# Quantitative Sobolev regularity of quasiregular maps

Francesco Di Plinio  Dipartimento di Matematica e Applicazioni, Università di
Napoli

Via Cintia, Monte S. Angelo 80126 Napoli, Italy<EMAIL_ADDRESS>,
A. Walton Green  and  Brett D. Wick  Department of Mathematics, Washington
University in Saint Louis

1 Brookings Drive, Saint Louis, Mo 63130, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We quantify the Sobolev space norm of the Beltrami resolvent
$(I-\mu\mathcal{S})^{-1}$, where $\mathcal{S}$ is the Beurling-Ahlfors
transform, in terms of the corresponding Sobolev space norm of the dilatation
$\mu$ in the critical and supercrticial ranges. Our estimate entails as a
consequence quantitative self-improvement inequalities of Caccioppoli type for
quasiregular distributions with dilatations in $W^{1,p}$, $p\geq 2$. Our proof
strategy is then adapted to yield quantitative estimates for the resolvent
$(I-\mu\mathcal{S}_{\Omega})^{-1}$ of the Beltrami equation on a sufficiently
regular domain $\Omega$, with $\mu\in W^{1,p}(\Omega)$. Here,
$\mathcal{S}_{\Omega}$ is the compression of $\mathcal{S}$ to a domain
$\Omega$. Our proofs do not rely on the compactness or commutator arguments
previously employed in related literature. Instead, they leverage the weighted
Sobolev estimates for compressions of Calderón-Zygmund operators to domains,
recently obtained by the authors, to extend the Astala-Iwaniec-Saksman
technique to higher regularities.

###### Key words and phrases:

Beltrami equation, quasiregular, quasiconformal, Sobolev regularity,
compression of singular integrals, $T\mathbf{1}$-theorems, weighted bounds,
Beurling-Ahlfors transform

###### 2010 Mathematics Subject Classification:

Primary: 30C62. Secondary: 42B20,42B37

F. Di Plinio was partially supported by the National Science Foundation under
the grant NSF-DMS-200510. This material is based upon work supported by the
National Science Foundation under Grant No. DMS-1929284 while this author was
in residence at the Institute for Computational and Experimental Research in
Mathematics in Providence, RI, during the Harmonic Analysis and Convexity
program of Fall 2022. F. Di Plinio is also partially supported by the FRA 2022
Program of University of Napoli Federico II, project ReSinAPAS - Regularity
and Singularity in Analysis, PDEs, and Applied Sciences.

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

B. D. Wick’s research partially supported in part by NSF grant NSF-
DMS-1800057, NSF-DMS-200510, NSF-DMS-2054863 as well as ARC DP 220100285.

##  1\. Introduction

For $K\geq 1$, a _$K$ -quasiregular_ map $f$ in
$W^{1,2}_{\mathrm{loc}}(\mathbb{C})$ is a solution of the Beltrami equation

(B) $\overline{\partial}f(z)=\mu(z)\partial f(z),\quad z\in\mathbb{C},$

where $\mu$, the dilatation, or Beltrami coefficient of $f$, satisfies

(1.1) $\left\|{\mu}\right\|_{\infty}=k=\frac{K-1}{K+1}<1.$

The assumption (1.1) forces (B) to be an elliptic PDE. Hence, there exists a
unique principal solution $f$ in $W^{1,2}_{\mathrm{loc}}(\mathbb{C})$ which is
normalized at $\infty$ by

$f(z)=z+O(z^{-1})$

and is in fact a homeomorphism. An important feature of $K$-quasiregular maps
is their self-improving regularity. First, they automatically belong to the
smaller Sobolev space $W^{1,p}_{\mathrm{loc}}(\mathbb{C})$ for $p$ up to, but
not including, the upper endpoint $p_{K}>2$ of the critical interval $I_{K}$,
cf. (1.2) below. On the other hand, if $f$ belongs to
$W^{1,q}_{\mathrm{loc}}(\mathbb{C})$ for some $q<2$, equation (B) may be
interpreted in the sense of distributions, in which case $f$ is termed _weakly
$K$-quasiregular_. And in fact, weakly $K$-quasiregular maps in
$W^{1,q}_{\mathrm{loc}}(\mathbb{C})$ are $K$-quasiregular if $q$ is larger
than or equal to the lower endpoint $q_{K}<2$, again cf. (1.2). Excluding the
endpoint case discussed below, both facts follow from Caccioppoli
inequalities, in turn a consequence of the invertibility of the Beltrami
resolvent on $L^{p}(\mathbb{C})$ for $p$ in the critical interval

(1.2) $I_{K}=(q_{K},p_{K}),\qquad q_{K}\coloneqq\frac{2K}{K+1},\qquad
p_{K}\coloneqq\frac{2K}{K-1}.$

which was established by Astala, Iwaniec, and Saksman in [astala01] relying
upon the following key results.

