $z\in\psi(Q_{I}).$ Then by change of variable,

$\left\langle f\right\rangle_{\psi(Q_{I})}=\left\langle
f\circ\psi\right\rangle_{v,Q_{I}}\leq M_{v}(f\circ\psi)(\psi^{-1}(z)),\qquad
v=\left|\psi^{\prime}\right|^{2},$

which immediately implies the pointwise inequality

$M_{\psi}f(z)\leq M_{v}(g)(\zeta),\quad g=f\circ\psi,\quad z\in\Omega.$

Fix $\sigma\in\mathrm{B}_{p}(\Omega)$, which equivalently means
$u=\sigma\circ\psi\in\mathrm{B}_{p}(\mathbb{D},v)$ (recall (1.6)). By (2.6)
and the obvious inequality

$M_{v}g\lesssim\sum_{j=1,2}\sum_{I\in\mathcal{D}_{j}}\langle
g\rangle_{v,Q_{I}}\chi_{Q_{I}},$

$M_{v}$ is bounded on $L^{p}(\mathbb{D},uv)$. Then notice, again using change
of variable, and assuming $f\in L^{p}(\Omega,\sigma)$,

$\displaystyle\|M_{\psi}f\|_{L^{p}(\Omega,\sigma)}$
$\displaystyle\leq\|[M_{v}g]\circ\psi^{-1}\|_{L^{p}(\Omega,\sigma)}$
$\displaystyle=\|M_{v}g\|_{L^{p}(\mathbb{D},uv)}$
$\displaystyle\lesssim\|g\|_{L^{p}(\mathbb{D},uv)}$
$\displaystyle=\|f\|_{L^{p}(\Omega,\sigma)}.$

∎

To conclude this section, we state a convenient version of the Koebe
distortion theorem.

######  Proposition 2.2.

There exists an absolute constant $K$ so that for all $I\in\mathbb{T}$, all
$z_{1},z_{2}\in T_{I}$, and any $\psi:\mathbb{D}\to\mathbb{C}$ which is
conformal, one has

(2.9) $K^{-1}\leq\frac{|\psi^{\prime}(z_{1})|}{|\psi^{\prime}(z_{2})|}\leq K.$

###### Proof.

A classical corollary to the Koebe distorition theorem [pommerenke-book]*Cor
1.1.5 is that (2.9) holds for any $z_{1},z_{2}\in\mathbb{D}$ and
$K=e^{6\lambda(z_{1},z_{2})}$ where $\lambda$ is the hyperbolic metric in
$\mathbb{D}$. However, there exists an absolute constant $r>0$ and points
$z_{I}\in\mathbb{D}$ such that
$T_{I}\subset\\{z\in\mathbb{D}:\lambda(z,z_{I})\leq r\\}$, whence (2.9)
follows with $K=e^{12r}$. ∎

####  2.2.1. Example

Next, to show Theorem E is a non-trivial qualitative extension of Theorem A,
let us construct a conformal map $\psi$ such that $|\psi^{\prime}|^{2}$
belongs to $\mathrm{B}_{p}(\mathbb{D})$ for all $p>1$, but not to
$\mathrm{B}_{1}(\mathbb{D})$. Define

$\psi_{0}(z)=\frac{z}{\log z}.$

This a conformal map on the disk
$D(\frac{1}{4},\frac{1}{4})=\\{z\in\mathbb{C}:|z-\frac{1}{4}|<\frac{1}{4}\\}$
whose image is drawn in Figure 1.

$\psi_{1}(\mathbb{D})$ Figure 1. $\psi_{1}(\mathbb{D})$

Compute

$|\psi_{0}^{\prime}(z)|^{2}=\left|\frac{\log z-1}{(\log
z)^{2}}\right|^{2}\sim\frac{1}{|\log z|^{2}}.$

But $|\log(z)|^{\alpha}$ is integrable over $D(\frac{1}{4},\frac{1}{4})$ for
every $\alpha\in\mathbb{R}$. Define $\psi_{1}(z)=\psi_{0}(\frac{z+1}{4})$ so
that $\psi_{1}(\mathbb{D})=\psi_{0}(D(\frac{1}{4},\frac{1}{4}))$. Then
$\psi_{1}$ is a conformal map on $\mathbb{D}$ with
$|\psi_{1}^{\prime}|^{2}\in\mathrm{B}_{p}(\mathbb{D})$ for each $p>1$.
However, $\psi_{1}^{\prime}(-1)=0$ so $|\psi_{1}^{\prime}|^{2}$ does not
belong to $\mathrm{B}_{1}(\mathbb{D})$.

####  2.2.2. Asymptotically conformally domains

Given a bounded domain $\Omega\subset\mathbb{C}$ and two points
$z_{1},z_{2}\in\partial\Omega$, define $\partial\Omega(z_{1},z_{2})$ to be the
smaller arc of $\partial\Omega$ between the points $z_{1}$ and $z_{2}$.
$\Omega$ is said to be asymptotically conformal if

$\max_{z\in\partial\Omega(z_{1},z_{2})}\frac{|z-z_{1}|+|z-z_{2}|}{|z_{1}-z_{2}|}\to
1\quad\mbox{as}\quad|z_{1}-z_{2}|\to 0.$

######  Lemma 2.3.

If $\Omega$ is asymptotically conformal, then
$\left|\psi^{\prime}\right|^{2}\in\mathrm{B}_{p}(\mathbb{D})$ for every $p>1$.

###### Proof.

In this case, $\log\psi^{\prime}$ belongs to the little Bloch space
[pommerenke-VMO]. But it is well-known [zhu2007, Theorem 8.15] that this is
equivalent to $\log\psi^{\prime}$ belonging to $\operatorname{{VMOA}}$ defined
with respect to the Bergman metric. More precisely,
$\log\psi^{\prime}\in\text{Hol}(\mathbb{D})$ and satisfies for any fixed
$r>0$:

$\lim_{R\rightarrow 0^{-}}\sup_{\begin{subarray}{c}z:\\\ 1>|z|\geq
R\end{subarray}}\frac{1}{|\beta(z,r)|}\int_{\beta(z,r)}|\log\psi^{\prime}-\log\psi^{\prime}_{z}|^{2}\,dA=0.$

Here $\beta(z,r)$ denotes a disk in the hyperbolic metric centered at $z$ and
of radius $r$, while for any function $f$, we write $f_{z}$ to indicate
$|\beta(z,r)|^{-1}\int_{\beta(z,r)}f\,dA$, omitting the implicit dependence on
$r$. We say $f\in\operatorname{{BMO}}$ if $f$ satisfies the weaker condition,
that

$\left\|f\right\|_{\operatorname{{BMO}}}:=\sup_{z\in\mathbb{D}}\frac{1}{|\beta(z,r)|}\int_{\beta(z,r)}|f-f_{z}|^{2}\,dA$

is finite. Different choices of $r$ give rise to equivalent norms, so this
space (along with $\operatorname{{VMOA}}$) is independent of $r>0$. Moreover,
any function in $\operatorname{{VMOA}}$ can be arbitrarily well-approximated
in the $\operatorname{{BMO}}$ norm by elements of $C(\overline{\mathbb{D}})$
[zhu2007, Proposition 8.12], and hence in particular by polynomials, which
belong to $\operatorname{{VMOA}}$. An additional key fact [zhu2007, pp.218]
that we will use is that any
$f\in\operatorname{{BMOA}}=\operatorname{{BMO}}\cap\text{ Hol}(\mathbb{D})$
satisfies the Lipschitz-like condition with respect to the hyperbolic metric
$\lambda$:

$|f(z_{1})-f(z_{2})|\leq
C\|f\|_{\operatorname{{BMO}}}\lambda(z_{1},z_{2}),\quad
z_{1},z_{2}\in\mathbb{D},$

where $C$ is an independent constant.

Let $p>1$ and write $\log\psi^{\prime}=g+h$, where $g$ is a polynomial and
$h\in\operatorname{{BMOA}}$ satisfying
$\|h\|_{\operatorname{{BMO}}}\leq\frac{\log(3/2)}{6Cr}\min\\{1,p-1\\}.$ We
will show that $|\psi^{\prime}|^{2}\in\mathrm{B}_{p}(\mathbb{D}).$ Fix a
Carleson tent $Q_{I}$ corresponding to $I\subset\mathbb{T}$ and let
$I_{j}^{k}$ denote the dyadic descendants of $I$ belonging to the $k$-th
generation, which (so $k\geq 1$), $1\leq j\leq 2^{k}.$ We then may choose
$r>0$ and points $z_{j,k}\in T_{I_{j}^{k}}$ so

$T_{I_{j}^{k}}\subset\beta(z_{j,k},r),\quad|T_{I_{j}^{k}}|\sim|\
\beta(z_{j,k},r)|\sim 2^{-2k}|Q_{I}|.$

Let $z_{I}$ be such a point corresponding to $T_{I}=T_{I_{1}^{0}}$. It is
straightforward to see then that if $z\in T_{I_{j}^{k}},$ there holds

$\lambda(z,z_{I})\leq(2k+1)r,\quad|h(z)-h(z_{I})|\leq
3Crk\|h\|_{\operatorname{{BMO}}}.$

Then, we can directly estimate:

$\displaystyle\frac{1}{|Q_{I}|}\int_{Q_{I}}|\psi^{\prime}|^{2}$
$\displaystyle\leq e^{2\|g\|_{\infty}}\cdot
e^{2\text{Re}\,h(z_{I})}\cdot\frac{1}{|Q_{I}|}\int_{Q_{I}}e^{2\text{Re}\,(h-h(z_{I}))}$
$\displaystyle\leq\frac{e^{2(\|g\|_{\infty}+\text{Re}\,h(z_{I}))}}{|Q_{I}|}\sum_{k\geq
0}\sum_{j=1}^{2^{k}}\int_{T_{I_{j}^{k}}}e^{2|h-h(z_{I})|}$
$\displaystyle\lesssim e^{2(\|g\|_{\infty}+\text{Re}\,h(z_{I}))}\sum_{k\geq
0}2^{-k}e^{6Crk\|h\|_{\operatorname{{BMO}}}}$
$\displaystyle=e^{2(\|g\|_{\infty}+\text{Re}\,h(z_{I}))}\sum_{k\geq
0}\left(\frac{e^{6Cr\|h\|_{\operatorname{{BMO}}}}}{2}\right)^{k}$
$\displaystyle\lesssim e^{2(\|g\|_{\infty}+\text{Re}\,h(z_{I}))}$

A similar computation shows that

$\left(\frac{1}{|Q_{I}|}\int_{Q_{I}}|\psi^{\prime}|^{-2/(p-1)}\right)^{p-1}\lesssim
e^{2(\|g\|_{\infty}-\text{Re}\,h(z_{I}))},$

completing the proof.