  1. (1)

Astala’s area distortion theorem, [astala94], that is, the principal solution
belongs to $W^{1,p}_{\mathrm{loc}}$ for $p<p_{K}$. As a consequence, suitable
powers of its Jacobian determinant are Muckenhoupt $\mathrm{A}_{p}$ weights.

  2. (2)

The Coifman-Fefferman theorem from harmonic analysis, [coifman74], namely that
Calderón-Zygmund singular integral operators are bounded on the weighted
$L^{p}(\omega)$ spaces, $1<p<\infty$, if and only if $\omega$ belongs to the
Muckenhoupt class $\mathrm{A}_{p}$.

The second point arises in connection with the Beurling-Ahlfors transform
$\mathcal{S}$, namely the Calderón-Zygmund operator defined for
$f\in\mathcal{C}^{\infty}_{0}(\mathbb{C})$ by

(1.3) $\mathcal{S}f(z)=-\frac{1}{\pi}\lim_{\varepsilon\to
0}\int_{\varepsilon<|z-w|<\frac{1}{\varepsilon}}\frac{f(w)}{(z-w)^{2}}\,\mathrm{d}w,\qquad
z\in\mathbb{C}.$

As a consequence of $L^{p}(\mathbb{C})$, $1<p<\infty$, as well as weak-$L^{1}$
estimates for maximal truncations of Calderón-Zygmund kernels, the above limit
exists for almost every $z\in\mathbb{C}$, when $f\in L^{p}(\mathbb{C})$ and
$1\leq p<\infty$. Furthermore $\mathcal{S}$ extends to an isometry on
$L^{2}(\mathbb{C})$, and intertwines the $\partial$ and $\overline{\partial}$
derivatives; see (2.1) below. Given $\mu$ satisfying (1.1), we refer to the
operator $(I-\mu\mathcal{S})^{-1}$ as the Beltrami resolvent. The general
argument of [astala01] is to recast the norm estimate of
$(I-\mu\mathcal{S})^{-1}$ as an a priori estimate for an inhomogeneous
Beltrami equation, which, by changing variables, amounts to a weighted
estimate for the $\overline{\partial}$ equation, with weight given by an
appropriate power of the Jacobian of the principal solution. Therefore,
utilizing $\mathcal{S}$, the a priori estimate is a consequence of (1) and
(2).

The operator $I-\mu\mathcal{S}$ fails to be invertible on $L^{p}(\mathbb{C})$
at the endpoints of the interval $(q_{K},p_{K})$. However, in essence
quantifying point (2) above, Petermichl and Volberg [petermichl02] obtained
the sharp estimate
$\left\|{\mathcal{S}}\right\|_{\mathcal{L}(L^{p}(\mathbb{C},\omega))}\lesssim[\omega]_{\mathrm{A}_{p}(\mathbb{C})}$
for $2\leq p<\infty$, which allows to approximate the endpoint case and obtain
that $I-\mu\mathcal{S}$ remains injective at the upper endpoint. By duality,
this implies that weakly $K$-quasiregular maps belonging to
$W^{1,q_{K}}_{\mathrm{loc}}(\mathbb{C})$ are in fact $K$-quasiregular.
Interestingly, the Petermichl-Volberg theorem was one of the motivating
factors behind the question of sharp dependence of the $L^{p}(\omega)$-norm of
a generic Calderón-Zygmund operator on the $\mathrm{A}_{p}$ weight
characteristic, culminating in Hytönen’s celebrated result [hytonen12].

###  1.1. Sobolev regularity of planar Beltrami equations

This work focuses on the case when $\mu$ has greater regularity, say in
addition to $\|\mu\|_{\infty}<1$,
$\left|{D\mu}\right|=|\partial\mu|+|\overline{\partial}\mu|$ belongs to
$L^{q}(\mathbb{C})$ for some $1<q<\infty$. Clop et. al. [clop-et-al] have
proved an analogue for Sobolev spaces which first generalizes the notion of a
weakly quasiregular map to a distributional quasiregular map, which means $f$
is only required to be in $L^{p}_{\mathrm{loc}}$ for some $1<p<\infty$ and
satisfies the distributional version of the Beltrami equation (B); see (1.6)
below. Their strategy is to prove the desired self-improvement estimates first
for quasiconformal $f$ which are then extended to distributional quasiregular
maps by factorization and the classical Weyl lemma, that holomorphic
distributions are holomorphic functions a.e.. We refer to [baison, clop-tolsa,
prats18, baison-frac, cruz2013beltrami] for related works on higher order
regularity of quasiregular and quasiconformal maps in the complex plane.

Our first result is a quantitative version of [clop-et-al] in the critical and
supercritical range, which we obtain as a consequence of weighted $W^{1,p}$
bounds for the Beurling-Ahlfors operator, in consonance with the strategy of
[astala01] for the zero-th order problem.

######  Theorem A.

Assume the dilatation $\mu\in L^{\infty}(\mathbb{C})\cap W^{1,2}(\mathbb{C})$
satisfies (1.1) for some $K\geq 1$. Then, for each $1<r<2$,

(1.4)
$\left\|(I-\mu\mathcal{S})^{-1}\right\|_{\mathcal{L}(W^{1,r}(\mathbb{C}))}\lesssim
1$

with implicit constant depending exponentially on $K$,
$\|\mu\|_{W^{1,2}(\mathbb{C})}$ and $\frac{1}{\min\left\\{2-r,r-1\right\\}}$.

If in addition $\mu\in W^{1,p}(\mathbb{C})$ for some $2<p<\infty$,

(1.5)
$\left\|(I-\mu\mathcal{S})^{-1}\right\|_{\mathcal{L}(W^{1,p}(\mathbb{C}))}\lesssim
1+\|\mu\|_{W^{1,p}(\mathbb{C})}^{2}$

with implicit constant depending exponentially on $K$,
$\|\mu\|_{W^{1,2}(\mathbb{C})}$ and $\max\left\\{\frac{1}{p-2},p\right\\}$.

In the critical case (1.4), one cannot hope for $I-\mu\mathcal{S}$ to be
invertible on $W^{1,2}(\mathbb{C})$ because in the corollary below, (1.10)
fails for $p=2$. Indeed, from [clop-et-al]*pp. 205-206, one can consider the
quasiregular distribution $\phi(z)=z(1-\log\left|{z}\right|)$ which does not
belong to $W^{2,2}_{\mathrm{loc}}(\mathbb{C})$, though its Beltrami
coefficient, $\mu(z)=\frac{z}{\bar{z}}\frac{1}{2\log\left|{z}\right|-1}$ does
in fact belong to $W^{2,2}_{\mathrm{loc}}(\mathbb{C})$.

Quantitative self-improvement of quasiregular maps is often expressed through
Caccioppoli inequalities (see the survey [sbordone] or [astala-book]*§5.4.1).
In our case, Theorem A implies the following Caccioppoli inequalities for
quasiregular distributions (see (1.7) - (1.10) below). Given an open set
$\Omega$, say $f\in L^{q}_{\mathrm{loc}}(\Omega)$ is a $K$-quasiregular
distribution if it satisfies the distributional Beltrami equation on $\Omega$,
namely for some $\mu$ satisfying (1.1),

(1.6) $\left\langle
f,\overline{\partial}\psi-\partial(\mu\psi)\right\rangle=0\quad\forall\psi\in\mathcal{C}^{\infty}_{0}(\Omega).$

######  Corollary A.1.

Let $\mu\in L^{\infty}(\Omega)\cap W^{1,2}_{\mathrm{loc}}(\Omega)$ satisfy
(1.1) for some $K\geq 1$. Then, for $2<q<\infty$, $f\in
L^{q}_{\mathrm{loc}}(\Omega)$ satisfying (1.6), and any
$\eta\in\mathcal{C}^{\infty}_{0}(\Omega)$,

(1.7) $\displaystyle\|\eta(Df)\|_{L^{q}}$
$\displaystyle\lesssim\|(D\eta)f\|_{L^{q}};$ (1.8) $\displaystyle\mbox{for all
}1<r<2,\quad\|\eta(D^{2}f)\|_{L^{r}}$
$\displaystyle\lesssim\|(D\eta)f\|_{L^{r}}+\|(D\eta)(Df)\|_{L^{r}}+\|(D^{2}\eta)f\|_{L^{r}}.$

In particular, $f\in W^{2,r}_{\mathrm{loc}}(\Omega)$ for every $r<2$. If
furthermore, $\mu\in W^{1,p}_{\mathrm{loc}}(\Omega)$ for some $p>2$, then, for
$\frac{p}{p-1}\leq q<\infty$, $f\in L^{q}_{\mathrm{loc}}(\Omega)$ satisfying
(1.6), and any $\eta\in\mathcal{C}^{\infty}_{0}(\Omega)$,

(1.9) $\displaystyle\|\eta(Df)\|_{L^{q}}$
$\displaystyle\lesssim\|(D\eta)f\|_{L^{q}};$ (1.10) $\displaystyle\mbox{for
all }1<r\leq p,\quad\|\eta(D^{2}f)\|_{L^{r}}$
$\displaystyle\lesssim\|(D\eta)f\|_{L^{r}}+\|(D\eta)(Df)\|_{L^{r}}+\|(D^{2}\eta)f\|_{L^{r}}.$

In particular, $f\in W^{2,p}_{\mathrm{loc}}(\Omega)$.