∎

###  2.3. Uniform Domains and the $\mathrm{D}_{p}(\Omega)$ class

We begin with the definition of uniform domains, which have also been called
$(\varepsilon,\infty)$ domains in the literature. Such domains were introduced
in [jones80, jones81] in the context of the extension problem for BMO and
Sobolev spaces. A domain $\Omega$ is called a _uniform domain_ if there exists
$\varepsilon>0$ so that for all points $x,y\in\Omega$, there exists a
rectifiable arc $\gamma:[0,1]\rightarrow\Omega$ with $\gamma(0)=x$ and
$\gamma(1)=y$ satisfying the following two properties:

  1. (1)

$\int_{0}^{1}|\gamma^{\prime}(t)|\,dt\leq\frac{1}{\varepsilon}|x-y|$

  2. (2)

$z=\gamma(t)$ for $t\in(0,1)$ satisfies

$\inf_{\zeta\in\Omega^{c}}|z-\zeta|\geq\varepsilon\frac{|x-z||y-z|}{|x-y|}.$

The next proposition gives two more equivalent definitions of uniform domains

######  Proposition 2.4.

Let $\Omega$ be a bounded simply connected domain and $\psi$ a conformal map
from $\mathbb{D}$ onto $\Omega$. The following are equivalent.

  * A.

$\Omega$ is a uniform domain.

  * B.

There exists $C>0$ such that for each $z_{1},z_{2}\in\partial\Omega$,

$\operatorname{diam}\partial\Omega(z_{1},z_{2})\leq C|z_{1}-z_{2}|.$

  * C.

There exists $\Psi:\mathbb{C}\to\mathbb{C}$ quasiconformal such that
$\Psi=\psi$ on $\mathbb{D}$.

###### Proof.

The equivalence of B. and C. is a well-known result due to Ahlfors
[pommerenke-book]*p. 94. The equivalence of A. and C. goes through the
extension problem for BMO functions, see [jones80]*p. 42. ∎

As discussed in the introduction, while the definition for
$\mathrm{D}_{p}(\Omega)$ in (1.12) is more intrinsic and geometric, the
$\mathrm{B}_{p}(\Omega)$ characteristic defined in (1.4) has many advantages
over it. Not only does it clearly connect to properties of the conformal map,
but it will also allow us to leverage the dyadic structure of $Q_{I}$ on
$\mathbb{D}$.

To prove the equivalence of these classes for uniform domains, we will use
part C. of Proposition 2.4. The two important properties of quasiconformal
maps $\Psi$ which we will need are

  1. (1)

$|J\Psi|:=|\partial\Psi|^{2}-|\overline{\partial}\Psi|^{2}$ belongs to
$\mathrm{A}_{\infty}(\mathbb{C})$;

  2. (2)

$\Psi$ is quasisymmetric.

Let us explain these two properties. First, there are many equivalent
definitions of the weight class $\mathrm{A}_{\infty}(\mathbb{C})$ [gc-
book]*Corollary IV.2.13 and Theorem IV.2.15. We will say a weight $v$ belongs
to $\mathrm{A}_{\infty}(\mathbb{C})$ if there exists $C,\delta>0$ such that
for each cube $Q\subset\mathbb{C}$ and each measurable set $E\subset Q$,

$\frac{|E|}{|Q|}\leq
C\left(\frac{\int_{E}v\,dA}{\int_{Q}v\,dA}\right)^{\delta}.$

Second, $f:\mathbb{C}\rightarrow\mathbb{C}$ is _quasisymmetric_ if there
exists an increasing function $\eta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$
such that for $z_{0},z_{1},z_{2}\in\mathbb{C}$,

$\frac{|f(z_{1})-f(z_{0})|}{|f(z_{2})-f(z_{0})|}\leq\eta\left(\frac{|z_{1}-z_{0}|}{|z_{2}-z_{0}|}\right).$

It follows by taking $z_{i}=f^{-1}(w_{i})$ that if $f$ is quasisymmetric, so
is $f^{-1}$, though with different $\eta.$

(1) is a well-known result due to Gehring [gehring], see also [astala-
book]*Theorem 13.4.2. For (2), see [astala-book]*Theorem 3.5.3. With these two
facts, we can prove the following lemma for $\psi$ which we will be crucial in
showing the sharpness of our condition as well as the equivalence of
$\mathrm{D}_{p}(\Omega)$ and $\mathrm{B}_{p}(\Omega)$.

######  Lemma 2.5.

Let $\Omega\subsetneq\mathbb{C}$ be a bounded, simply connected uniform domain
with $\psi:\mathbb{D}\rightarrow\Omega$ conformal and fix
$I\subseteq\mathbb{T}.$ There exists $\delta>0$ such that for all measurable
$E\subseteq Q_{I}$,

$\frac{|E|}{|Q_{I}|}\lesssim\left(\frac{|\psi(E)|}{|\psi(Q_{I})|}\right)^{\delta}.$

###### Proof.

Let $I$ be an interval in $\mathbb{T}$. There exists a Euclidean cube
$P\supseteq Q_{I}$ with $|P|\sim|Q_{I}|$. Since
$|J\Psi|\in\mathrm{A}_{\infty}(\mathbb{C})$, there exists $\delta>0$ such that
for any $E\subset Q_{I}\subset P$,

$\displaystyle\frac{|E|}{|Q_{I}|}$
$\displaystyle\lesssim\frac{|E|}{|P|}\lesssim\left(\frac{\int_{E}|J\Psi|\,dA}{\int_{P}|J\Psi|\,dA}\right)^{\delta}$
$\displaystyle\leq\left(\frac{\int_{E}|J\Psi|\,dA}{\int_{Q_{I}}|J\Psi|\,dA}\right)^{\delta}=\left(\frac{\int_{E}|\psi^{\prime}|^{2}\,dA}{\int_{Q_{I}}|\psi^{\prime}|^{2}}\,dA\right)^{\delta}$
$\displaystyle=\left(\frac{|\psi(E)|}{|\psi(Q_{I})|}\right)^{\delta}.$

∎

We are ready to prove the promised equivalence.

######  Proposition 2.6.

Suppose that $\Omega$ is a bounded, simply connected uniform domain and $\psi$
maps $\mathbb{D}$ onto $\Omega$ conformally. Then, for any weight $\sigma$,

$\left[\sigma\right]_{\mathrm{B}_{p}(\Omega)}\sim\left[\sigma\right]_{\mathrm{D}_{p}(\Omega)}.$

###### Proof.

The upper inequality will follow from the fact that $\Psi$ is quasisymmetric
and the lower inequality from a similar argument for $\Psi^{-1}$. To prove the
upper bound, let $Q_{I}\subset\mathbb{D}$, and let $z_{I}$ be a point in
$Q_{I}$ satisfying $|z_{I}-z|\sim\ell(I)$ for every $z\in\partial Q_{I}$. Let
$r=\min_{z\in\partial Q_{I}}|\psi(z)-\psi(z_{I})|$ and $R=\max_{z\in\partial
Q_{I}}|\psi(z)-\psi(z_{I})|$. The quasisymmetry of $\Psi$ shows that $r\sim R$
and clearly

$D(\psi(z_{I}),r)\subset\psi(Q_{I})\subset D(\psi(z_{I}),R).$

Pick some $p\in\partial\Omega\cap D(\psi(z_{I}),R)$, and by the triangle
inequality, $D(\psi(z_{I}),R)\subset D(p,2R)$. All in all,

$\psi(Q_{I})\subset D(p,2R)\cap\Omega$

and

$\displaystyle|\psi(Q_{I})|$ $\displaystyle\geq|D(\psi(z_{I}),r)|\sim
r^{2}\sim R^{2}\sim|D(p,2R)|$ $\displaystyle\geq|D(p,2R)\cap\Omega|.$

This computation shows that every $\psi(Q_{I})$ is contained in a disk $D$
centered on $\partial\Omega$ of comparable area, so
$[\sigma]_{\mathrm{B}_{p}(\Omega)}\lesssim[\sigma]_{\mathrm{D}_{p}(\Omega)}.$

On the other hand, let $p\in\partial\Omega$ and $s>0$. We want to find $Q_{I}$
such that

(2.10)
$D(p,s)\cap\Omega\subset\psi(Q_{I}),\qquad\left|D(p,s)\cap\Omega\right|\gtrsim\left|\psi(Q_{I})\right|,$

from which
$[\sigma]_{\mathrm{D}_{p}(\Omega)}\lesssim[\sigma]_{\mathrm{B}_{p}(\Omega)}$
immediately follows. As before, set
$r=\min_{|z-p|=s}|\Psi^{-1}(z)-\Psi^{-1}(p)|$ and
$R=\max_{|z-p|=s}|\Psi^{-1}(z)-\Psi^{-1}(p)|$. Again, since $\Psi$ is
quasisymmetric, $r\sim R$ and with $q=\Psi^{-1}(p)\in\mathbb{T}$,

$D(q,r)\subset\Psi^{-1}(D(p,s))\subset D(q,R).$

Simple geometry shows that there exists $Q_{I}\supset D(q,R)\cap\mathbb{D}$
with $|Q_{I}|\sim R^{2}$. Clearly,

$D(p,s)\cap\Omega\subset\psi(Q_{I}).$

To establish (2.10), it remains to show
$|\psi(Q_{I})|\lesssim|D(p,s)\cap\Omega|.$ Thus, the proof is concluded by
applying Lemma 2.5 to obtain

$1\lesssim\left(\frac{|D(q,r)\cap\mathbb{D}|}{|Q_{I}|}\right)^{1/\delta}\lesssim\frac{|\psi(D(q,r)\cap\mathbb{D})|}{|\psi(Q_{I})|}\lesssim\frac{|D(p,s)\cap\Omega|}{|\psi(Q_{I})|}.$

∎

##  3\. Limited range weighted estimates for $\Pi_{\Omega}$

In addition to the weight classes $\mathrm{B}_{p}(\Omega)$ defined in (1.4),
for $q_{0}\geq 1$, define

$\mathrm{B}_{q_{0}^{+}}(\mathbb{D})=\bigcap_{q>q_{0}}\mathrm{B}_{q}(\mathbb{D}),$

and for $1<s<\infty$, define the reverse Hölder class
$\mathrm{RH}_{s}(\Omega)$ to be the weights $\sigma$ on $\Omega$ such that the
characteristic

$\left[\sigma\right]_{\mathrm{RH}_{s}(\Omega)}=\sup_{I\subseteq\mathbb{T}}\left\langle\sigma\right\rangle_{s,\psi(Q_{I})}\left(\left\langle\sigma\right\rangle_{\psi(Q_{I})}\right)^{-1}$

is finite. The goal of this section is to prove the following of which Theorem
B from the introduction is the special case $p_{0}=q_{0}=1$.

######  Theorem E.

Let $\Omega$ be a bounded, simply connected domain and $\psi$ a conformal map
from $\mathbb{D}$ onto $\Omega$. Let $1\leq p_{0}<2$ and set
$q_{0}=\frac{p_{0}}{2-p_{0}}$. If
$\left|\psi^{\prime}\right|^{2}\in\mathrm{B}_{q_{0}^{+}}(\mathbb{D})$,
$p_{0}<p<p_{0}^{\prime}$, and

(3.1)
$\sigma\in\left\\{\begin{array}[]{ll}\mathrm{B}_{p}(\Omega)&p_{0}=1,\\\\[5.69054pt]
\mathrm{B}_{\frac{p}{p_{0}}}(\Omega)\cap\mathrm{RH}_{\frac{p_{0}^{\prime}}{p_{0}^{\prime}-p}}(\Omega),&p_{0}>1,\end{array}\right.$

then $\Pi_{\Omega}:L^{p}(\Omega,\sigma)\to L^{p}(\Omega,\sigma)$.

If $\left|\psi^{\prime}\right|^{2}$ belonged to the smaller class
$\mathrm{B}_{q_{0}}(\mathbb{D})$, this theorem would be more or less trivial
for the same reasons leading to Theorem A above. We outline that argument
below, after the statement of Theorem F, the main technical result from which
Theorem E follows.

For two weights $u,v$ on $\mathbb{D}$ and a dyadic grid $\mathcal{D}$ define
the weighted characteristic

$\left[u\right]_{\mathrm{B}_{p}(\mathcal{D},v)}=\sup_{I\in\mathcal{D}}\left\langle
u\right\rangle_{v,Q_{I}}\left\langle
u^{-1}\right\rangle_{v,\frac{1}{p-1},Q_{I}}.$

We say $u$ belongs to $\mathrm{B}_{p}(\mathcal{D},v)$ if the above
characteristic is finite, and we drop the dependence on $v$ when $v\equiv 1$.
Similarly, we let

$\left[u\right]_{\mathrm{RH}_{s}(\mathcal{D},v)}=\sup_{I\in\mathcal{D}}\left\langle
u\right\rangle_{s,v,Q_{I}}\left(\left\langle
u\right\rangle_{v,Q_{I}}\right)^{-1}.$

######  Theorem F.

Let $\psi:\mathbb{D}\to\mathbb{C}$ be conformal, $\mathcal{D}$ a dyadic grid
on $\mathbb{T}$, and $q_{0}\geq 1$. Suppose that
$v=\left|\psi^{\prime}\right|^{2}\in\mathrm{B}_{q_{0}^{+}}(\mathbb{D})$. If

(3.2)
$u\in\left\\{\begin{array}[]{ll}\mathrm{B}_{2}(\mathcal{D},v),&q_{0}=1;\\\\[5.69054pt]
\mathrm{B}_{\frac{q_{0}+1}{q_{0}}}(\mathcal{D},v)\cap\mathrm{RH}_{q_{0}}(\mathcal{D},v)&1<q_{0}<\infty,\end{array}\right.$

then $u\in\mathrm{B}_{2}(\mathcal{D},v)$.

If $v$ belonged to the smaller class $\mathrm{B}_{q_{0}}(\mathbb{D})$, this
result would be trivial. Indeed, for $q_{0}=1$, consult (1.9). For $q_{0}>1$,
$r=\frac{1+q_{0}}{q_{0}}$ satisfies $-q_{0}=\frac{1}{1-r}$. Therefore,

(3.3) $\left\langle u\right\rangle_{Q_{I}}\left\langle
u^{-1}\right\rangle_{Q_{I}}\leq\left[v\right]_{\mathrm{B}_{q_{0}}}^{\frac{2}{q_{0}}}\left\langle
u\right\rangle_{q_{0},v,Q_{I}}\left\langle
u^{-1}\right\rangle_{q_{0},v,Q_{I}}\leq\left[v\right]_{\mathrm{B}_{q_{0}}}^{\frac{2}{q_{0}}}\left[u\right]_{\mathrm{RH}_{q_{0}}(v)}\left[u\right]_{\mathrm{B}_{r}(v)}.$

The key idea in the proof of Theorem F is to use dyadic versions of weight
characteristics and apply a dyadic regularization. We remark a similar idea
appears in [huowick2022] when proving weighted inequalities for the vector-
valued Bergman projection with matrix weights.

Recall the weighted dyadic maximal operators $M_{v}^{\mathcal{D}}$ from (2.3)
and the following paragraph. Define the class
$\mathrm{B}_{\infty}(\mathcal{D},v)$ to be those weights $u$ satisfying

(3.4)
$\left[u\right]_{\mathrm{B}_{\infty}(\mathcal{D},v)}:=\sup_{I\in\mathcal{D}}\frac{\left\langle
M_{v}^{\mathcal{D}}(u\chi_{Q_{I}})\right\rangle_{v,Q_{I}}}{\left\langle
u\right\rangle_{v,Q_{I}}}<\infty.$

######  Proposition 3.1.

If there exists $C>0$ and $0<s<1$ such that

(3.5) $\left\langle u\right\rangle_{v,Q_{I}}\leq C\left\langle
u\right\rangle_{s,v,Q_{I}},\qquad\forall I\in\mathcal{D},$

then $u\in\mathrm{B}_{\infty}(\mathcal{D},v)$.

###### Proof.

Let $I\in\mathcal{D}$ and $z\in Q_{I}$. Then,

$M_{v}^{\mathcal{D}}(u\chi_{Q_{I}})(z)=\sup_{\begin{subarray}{c}J\in\mathcal{D}:\\\
J\subset I\\\ z\in Q_{J}\end{subarray}}\left\langle
u\right\rangle_{v,Q_{J}}\leq
C\left[M_{v}^{\mathcal{D}}(u^{s}\chi_{Q_{I}})(z)\right]^{\frac{1}{s}}.$

Multiplying the above display by $v(z)$, integrating over $z\in Q_{I}$, and
using the fact that $M_{v}^{\mathcal{D}}$ is bounded on
$L^{1/s}(\mathbb{D},v)$ concludes the proof. ∎

An immediate consequence of Proposition 3.1 is

(3.6)
$\bigcup_{1<p<\infty}\left\\{u:u\in\mathrm{B}_{p}(\mathcal{D},v)\right\\}\cup\left\\{w^{p}:w\in\mathrm{RH}_{p}(\mathcal{D},v)\right\\}\subset\mathrm{B}_{\infty}(\mathcal{D},v).$

We say a weight $u$ satisfies the APR($\mathcal{D}$) condition if there exists
$C>0$ such that for each $I\in\mathcal{D}$,

(3.7) $C^{-1}u(z_{2})\leq u(z_{1})\leq Cu(z_{2})\qquad\forall z_{1},z_{2}\in
T_{I}.$

This is a special case of the condition introduced in [APR] with
$\rho=\frac{1}{2}$ (though the condition in [APR] is qualitatively independent
of $\rho$ and ranges over all intervals).

######  Proposition 3.2.

Suppose that $v$ is a doubling weight on $\mathcal{D}$, and
$w\in\mathrm{B}_{\infty}(\mathcal{D},v)$ satisfies the APR($\mathcal{D}$)
condition. Then there exists $\tau>1$ only depending on
$[w]_{\mathrm{B}_{\infty}(\mathcal{D},v)}$ and $c_{v}$ so that
$w\in\mathrm{RH}_{\tau}(\mathcal{D},v)$.