Experts in the area will readily observe that without the precise dependence
on $\|\mu\|_{W^{1,2}(\mathbb{C})}$ we provide, the inequalities (1.7) and
(1.9) follow from the fact that $W^{1,2}(\mathbb{C})$ embeds into the space of
functions with vanishing mean oscillation, or
$\operatorname{{VMO}}(\mathbb{C})$, together with the invertibility of
$I-\mu\mathcal{S}$ on $L^{p}(\mathbb{C})$ for every $1<p<\infty$ whenever
$\mu\in\operatorname{{VMO}}(\mathbb{C})$, see [astala01]*Theorem 5. However,
by sharpening the assumption to $\mu\in W^{1,2}(\mathbb{C})$, we establish, in
Lemma 2.3 below, a quantitative version of this result which is a crucial step
in Theorem A and Corollary A.1. The conclusion that quasiregular distributions
in $L^{q}_{\mathrm{loc}}(\Omega)$ self-improve to membership in
$W^{2,r}_{\mathrm{loc}}(\Omega)$ for $q$ and $r$ in the above specified ranges
is one of the main results of Clop et. al. in [clop-et-al]. However, the
Caccioppoli inequalities (1.8) and (1.10), with the precise implicit
dependence on the local regularity of $\mu$ inherited from (1.4) and (1.5) in
Theorem A are new to the best of our knowledge. Theorem A and the Caccioppoli
inequalities are proved in §3, and rely on a Moser-Trudinger estimate for the
Jacobian of the principal solution to (B) with $\mu\in W^{1,2}(\mathbb{C})$.

###  1.2. Global Sobolev regularity of Beltrami equations on domains $\Omega$

Theorem A was actually uncovered in our attempts to address a more delicate
problem, the invertibility of $I-\mu\mathcal{S}_{\Omega}$, where
$\mathcal{S}_{\Omega}$ is the compression of the Beurling-Ahlfors transform to
a bounded Lipschitz domain $\Omega\subset\mathbb{C}$ defined by

$\left\langle{\mathcal{S}_{\Omega}f},{g}\right\rangle=\left\langle{\mathcal{S}\left(f\mathbf{1}_{\overline{\Omega}}\right)},{\mathbf{1}_{\overline{\Omega}}g}\right\rangle,\qquad
f,g\in\mathcal{C}^{\infty}_{0}(\mathbb{C}).$

In [diplinio23domains], we developed new $T(1)$-type theorems and weighted
Sobolev space estimates for Calderón-Zygmund operators on domains, a broad
class which includes compressions of global CZ operators. In particular,
together with past work of e.g. Tolsa [tolsa2013regularity], the estimates of
[diplinio23domains] uncover the precise connection between boundary regularity
of $\Omega$ and weighted Sobolev estimates for $\mathcal{S}_{\Omega}$. In this
article, this connection is exploited to extend the resolvent strategy of
[astala01] to the Sobolev case and obtain the first quantitative Sobolev
estimate for $(I-\mu\mathcal{S}_{\Omega})^{-1}$.

The compressed Beltrami resolvent $(I-\mu\mathcal{S}_{\Omega})^{-1}$ is
connected to the Beltrami equation (B) for dilatations $\mu$ whose support is
contained in $\overline{\Omega}$ and belonging to $W^{1,p}(\Omega)$ for some
$p>2$. The Caccioppoli inequalities of Corollary A.1 imply that any solution
$f$ to (B) with $\mu$ of this form belongs to
$W^{2,p}_{\mathrm{loc}}(\Omega)$. Thus, the interest is in global regularity,
i.e. whether $f$ belongs to $W^{2,p}(\Omega)$. This problem is even of
interest when $f$ is the principal solution, in which case one has the
representation from [astala-book]*p. 165,

$\overline{\partial}f=(I-\mu\mathcal{S}_{\Omega})^{-1}\mu.$

Furthermore, since $\partial f=\mathcal{S}_{\Omega}(\overline{\partial}f)$ by
(2.1) below,