###### Proof.

This is a straightforward adaptation of the argument in
[stockdale2023weighted]*Lemma 2.8. First note that, as a consequence of the
APR($\mathcal{D}$) condition for $w$, for any Carleson box $Q_{I}$ with
$I\in\mathcal{D}$ and $z\in Q_{I}$, we have $w(z)\lesssim
M_{v}^{\mathcal{D}}(w\chi_{Q_{I}})(z).$ Indeed, note that if $z\in Q_{I}$,
there is a unique interval $J\subset I$ in $\mathcal{D}$ so $z\in T_{J}.$ We
then estimate, using (3.7) and the second inequality in (2.2):

$w(z)\lesssim\left\langle w\right\rangle_{T_{J},v}\lesssim\left\langle
w\right\rangle_{Q_{J},v}\leq M^{\mathcal{D}}_{v}(w\chi_{Q_{I}})(z).$

Therefore, we have reduced the problem to showing that for some $\tau>1$,

(3.8) $\left\langle
M_{v}^{\mathcal{D}}(w\chi_{Q_{I}})\right\rangle_{\tau,v,Q_{I}}\lesssim\left\langle
w\right\rangle_{v,Q_{I}}.$

Fix $I\in\mathcal{D}$ and for each $\lambda>0,$ let $\Omega_{\lambda}=\\{z\in
Q_{I}:M_{v}^{\mathcal{D}}(w\chi_{Q_{I}})(z)>\lambda\\}.$ For each $\lambda>0,$
write $\Omega_{\lambda}$ as a disjoint union of maximal $Q_{\lambda,j}$ and
let ${\widehat{Q}}_{\lambda,j}$ denote the corresponding dyadic parents. Note
that by maximality

(3.9) $\displaystyle\left\langle
w\chi_{Q_{I}}\right\rangle_{v,{\widehat{Q}}_{\lambda,j}}\leq\lambda,$ (3.10)
$\displaystyle
M_{v}^{\mathcal{D}}(w\chi_{Q_{I}})(z)=M_{v}^{\mathcal{D}}(w\chi_{Q_{\lambda,j}})(z)\quad\mbox{for}\quad
z\in Q_{\lambda,j}.$

Using the distribution function, for any $\tau>1$, split

(3.11) $\displaystyle\int_{Q_{I}}$
$\displaystyle[M_{v}(w\chi_{Q_{I}})]^{\tau}v\,dA$
$\displaystyle=\int_{0}^{\left\langle
w\right\rangle_{v,Q_{I}}}(\tau-1)\lambda^{\tau-2}\left(\int_{\Omega_{\lambda}}M_{v}^{\mathcal{D}}(w\chi_{Q_{I}})v\,dA\right)\,d\lambda$
$\displaystyle+\int_{\left\langle
w\right\rangle_{v,Q_{I}}}^{\infty}(\tau-1)\lambda^{\tau-2}\left(\int_{\Omega_{\lambda}}M_{v}^{\mathcal{D}}(w\chi_{Q_{I}})v\,dA\right)\,d\lambda.$

By virtue of the containment $\Omega_{\lambda}\subset Q_{I}$ and (3.4), the
first integral is trivially bounded by
$\left[w\right]_{\mathrm{B}_{\infty}(\mathcal{D},v)}\left\langle
w\right\rangle_{v,Q_{I}}^{\tau}v(Q_{I})$. To handle the second integral,
notice that for each $\lambda$ and $j$, by (3.10), (3.4), (3.9), and (2.2),

$\int_{Q_{\lambda,j}}M_{v}^{\mathcal{D}}(w\chi_{Q_{I}})v\,dA\leq[w]_{\mathrm{B}_{\infty}(\mathcal{D},v)}\int_{Q_{\lambda,j}}wv\,dA\leq[w]_{\mathrm{B}_{\infty}(\mathcal{D},v)}c_{v}\lambda\int_{Q_{\lambda,j}}v\,dA.$

Summing this over $j$, we obtain

$\displaystyle\int_{\left\langle w\right\rangle_{v,Q_{I}}}^{\infty}$
$\displaystyle(\tau-1)\lambda^{\tau-2}\left(\int_{\Omega_{\lambda}}M_{v}^{\mathcal{D}}(w\chi_{Q_{I}})v\,dA\right)\,d\lambda$
$\displaystyle\leq[w]_{\mathrm{B}_{\infty}(\mathcal{D},v)}\times
c_{v}\times\frac{\tau-1}{\tau}\int_{\left\langle
w\right\rangle_{v,Q_{I}}}^{\infty}\tau\lambda^{\tau-1}v(\Omega_{\lambda})\,d\lambda.$

Choosing
$\tau=\frac{2c_{v}[w]_{\mathrm{B}_{\infty}(\mathcal{D},v)}}{2c_{v}[w]_{\mathrm{B}_{\infty}(\mathcal{D},v)}-1}$,
the final term is bounded by $\frac{1}{2}$ times the LHS of (3.11); therefore
(3.8) is established. ∎

Given any two weights $u$, $v$, we introduce the following regularization
associated to a dyadic grid $\mathcal{D}$:

(3.12) $u_{\mathcal{D},v}:=\sum_{I\in\mathcal{D}}\langle
u\rangle_{v,T_{I}}\chi_{T_{I}}.$

When $v\equiv 1$, we simply write $u_{\mathcal{D}}.$ Let us list some
elementary properties of the regularizations.

######  Lemma 3.3.

  * (i)

$u_{\mathcal{D},v}$ satisfies the APR($\mathcal{D}$) condition.

  * (ii)

For each $I\in\mathcal{D}$, $\left\langle
u\right\rangle_{v,Q_{I}}=\left\langle
u_{\mathcal{D},v}\right\rangle_{v,Q_{I}}$.

  * (iii)

For $0<s\leq 1$, $(u^{s})_{\mathcal{D},v}\leq u_{\mathcal{D},v}^{s}$.

  * (iv)

If $u$ satisfies (3.5), then so does $u_{\mathcal{D},v}$.

###### Proof.

Statements (i) and (ii) are trivial. Statement (iii) follows from Hölder’s
inequality. Statement (iv) follows by applying (ii), (3.5), (ii) again, and
(iii) to obtain

$\left\langle u_{\mathcal{D},v}\right\rangle_{v,Q_{I}}=\left\langle
u\right\rangle_{v,Q_{I}}\leq C\left\langle
u\right\rangle_{s,v,Q_{I}}=C\left\langle(u^{s})_{\mathcal{D},v}\right\rangle_{v,Q_{I}}^{\frac{1}{s}}\leq
C\left\langle u_{\mathcal{D},v}\right\rangle_{s,v,Q_{I}}.$

∎

We are now ready for the final ingredient to the proof of Theorem F.

######  Proposition 3.4.

Let $v=\left|\psi^{\prime}\right|^{2}$ for $\psi$ which is conformal on
$\mathbb{D}$. Let $1\leq q<\infty$ such that $u^{q}$ satisfies (3.5). Then
there exists $r>q$ such that

$\left\langle
u\right\rangle_{Q_{I}}\lesssim[v]_{\mathrm{B}_{r}(\mathbb{D})}^{\frac{1}{r}}\left\langle
u\right\rangle_{q,v,Q_{I}},\qquad\forall I\in\mathcal{D}.$

###### Proof.

Let $I\in\mathcal{D}$ and notice for any $r>1$,

(3.13) $\displaystyle\left\langle u\right\rangle_{Q_{I}}$
$\displaystyle=\left\langle u_{\mathcal{D}}\right\rangle_{Q_{I}}$
$\displaystyle=\left\langle
u_{\mathcal{D}}v^{\frac{1}{r}}v^{-\frac{1}{r}}\right\rangle_{Q_{I}}$
$\displaystyle\leq\left\langle
u_{\mathcal{D}}^{r}v\right\rangle_{Q_{I}}^{\frac{1}{r}}\left\langle
v^{-\frac{1}{r-1}}\right\rangle_{Q_{I}}^{\frac{r-1}{r}}$
$\displaystyle\leq\left[v\right]_{\mathrm{B}_{r}}^{\frac{1}{r}}\left\langle
u_{\mathcal{D}}\right\rangle_{r,v,Q_{I}}.$

However, since $v=\left|\psi^{\prime}\right|^{2}$ and $\psi$ is conformal, by
the Koebe distortion theorem (Proposition 2.2), $\left\langle
u\right\rangle_{T_{I}}\sim\left\langle u\right\rangle_{v,T_{I}}$. Thus
$u_{\mathcal{D}}\sim u_{v,\mathcal{D}}$ which combined with (3.13) implies

$\left\langle
u\right\rangle_{Q_{I}}\lesssim\left[v\right]_{\mathrm{B}_{r}}^{\frac{1}{r}}\left\langle
u_{\mathcal{D},v}\right\rangle_{r,v,Q_{I}}.$

By Lemma 3.3 (i) and (iv), $(u^{q})_{\mathcal{D},v}$ satisfies the
APR($\mathcal{D}$) condition and (3.5), so Propositions 3.1 and 3.2 provide
$\tau>1$ such that

$\left\langle
u_{\mathcal{D},v}\right\rangle_{q\tau,v,Q_{I}}^{q}\leq\left\langle(u^{q})_{\mathcal{D},v}\right\rangle_{\tau,v,Q_{I}}\lesssim\left\langle(u^{q})_{\mathcal{D},v}\right\rangle_{v,Q_{I}}=\left\langle
u\right\rangle_{q,v,Q_{I}}^{q}.$

Taking $r=q\tau>q$ concludes the proof. ∎

Now the proof of Theorem F is immediate. Let $q_{0}\geq 1$ and let $u$ belong
to the RHS of (3.2). By (3.6), $u^{q_{0}}$ and $u^{-q_{0}}$ both satisfy
(3.5). Therefore, applying Proposition 3.4 to $u$ and $u^{-1}$ with exponent
$q=q_{0}$, gives $r_{1},r_{2}>q_{0}$ such that (3.13) holds for the respective
weights and parameters. Therefore, setting
$r^{*}=\min\\{r_{1},r_{2}\\}>q_{0}$, for each $I\in\mathcal{D}$,

(3.14) $\displaystyle\left\langle u\right\rangle_{Q_{I}}\left\langle
u^{-1}\right\rangle_{Q_{I}}$
$\displaystyle\lesssim\left[v\right]_{\mathrm{B}_{r^{*}}(\mathbb{D})}^{\frac{2}{r^{*}}}\left\langle
u\right\rangle_{q_{0},v,Q_{I}}\left\langle
u^{-1}\right\rangle_{q_{0},v,Q_{I}}$
$\displaystyle\leq\left[v\right]_{\mathrm{B}_{r^{*}}(\mathbb{D})}^{\frac{2}{r^{*}}}\left\\{\begin{array}[]{ll}\left[u\right]_{\mathrm{RH}_{q_{0}}(\mathcal{D},v)}\left[u\right]_{\mathrm{B}_{\frac{q_{0}+1}{q_{0}}}(\mathcal{D},v)},&q_{0}>1;\\\
\left[u\right]_{\mathrm{B}_{2}(\mathcal{D},v)},&q_{0}=1.\end{array}\right.$

To derive Theorem E from Theorem F, let $\sigma$ be as in (3.1) with $p=2$.
Setting $u=\sigma\circ\psi$ and $v=|\psi^{\prime}|^{2}$, (1.6) implies that
$u$ belongs to the RHS of (3.2) for any dyadic grid $\mathcal{D}$. Therefore,
Theorem F places $u=\sigma\circ\psi\in\mathrm{B}_{2}(\mathcal{D},v)$, and
taking $\mathcal{D}=\mathcal{D}_{j}$ for $j=1,2$ yields
$u\in\mathrm{B}_{2}(\mathbb{D})$. By (1.3) $\Pi_{\mathbb{D}}$ is bounded on
$L^{2}(\mathbb{D},u)$, and applying (1.8) yields the conclusion when $p=2$.
The case of general $p\in(p_{0},p_{0}^{\prime})$ follows by extrapolation
[cmp2011]*Theorems 3.22 and 3.31 since $\\{\psi(Q_{I})\\}$ is a Muckenhoupt
basis by Proposition 2.1.

Theorem E can also be proved without extrapolation. A more general statement
than (3.2) holds in Theorem F, namely that if

(3.15)
$v=\left|\psi^{\prime}\right|^{2}\in\mathrm{B}_{q_{0}^{+}}(\mathbb{D}),\
\mbox{and}\
u\in\left\\{\begin{array}[]{ll}\mathrm{B}_{\frac{p}{p_{0}}}(\mathcal{D},v)\cap\mathrm{RH}_{\frac{p_{0}^{\prime}}{p_{0}^{\prime}-p}}(\mathcal{D},v)&p_{0}>1,\\\
\mathrm{B}_{p}(\mathcal{D},v)&p_{0}=1,\end{array}\right.$

then $uv^{1-p/2}\in\mathrm{B}_{p}(\mathcal{D})$, where $p_{0}$ is related to
$q_{0}$ by $\frac{p_{0}^{\prime}}{2}=q_{0}^{\prime}$. Assuming this, Theorem E
can be proved in the same way as $p=2$ above. To prove that (3.15) implies
$uv^{1-p/2}\in\mathrm{B}_{p}(\mathcal{D})$, all that is needed is to tediously
check many exponents and use the following generalization of Proposition 3.4.

######  Proposition 3.5.

Let $v=\left|\psi^{\prime}\right|^{2}$ for $\psi$ which is conformal on
$\mathbb{D}$ and $\theta<\frac{1}{2}$. Let $q\geq 1$ such that
$(1-\theta)q^{\prime}\geq 1$ and $u^{q}$ satisfies (3.5). Then there exists
$r>\left((1-\theta)q^{\prime}\right)^{\prime}$ such that

$\left\langle
uv^{\theta}\right\rangle_{Q_{I}}\lesssim[v]_{\mathrm{B}_{r}(\mathbb{D})}^{\frac{1}{r}}\left\langle
u\right\rangle_{q,v,Q_{I}}\left\langle
v^{-1}\right\rangle_{\frac{1}{r-1},Q_{I}}^{-\theta},\qquad\forall
I\in\mathcal{D}.$

Proposition 3.4 is the special case of $\theta=0$. Its proof is the same,
using the regularization $u_{\mathcal{D},v^{\theta}}$ rather than
$u_{\mathcal{D}}$. Now assuming (3.15), apply Proposition 3.5 to
$(u,q_{1},\theta_{1})$ and $(u^{-\frac{1}{p-1}},q_{2},\theta_{2})$ with

$\textstyle q_{1}=\frac{p_{0}^{\prime}}{p_{0}^{\prime}-p},\quad
q_{2}=\frac{p-1}{\frac{p}{p_{0}}-1},\quad\theta_{1}=1-\frac{p}{2},\quad\theta_{2}=\frac{-1}{p-1}\left(1-\frac{p}{2}\right).$

The key computation is that $((1-\theta_{j})q_{j}^{\prime})^{\prime}=q_{0}$
for $j=1,2$, and then a similar argument to (3.14) shows
$uv^{1-p/2}\in\mathrm{B}_{p}(\mathcal{D})$.

##  4\. Converse to Theorem E on uniform domains

In this section, we will prove the following result showing that on uniform
domains, Theorem E is sharp. Define the reverse Hölder characteristic with
respect to disks centered on the boundary by

$[\sigma]_{\mathrm{RHD}_{s}(\Omega)}=\sup_{D}\left\langle\sigma\right\rangle_{s,D\cap\Omega}\left(\left\langle\sigma\right\rangle_{D\cap\Omega}\right)^{-1}.$

######  Theorem G.

Let $\Omega$ be a bounded, simply connected uniform domain and $\psi$ a
conformal map from $\mathbb{D}$ onto $\Omega$. Suppose $1\leq p_{0}<2$ and set
$q_{0}=\frac{p_{0}}{2-p_{0}}$. The following are equivalent.

  * A.

$\left|\psi^{\prime}\right|^{2}\in\mathrm{B}_{q_{0}^{+}}(\mathbb{D})$.

  * B.

For $p_{0}<p<p_{0}^{\prime}$, if

$\sigma\in\left\\{\begin{array}[]{ll}\mathrm{B}_{p}(\Omega)&p_{0}=1,\\\\[5.69054pt]
\mathrm{B}_{\frac{p}{p_{0}}}(\Omega)\cap\mathrm{RH}_{\frac{p_{0}^{\prime}}{p_{0}^{\prime}-p}}(\Omega),&p_{0}>1,\end{array}\right.$

then $\Pi_{\Omega}:L^{p}(\Omega,\sigma)\to L^{p}(\Omega,\sigma)$.

  * C.

For $p_{0}<p<p_{0}^{\prime}$, if

$\sigma\in\left\\{\begin{array}[]{ll}\mathrm{D}_{p}(\Omega)&p_{0}=1,\\\\[5.69054pt]
\mathrm{D}_{\frac{p}{p_{0}}}(\Omega)\cap\mathrm{RHD}_{\frac{p_{0}^{\prime}}{p_{0}^{\prime}-p}}(\Omega),&p_{0}>1,\end{array}\right.$

then $\Pi_{\Omega}:L^{p}(\Omega,\sigma)\to L^{p}(\Omega,\sigma)$.

B. and C. are equivalent by Proposition 2.6 (the same argument there works for
$\mathrm{RH}_{s}(\Omega)$ and $\mathrm{RHD}_{s}(\Omega)$). A. implies B. by
Theorem E so it remains to show that B. implies A. We will need the following
two lemmata to establish this remaining implication. The first is a
consequence of the Koebe Distortion Theorem (Proposition 2.2).

######  Lemma 4.1.

Let $I_{1},I_{2}$ be neighboring intervals in $\mathbb{T}$ and
$I_{0}=I_{1}\cup I_{2}$. Then

$|\psi(Q_{I_{0}})|\lesssim|\psi(Q_{I_{1}})|+|\psi(Q_{I_{2}})|.$

The second concerns the maximal function

$M_{\psi}f(z)=\sup_{I\subset\mathbb{T}}\chi_{\psi(Q_{I})}\left\langle
f\right\rangle_{\psi(Q_{I})},\qquad f\in L^{1}(\Omega).$

######  Lemma 4.2.

Let $f\in L^{1}(\Omega)$ and $q<1$. Then,

$(M_{\psi}f)^{q}\in\mathrm{B}_{1}(\Omega).$

The proofs of Lemmata 4.2 and 4.1 are postponed to Section 4.1. For now, let
us prove following proposition, which will quickly show that B. implies A. It
is loosely based on the necessity argument from [jones-hom] concerning the
homeomorphisms preserving $\operatorname{BMO}(\mathbb{R})$.

######  Proposition 4.3.

Suppose $1\leq s<\infty$ has the property that every
$w\in\mathrm{B}_{1}(\Omega)\cap\mathrm{RH}_{s}(\Omega)$ has the additional
property that $w\circ\psi$ is weakly doubling. Then for
$v=\left|\psi^{\prime}\right|^{2}$, any $g\in L^{1}(\mathbb{D},v)$, and any
$q<\frac{1}{s}$,

(4.1) $\left\langle
M_{v}(\chi_{Q_{I}}g)\right\rangle_{q,Q_{I}}\lesssim\left\langle
g\right\rangle_{v,Q_{I}}.$

###### Proof.

Let $sq<1$, and $g\in L^{1}(\mathbb{D},v)$. Set $\phi=\psi^{-1}$ and
$f=g\circ\phi$. Fix an interval $I$. Partition $I=I_{1}\cup I_{2}$ where
$|\psi(Q_{I_{1}})|=|\psi(Q_{I_{2}})|$. By Lemma 4.1 these quantities are
comparable to $|\psi(Q_{I})|$. Let $e^{i\theta_{j}}$ be the center of $I_{j}$,
and partition $Q_{I}=Q_{1}\cup Q_{2}$ where

$Q_{j}=\left\\{re^{i\theta}\in
Q_{I}:|\theta-\theta_{j}|\leq\frac{\ell(I_{j})}{2}\right\\}.$

The sublinearity of the maximal function and the fact that $q<1$ implies that
for some $j\in\\{1,2\\}$,

(4.2)
$2\int_{\psi(Q_{I})}M_{\psi}(\chi_{\psi(Q_{j})}f)^{q}\left|\phi^{\prime}\right|^{2}\geq\int_{\psi(Q_{I})}M_{\psi}(\chi_{\psi(Q_{I})}f)^{q}\left|\phi^{\prime}\right|^{2}.$

Without loss of generality, assume (4.2) holds for $j=1$ and set
$w=M_{\psi}(\chi_{\psi(Q_{1})}f)^{q}$. Then, by Lemma 4.2,
$w^{s}\in\mathrm{B}_{1}(\Omega)$ so
$w\in\mathrm{B}_{1}(\Omega)\cap\mathrm{RH}_{s}(\Omega)$. Take $J$ to be a
neighbor of $I_{2}$ and $I$ with $\ell(J)=\ell(I)$, and we claim that

(4.3) $w\lesssim\left(\left\langle
f\right\rangle_{\psi(Q_{I})}\right)^{q}\quad\mbox{ on }\psi(Q_{J}).$

Indeed, for $z\in\psi(Q_{J})$ and any $K$ such that $z\in\psi(Q_{K})$ and $K$
has nonempty intersection with $I_{1}$, $I_{2}$ must be contained in $K$.
Therefore, $|\psi(Q_{K})|\geq|\psi(Q_{I_{2}})|\gtrsim|\psi(Q_{I})|$, which
establishes (4.3).

Now we pull back with $\psi$. Set $u=w\circ\psi$ and notice that

$\left\langle u\right\rangle_{Q_{J\cup
I}}\gtrsim\frac{1}{|Q_{I}|}\int_{Q_{I}}w\circ\psi\gtrsim\frac{1}{|Q_{I}|}\int_{\psi(Q_{I})}M_{\psi}(\chi_{\psi(Q_{I})}f)^{q}\left|\phi^{\prime}\right|^{2},$

where the last inequality follows by changing variables and (4.2). On the
other hand, since we assume that $u$ is weakly doubling, by (4.3),

$\left\langle u\right\rangle_{Q_{J\cup I}}\lesssim\left\langle
u\right\rangle_{Q_{J}}\lesssim\left(\left\langle
f\right\rangle_{\psi(Q_{I})}\right)^{q}.$

Combining these two displays, we obtain

$\left(\frac{1}{|Q_{I}|}\int_{\psi(Q_{I})}M_{\psi}(\chi_{\psi(Q_{I})}f)^{q}\left|\phi^{\prime}\right|^{2}\right)^{\frac{1}{q}}\lesssim\left\langle
f\right\rangle_{\psi(Q_{I})}.$

Changing variables, we obtain (4.1). ∎

###### Proof of B. implies A. in Theorem G.

Let $1\leq p_{0}<2$, and let

$w\in\left\\{\begin{array}[]{ll}\mathrm{B}_{1}(\Omega)&p_{0}=1;\\\\[5.69054pt]
\mathrm{B}_{1}(\Omega)\cap\mathrm{RH}_{q_{0}}(\Omega),&p_{0}>1.\end{array}\right.$

Then, since
$\mathrm{B}_{1}(\Omega)\subset\mathrm{B}_{\frac{2}{p_{0}}}(\Omega)$, part B.
implies that $\Pi_{\Omega}$ is bounded on $L^{2}(\Omega,w)$, or equivalently
by (1.8) and (1.3), $w\circ\psi\in\mathrm{B}_{2}(\mathbb{D}).$ Since any
weight in $\mathrm{B}_{2}(\mathbb{D})$ is weakly doubling, we have inequality
(4.1) for any $q<\frac{1}{q_{0}}$. Now, let $p>q_{0}$ and select
$g=v^{-\frac{p}{p-1}}$ so that on the one hand

$\left\langle
g\right\rangle_{v,Q_{I}}=\frac{\int_{Q_{I}}v^{-\frac{p}{p-1}+1}\,dA}{\int_{Q_{I}}v\,dA}=\frac{\int_{Q_{I}}v^{-\frac{1}{p-1}}\,dA}{\int_{Q_{I}}v\,dA}.$

On the other hand, we claim to have the pointwise domination

$M_{v}(\chi_{Q_{I}}g)(\zeta)\gtrsim g(\zeta),\qquad\zeta\in Q_{I}.$

To see this, fix $\zeta\in Q_{I}$ and find $J\subseteq I$ so that $\zeta\in
T_{J}.$ Then, using the doubling property of $v$ from Lemma 2.5 together with
Proposition 2.2, we have

$\displaystyle g(\zeta)\sim\left\langle
g\right\rangle_{v,T_{J}}\lesssim\langle g\rangle_{v,Q_{J}}\lesssim
M_{v}(\chi_{Q_{I}}g)(\zeta).$

Therefore, applying (4.1) with $q=\frac{1}{p}<\frac{1}{q_{0}}$, we obtain

$\frac{\int_{Q_{I}}v^{-\frac{1}{p-1}}\,dA}{\int_{Q_{I}}v\,dA}\gtrsim\langle
M_{v}(\chi_{Q_{I}}g)\rangle_{q,Q_{I}}\gtrsim\left(\frac{1}{|Q_{I}|}\int_{Q_{I}}v^{-\frac{1}{p-1}}\,dA\right)^{p}.$

Simple algebra and taking a supremum over all intervals $I$ then implies
$v\in\mathrm{B}_{p}(\mathbb{D})$, and since $p>q_{0}$ was arbitrary we
conclude $v\in\mathrm{B}_{q_{0}^{+}}(\mathbb{D})$. ∎

######  Remark 4.4.

Notice that the only place we used that $\Omega$ was a uniform domain was in
the Proof of B. implies A. to bound $g$ pointwise by the maxmal function,
which relied on $v$ being doubling. So even in the case of non-uniform domain,
we obtain the necessary condition

$\left\langle
M_{v}(\chi_{Q_{I}}v^{-\frac{p}{p-1}})\right\rangle_{\frac{1}{p},Q_{I}}\left\langle
v\right\rangle_{Q_{I}}\lesssim\left\langle
v^{-\frac{1}{p-1}}\right\rangle_{Q_{I}},\quad p>q_{0},$

which reduces to the $\mathrm{B}_{q_{0}^{+}}(\mathbb{D})$ condition if
$M_{v}(\chi_{Q_{I}}v^{-\frac{p}{p-1}})\gtrsim v^{-\frac{p}{p-1}}$, which holds
in particular if $v$ is doubling.

###  4.1. Proofs of Lemmata 4.1 and 4.2

###### Proof of Lemma 4.1.

We may assume $\ell(I_{1})\geq\ell(I_{2})$ and $I_{1}$ is the right neighbor
of $I_{2}$. Let $\theta_{j}$ be the center of $I_{j}$. Then $Q_{I_{0}}$ can be
partitioned into 4 regions: $Q_{I_{1}}$, $Q_{I_{2}}$,

$B=\left\\{re^{i\theta}:1-\ell(I_{1})\leq
r<1-\ell(I_{2}),|\theta-\theta_{2}|\leq\frac{\ell(I_{2})}{2}\right\\},$

and

$T=\left\\{re^{i\theta}:1-\ell(I_{0})\leq
r<1-\ell(I_{1}),|\theta-\theta_{0}|\leq\frac{\ell(I_{0})}{2}\right\\}.$

$Q_{I_{1}}$$Q_{I_{2}}$$B$$T$$T_{J}$

$Q_{I_{1}}$$Q_{I_{2}}$$B$$T$$T_{J_{1}}$$T_{J_{2}}$$T_{J_{3}}$

Figure 2. $Q_{I_{0}}=Q_{I_{1}}\cup Q_{I_{2}}\cup B\cup T$ and
$B\subset\cup_{k=1}^{K}T_{J_{k}}$.

First, let $J$ be an interval with the same right endpoint as $I_{1}$ but with
$\ell(J)=\frac{\ell(I_{1})+\ell(I_{0})}{2}$. There exists $z_{1}\in T\cap
T_{J}$ and $z_{2}\in T_{I_{1}}\cap T_{J}$. Furthermore, since $T\subset
T_{I_{0}}$, by Proposition 2.2, for any $z_{3}\in T$ and $z_{4}\in T_{I_{1}}$,

$\left|\psi^{\prime}(z_{3})\right|\sim\left|\psi^{\prime}(z_{1})\right|\sim\left|\psi^{\prime}(z_{2})\right|\sim\left|\psi^{\prime}(z_{4})\right|.$

Since $|T|\lesssim|T_{I_{1}}|$ the above display implies
$\left|\psi(T)\right|\lesssim\left|\psi(T_{I_{1}})\right|\leq\left|\psi(Q_{I_{1}})\right|$.

To handle $B$, for each $k\in\mathbb{N}$, let $J_{k}$ be the interval with the
same left endpoint as $I_{2}$, but with $\ell(J_{k})=2^{k}\ell(I_{2})$. Then,
$B\subset\cup_{k=1}^{K}T_{J_{k}}$ where
$K=\lceil\log_{2}\frac{\ell(I_{1})}{\ell(I_{2})}\rceil$.
$\\{T_{J_{k}}\\}_{k=1}^{K}$ is pairwise disjoint and

$|T_{J_{k}}|\lesssim|(Q_{I_{1}}\cup T)\cap T_{J_{k}}|.$

Therefore, applying again Proposition 2.2 and summing over $k$,

$|\psi(B)|\leq\sum_{k=1}^{K}|\psi(T_{J_{k}})|\lesssim\sum_{k=1}^{K}|\psi((Q_{I_{1}}\cup
T)\cap T_{J_{k}})|\lesssim\left|\psi(Q_{I})\right|.$

∎

To prove Lemma 4.2, we will use the following Vitali covering lemma for
$\psi(Q_{I})$.

######  Lemma 4.5.

Let $\mathcal{I}$ be a finite collection intervals in $\mathbb{T}$. Then,
there exist disjoint intervals $\\{I_{k}\\}_{k=1}^{K}\subset\mathcal{I}$, such
that

$\left|\bigcup_{I\in\mathcal{I}}\psi(Q_{I})\right|\lesssim\sum_{k=1}^{K}\left|\psi(Q_{I_{k}})\right|,$

where the implicit constant is absolute.

###### Proof.

For any non-empty collection of intervals $\mathcal{J}$, let $I(\mathcal{J})$
to be an interval in $\mathcal{J}$ with
$|\psi(Q_{I(\mathcal{J})})|=\max_{I\in\mathcal{J}}\left|\psi(Q_{I})\right|$.
Let $\mathcal{I}_{0}=\mathcal{I}$ and inductively define for $k\geq 0$,

$I_{k}=I(\mathcal{I}_{k}),\quad\mathcal{I}_{k+1}=\\{I\in\mathcal{I}_{k}:I\cap
I_{k}=\varnothing\\}.$

Since $\mathcal{I}$ is a finite collection, this process terminates after say
$K$ steps once $\mathcal{I}_{K+1}=\varnothing$. By construction
$\\{I_{k}\\}_{k=0}^{K}$ are disjoint. Fix $k$ and let $J_{k}^{1}$ and
$J_{k}^{2}$ be the left and right neighbors of $I_{k}$ such that

$|\psi(Q_{J_{k}^{1}})|=|\psi(Q_{I_{k}})|=|\psi(Q_{J_{k}^{2}})|.$

Setting $J_{k}=J_{k}^{1}\cup I_{k}\cup J_{k}^{2}$, Lemma 4.1 implies
$\left|\psi(Q_{J_{k}})\right|\lesssim\left|\psi(Q_{I_{k}})\right|$. The lemma
will be proved if we can show that for each $I\in\mathcal{I}$, there exists
$k$ such that $I\subset J_{k}$.

To this end, given $I\in\mathcal{I}$, let $k$ be the first index such that
$I\not\in\mathcal{I}_{k+1}$. Then $I\in\mathcal{I}_{k}$ and $I\cap
I_{k}\neq\varnothing$. Suppose toward a contradiction that $I\backslash
J_{k}\neq\varnothing$. Then $I\supsetneq J_{k}^{j}$ for some $j\in\\{1,2\\}$,
but since $I\in\mathcal{I}_{k}$,

$\left|\psi(Q_{I})\right|\leq\left|\psi(Q_{I_{k}})\right|=\left|\psi(Q_{J_{k}^{j}})\right|<\left|\psi(Q_{I})\right|,$

which is a contradiction. ∎

As a consequence,

######  Lemma 4.6.

$M_{\psi}$ is of weak type $(1,1)$.

###### Proof.

Let $f\in L^{1}(\Omega)$, $\lambda>0$, and
$E=\\{z\in\Omega:M_{\psi}f(z)>\lambda\\}$. For each $z\in E$, there exists
$I(z)$ such that $z\in\psi(Q_{I(z)})$ and $\left\langle
f\right\rangle_{\psi(Q_{I(z)})}>\lambda$. Therefore
$\\{\psi(Q_{I(z)})\\}_{z\in E}$ covers $E$. Now let $F$ be an arbitrary
compact subset of $E$. $F$ can be covered by a finite subcollection of
$\\{\psi(Q_{I(z)})\\}_{z\in E}$, say $\\{\psi(Q_{I})\\}_{I\in\mathcal{I}}$.
Applying Lemma 4.5 above, there exists a pairwise disjoint subcollection
$\\{I_{k}\\}_{k=1}^{K}\subset\mathcal{I}$ such that

$|F|\lesssim\sum_{k=1}^{K}|\psi(Q_{I_{k}})|\leq\frac{1}{\lambda}\sum_{k=1}^{K}\int_{\psi(Q_{I_{k}})}|f|\,dA\leq\frac{1}{\lambda}\left\|f\right\|_{L^{1}(\Omega)}.$

We conclude by taking the supremum over all $F$ which are compact subsets of
$E$. ∎

###### Proof of Lemma 4.2.

The $\mathrm{B}_{1}(\Omega)$ condition can equivalently be stated as follows.
There exists $C>0$ such that for each $z\in\Omega$ and $I$ such that
$z\in\psi(Q_{I})$,

$\left\langle M_{\psi}f\right\rangle_{q,\psi(Q_{I})}\leq
C^{\frac{1}{q}}M_{\psi}f(z).$

Fix such $z$ and $I$. Let $I_{1},I_{2}$ be the right and left neighbors of $I$
such that $|\psi(Q_{I_{j}})|=|\psi(Q_{I})|$. Setting $J={I_{1}}\cup
I\cup{I_{2}}$, Lemma 4.1 implies that

(4.4) $|\psi(Q_{I})|\sim|\psi(Q_{J})|.$

Split $f=f_{1}+f_{2}$ where $f_{1}=f\chi_{\psi(Q_{J})}$. Using the
distribution function and the weak type estimate, for any $T>0$,

$\displaystyle\int_{\psi(Q_{I})}(M_{\psi}f_{1})^{q}\,dA$
$\displaystyle\leq\int_{0}^{T}+\int_{T}^{\infty}q\lambda^{q-1}|\\{z\in\psi(Q_{I}):M_{\psi}f(z)>\lambda\\}|\,d\lambda$
$\displaystyle\leq
T^{q}|\psi(Q_{I})|+\left\|f_{1}\right\|_{L^{1}(\Omega)}\frac{q}{1-q}T^{q-1}.$

Taking $T=\left\|f_{1}\right\|_{L^{1}(\Omega)}|\psi(Q_{I})|^{-1}$, we obtain

$\left\langle
M_{\psi}f\right\rangle_{q,\psi(Q_{I})}\lesssim\left(\frac{\left\|f_{1}\right\|_{L^{1}(\Omega)}}{|\psi(Q_{I})|}\right)^{q}\lesssim\left\langle
f\right\rangle_{\psi(Q_{J})}\leq M_{\psi}f(z),$

where the second inequality comes from (4.4) and the support of $f_{1}$. For
$f_{2}$, let $w\in\psi(Q_{I})\cap\psi(Q_{K})$. If $K\subset J$, then
$\left\langle f\right\rangle_{\psi(Q_{K})}=0$. Otherwise, $K$ exits $J$, but
since it also has nonempty intersection with $I$, either $I_{1}$ or $I_{2}$ is
contained in $K$. WLOG assume it is $I_{1}$. Then, setting $L=K\cup I$ and
$K_{1}=K\backslash I$, Lemma 4.1, applied to $K_{1}$ and $I$, implies

$|\psi(Q_{L})|\lesssim|\psi(Q_{K_{1}})|+|\psi(Q_{I})|=|\psi(Q_{K_{1}})|+|\psi(Q_{I_{1}})|\leq
2|\psi(Q_{K})|.$

Therefore, $\left\langle f_{2}\right\rangle_{\psi(Q_{K})}\lesssim\left\langle
f\right\rangle_{\psi(Q_{L})}$ which implies $M_{\psi}f_{2}(w)\lesssim
M_{\psi}f(z)$. ∎

##  5\. Proof of Theorem D

Let $\sigma\in\mathrm{B}_{1}(\Omega)$. Set $u=\sigma\circ\psi$,
$v=|\psi^{\prime}|^{2}$ and $w=|\psi^{\prime}|$. The weights $u$, $v$, and $w$
are defined on $\mathbb{D}$ so we omit the explicit reference to $\mathbb{D}$
in the $\mathrm{B}_{p}$ characteristics that follow. We also identify weights
with the absolutely continuous measure they induce via the notation
$u(E)=\int_{E}u\,dA$ for measurable sets $E$. The following weak-type estimate
for the weighted dyadic maximal function can be proven using a standard
maximal covering argument that is independent of the underlying measure
$w\,dA$, which we omit.

######  Proposition 5.1.

Suppose $u,v,w$ are weights on $\mathbb{D}$ such that $uw\in\mathrm{B}_{1}(w)$
and $v=w^{2}$. Then, for all $f\in L^{1}(\mathbb{D},uv),$

(5.1)
$uv\left(\left\\{M_{w}^{\mathcal{D}}f>\lambda\right\\}\right)\leq\frac{[uw]_{\mathrm{B}_{1}(w)}}{\lambda}\int_{\mathbb{D}}|f|\,uv\,dA.$

Next, we compute the analogue of (1.8) for the weak-type estimate.

######  Proposition 5.2.

For each $\lambda>0$,

(5.2)
$\displaystyle\sup_{\left\|f\right\|_{L^{1}(\Omega,\sigma)}=1}\sigma\left(\left\\{z\in\Omega:\left|\Pi_{\Omega}f(z)\right|>\lambda\right\\}\right)$
$\displaystyle=$
$\displaystyle\sup_{\left\|g\right\|_{L^{1}(\mathbb{D},uw)}=1}uv\left(\left\\{\zeta\in\mathbb{D}:\left|w(\zeta)^{-1}\Pi_{\mathbb{D}}g(\zeta)\right|>\lambda\right\\}\right)$

###### Proof.

Let $\lambda>0$, $f\in L^{1}(\Omega,\sigma)$ with unit norm, and set

$E=\left\\{z\in\Omega:|\Pi_{\Omega}f(z)|>\lambda\right\\}.$

Let $g:\mathbb{D}\rightarrow\mathbb{C}$ be defined by
$g=(f\circ\psi)\cdot\psi^{\prime}$. Thus, by change of variable,

$1=\int_{\Omega}\left|f\right|\sigma\,dA=\int_{\mathbb{D}}\left|g\right|u\left|\psi^{\prime}\right|\,dA,\quad\sigma\left(E\right)=\int_{\psi^{-1}(E)}u\left|\psi^{\prime}\right|^{2}\,dA.$

Using the transformation law for the Bergman kernel (see for example [lanzani-
stein-2004, p. 72]), one can conclude

$\psi^{-1}(E)=\left\\{\zeta\in\mathbb{D}:\,\frac{|\Pi_{\mathbb{D}}g(\zeta)|}{\left|\psi^{\prime}(\zeta)\right|}>\lambda\right\\}.$

∎

Equipped with Propositions 5.2 and 5.1, we are ready to prove Theorem D.
First, we claim

(5.3)
$\max\left\\{[uw]_{\mathrm{B}_{1}(w)},[uw]_{\mathrm{B}_{1}}\right\\}\leq[v]_{\mathrm{B}_{1}}[\sigma]_{\mathrm{B}_{1}(\Omega)}.$

The estimate in (5.3) for $[uw]_{\mathrm{B}_{1}}$ follows from (1.9) with
$p=1$ and a slight modification will show the estimate for
$[uw]_{\mathrm{B}_{1}(w)}.$ Indeed, there holds for each $Q_{I}$,

$\displaystyle[\sigma]_{\mathrm{B}_{1}(\Omega)}[v]_{\mathrm{B}_{1}}$
$\displaystyle\geq\frac{[v]_{\mathrm{B}_{1}}}{\int_{Q_{I}}v\,dA}\int_{Q_{I}}uv\,dA\cdot\|u^{-1}\|_{L^{\infty}(Q_{I})}$
$\displaystyle\geq\frac{1}{\inf_{Q_{I}}v}\left\langle
uv\right\rangle_{Q_{I}}\|u^{-1}\|_{L^{\infty}(Q_{I})}$
$\displaystyle\geq\frac{1}{(\inf_{Q_{I}}v)^{1/2}}\left\langle
uv\right\rangle_{Q_{I}}\|u^{-1}v^{-1/2}\|_{L^{\infty}(Q_{I})}$
$\displaystyle=\frac{\left\langle
w\right\rangle_{Q_{I}}}{\inf_{Q_{I}}w}\left\langle
uw\right\rangle_{w,Q_{I}}\|u^{-1}w^{-1}\|_{L^{\infty}(Q_{I})}$
$\displaystyle\geq\left\langle
uw\right\rangle_{w,Q_{I}}\|u^{-1}w^{-1}\|_{L^{\infty}(Q_{I})}.$

Taking the supremum over $I$ establishes (5.3). Therefore, by (5.3),
Proposition 5.2, and the trivial estimates
$c_{w}\leq\left[w\right]_{\mathrm{B}_{1}}\leq\left[v\right]_{\mathrm{B}_{1}}$,
it is enough to show

(5.4)
$uv\left(\left\\{\zeta\in\mathbb{D}:\,\frac{|\Pi_{\mathbb{D}}g(\zeta)|}{w(\zeta)}>\lambda\right\\}\right)\lesssim\frac{\left(c_{w}[w]_{\mathrm{B}_{1}}\right)^{3}[uw]_{\mathrm{B}_{1}(w)}^{2}[uw]_{\mathrm{B}_{1}}}{\lambda}\int_{\mathbb{D}}|g|\,uw$

Moreover, $\Pi_{\mathbb{D}}$ is pointwise equivalent to
$\mathcal{A}_{\mathcal{D}_{1}}+\mathcal{A}_{\mathcal{D}_{2}}$, defined by
(2.5) with $v\equiv 1$ (see [rtw2017, Lemma 5]), so it is enough to prove
(5.4) for $\mathcal{A}_{\mathcal{D}}$ in place of $\Pi_{\mathbb{D}}$ for a
generic dyadic grid $\mathcal{D}$ and for $g$ positive.

To this end, let
$E_{\lambda}=\\{\zeta\in\mathbb{D}:M^{\mathcal{D}}_{w}(gw^{-1})(\zeta)>\lambda\\}$
and write $E_{\lambda}$ as a union of disjoint maximal Carleson boxes
$Q_{\lambda,k}$ and let $\hat{Q}_{\lambda,k}$ denote the dyadic parents. Write

$g=g_{1}+g_{2},\quad g_{1}=g\chi_{E_{\lambda}^{c}},\quad
g_{2}=g\chi_{E_{\lambda}}.$

Notice that if $I\in\mathcal{D}$, then either there exists $k$ so that
$Q_{I}\subset Q_{\lambda,k}$ or $Q_{I}$ intersects $E_{\lambda}^{c}.$ In the
latter case, note by definition of $Q_{\lambda,k}$ and the doubling of $w$, we
have

(5.5) $\left\langle g_{1}w^{-1}\right\rangle_{w,T_{I}}\leq c_{w}\left\langle
g_{1}w^{-1}\right\rangle_{w,Q_{I}}\leq c_{w}\lambda,$

In the former case, $\left\langle g_{1}w^{-1}\right\rangle_{w,T_{I}}=0$ since
$Q_{\lambda,k}\subset E_{\lambda}$, so (5.5) holds for all $I\in\mathcal{D}$.
Similarly, for the function $g_{2}$, by maximality of $Q_{\lambda,k}$ and
doubling of $w$,

(5.6) $\left\langle g_{2}w^{-1}\right\rangle_{w,Q_{\lambda,k}}\leq
c_{w}\left\langle
g_{2}w^{-1}\right\rangle_{w,\hat{Q}_{\lambda,k}}<c_{w}\lambda.$

Write

$\displaystyle
uv\left(\left\\{\zeta\in\mathbb{D}:\,\frac{|\mathcal{A}_{\mathcal{D}}g(\zeta)|}{w(\zeta)}>\lambda\right\\}\right)$
$\displaystyle\leq
uv\left(\left\\{\zeta\in\mathbb{D}:\,\frac{|\mathcal{A}_{\mathcal{D}}g_{1}(\zeta)|}{w(\zeta)}>\frac{\lambda}{2}\right\\}\right)$
$\displaystyle\quad+uv\left(\left\\{\zeta\in
E_{\lambda}^{c}:\,\frac{|\mathcal{A}_{\mathcal{D}}g_{2}(\zeta)|}{w(\zeta)}>\frac{\lambda}{2}\right\\}\right)$
$\displaystyle\quad+uv(E_{\lambda})$ $\displaystyle:=(I)+(II)+(III).$

$(III)$ is controlled by
$\frac{[uw]_{\mathrm{B}_{1}(w)}}{\lambda}\int_{\mathbb{D}}|g|\,uw$ using
Proposition 5.1. To control $(I)$ and $(II)$, note that using the
$\mathrm{B}_{1}$ condition for $w$,

$\mathcal{A}_{\mathcal{D}}g_{1}(\zeta)w^{-1}(\zeta)\leq[w]_{\mathrm{B}_{1}}\sum_{I\in\mathcal{D}}\frac{\int_{Q_{I}}g_{1}\,dA}{\int_{Q_{I}}w\,dA}\chi_{Q_{I}}(\zeta)=[w]_{\mathrm{B}_{1}}\mathcal{A}_{\mathcal{D},w}(g_{1}w^{-1})(\zeta)$

Recalling the dyadic regularization from (3.12), set
$\widetilde{g_{1}}=(g_{1})_{\mathcal{D}}$. By Lemma 3.3.ii,
$\int_{Q_{I}}\tilde{g}_{1}\,dA=\int_{Q_{I}}g_{1}\,dA$ for each
$I\in\mathcal{D}$ whence

$\mathcal{A}_{\mathcal{D},w}(g_{1}w^{-1})=\mathcal{A}_{\mathcal{D},w}(\widetilde{g_{1}}w^{-1}).$

Furthermore, for $I\in\mathcal{D}$ and $\zeta\in T_{I}\subset Q_{I}$,
$w^{-1}(\zeta)\leq[w]_{\mathrm{B}_{1}}\left(\int_{T_{I}}w\,dA\right)^{-1}$,
which, combined with (5.5), yields

$\widetilde{g_{1}}w^{-1}\leq[w]_{\mathrm{B}_{1}}\sum_{I\in\mathcal{D}}\langle
g_{1}w^{-1}\rangle_{w,T_{I}}\chi_{T_{I}}\leq
c_{w}\times[w]_{\mathrm{B}_{1}}\times\lambda.$

On the other hand, defining $\widetilde{g_{2}}=\sum_{k}\langle
g_{2}\rangle_{Q_{\lambda,k}}\chi_{Q_{\lambda,k}}$, (5.6) gives
$\widetilde{g_{2}}w^{-1}\leq c_{w}[w]_{\mathrm{B}_{1}}\lambda$ and by the
support condition on $g_{2}$, for $\zeta\in E_{\lambda}^{c}$ there holds

$\displaystyle\mathcal{A}_{\mathcal{D}}g_{2}(\zeta)w^{-1}(\zeta)$
$\displaystyle\leq[w]_{\mathrm{B}_{1}}\sum_{I\in\mathcal{D}}\frac{\int_{Q_{I}}g_{2}}{\int_{Q_{I}}w}\chi_{Q_{I}}(\zeta)$
$\displaystyle\leq[w]_{\mathrm{B}_{1}}\sum_{k}\sum_{\begin{subarray}{c}I\in\mathcal{D}\\\
Q_{I}\supset
Q_{\lambda_{k}}\end{subarray}}\frac{\int_{Q_{I}}g_{2}}{\int_{Q_{I}}w}\chi_{Q_{I}}(\zeta)$
$\displaystyle\leq[w]_{\mathrm{B}_{1}}\mathcal{A}_{\mathcal{D},w}(\widetilde{g_{2}}w^{-1})(\zeta).$

Therefore,

$\displaystyle(I)+(II)$
$\displaystyle\lesssim\sum_{j=1,2}\frac{\left([w]_{\mathrm{B}_{1}}\right)^{2}}{\lambda^{2}}\int_{\mathbb{D}}\left|\mathcal{A}_{\mathcal{D},w}(\widetilde{g_{j}}w^{-1})\right|^{2}\,uv\,dA$
$\displaystyle\lesssim\sum_{j=1,2}\frac{\left(c_{w}[w]_{\mathrm{B}_{1}}[uw]_{\mathrm{B}_{2}(w)}\right)^{2}}{\lambda^{2}}\int_{\mathbb{D}}\left|\widetilde{g_{j}}w^{-1}\right|^{2}\,uv\,dA$
$\displaystyle\lesssim\sum_{j=1,2}\frac{\left(c_{w}[w]_{\mathrm{B}_{1}}\right)^{3}\left([uw]_{\mathrm{B}_{2}(w)}\right)^{2}}{\lambda}\int_{\mathbb{D}}\left|\widetilde{g_{j}}w^{-1}\right|\,uv\,dA,$

where the second inequality follows from (2.6) with $p=2$, $w\mapsto uw$, and
$v\mapsto w$. The proof of (5.4) is concluded by again invoking the fact
$uw\in\mathrm{B}_{1}$ to obtain

$\int_{F}\left\langle
h\right\rangle_{F}uw\,dA\leq\left[uw\right]_{\mathrm{B}_{1}}\int_{F}|h|uw\,dA,\quad
F\in\\{Q_{I},T_{I}\\}_{I\in\mathcal{D}},$

for any $h\in L^{1}(uw)$, whence

$\int_{\mathbb{D}}\left|\widetilde{g_{j}}w^{-1}\right|\,uv\,dA\lesssim\left[uw\right]_{\mathrm{B}_{1}}\int_{\mathbb{D}}\left|g\right|\,uw\,dA.$

## References