(1.11)
$\left\|{f}\right\|_{W^{2,p}(\Omega)}\lesssim\left(1+\left\|\mathcal{S}_{\Omega}\right\|_{\mathcal{L}(W^{1,p}(\Omega))}\right)\left\|(I_{\Omega}-\mu\mathcal{S}_{\Omega})^{-1}\right\|_{\mathcal{L}(W^{1,p}(\Omega))}\left\|\mu\right\|_{W^{1,p}(\Omega)}.$

The first factor, the norm of $\mathcal{S}_{\Omega}$ on the Sobolev space
$W^{1,p}(\Omega)$ is now well-understood in the supercritical range in terms
of the boundary regularity of $\Omega$, see [cruz-tolsa, tolsa2013regularity,
prats17]. In fact, by these results, it is quantitatively equivalent to the
Besov space $B^{1-\frac{1}{p}}_{p,p}(\partial\Omega)$ norm of the boundary
normal of $\Omega$; see Definition 4.1 below. Accordingly, we say $\Omega$ is
a $\mathcal{B}_{p}$ domain if this boundary regularity condition is satisfied.

The second factor in (1.11) is our chief object of interest. While
quantitative estimates of this norm appear to be unavailable in past
literature, several results of qualitative nature have been obtained through
methods in antithesis with those developed herein. Initially, invertibility of
$I-\mu\mathcal{S}_{\Omega}$ was studied in the Hölder scale in [mateu-extra]
and subsequently extended to the Sobolev and Triebel-Lizorkin scales,
[cruz2013beltrami, prats17, prats19, astala-prats-saksman]. These works all
share the Neumann series blueprint initially introduced by Iwaniec in
[iwaniec]. The main ingredients are unweighted bounds for
$\mathcal{S}_{\Omega}$ on smoothness spaces, established by means of
_unweighted_ $T(1)$-type theorems,

The apex of this line of attack was a pair of papers by M. Prats in [prats17,
prats19], sharpening the result of [cruz2013beltrami], and establishing among
other results the remarkable qualitative fact that $I-\mu\mathcal{S}_{\Omega}$
is invertible on $W^{1,p}(\Omega)$ assuming only that $\Omega$ is a
$\mathcal{B}_{p}$ domain.

######  Theorem B.

Let $p\geq r>2$, and $\Omega\subset\mathbb{C}$ be a bounded simply connected
$\mathcal{B}_{p}$ domain. Let $\mu$ be supported in $\overline{\Omega}$,
satisfying (1.1) for some $K\geq 1$, and in addition $\mu\in W^{1,p}(\Omega)$.
Let $f$ be the principal solution to (B). Then, for $O=f(\Omega)$ and
$\omega=\left|{Jf^{-1}}\right|^{1-p}$,

$\displaystyle\|(I-\mu\mathcal{S}_{\Omega})^{-1}\|_{\mathcal{L}(W^{1,p}(\Omega))}\lesssim\mathscr{O}\left[\left\|\mathcal{S}_{O}\right\|_{\mathcal{L}(W^{1,p}(O,\omega))}+\mathscr{O}^{3}\left(1+\|\mu\|_{W^{1,p}(\Omega)}^{6}\right)\right],$
$\displaystyle\mathscr{O}=1+\left\|{O}\right\|_{\mathcal{B}_{p}}+\left\|{\Omega}\right\|_{\mathcal{B}_{p}}.$

The implicit constant depends exponentially on
$\left\|{\mu}\right\|_{W^{1,r}(\Omega)}$ and double exponentially on
$\max\left\\{\frac{1}{r-2},p\right\\}$, $K$,
$\left\|{\mu}\right\|_{W^{1,2}(\Omega)}$, and the Dini character of $O$ and
$\Omega$.

Using the novel weighted $T(1)$ theorems on domains established by the authors
in [diplinio23domains], and the relationship between these testing conditions
and boundary smoothness developed in [cruz-tolsa, tolsa2013regularity],
$\left\|{\mathcal{S}_{O}}\right\|_{W^{1,p}(O,\omega)}$ can be quantitatively
controlled by $\left\|{O}\right\|_{\mathcal{B}_{p+\varepsilon}}$ (see Lemma
4.3.iii below) for any $\varepsilon>0$, yielding the following corollary.

######  Corollary B.1.

Let $p$, $r$, $\Omega$, $\mu$, $f$, $O$, and $\mathscr{O}$ be as in Theorem B.
Then, for any $\varepsilon>0$,

$\|(I-\mu\mathcal{S}_{\Omega})^{-1}\|_{\mathcal{L}(W^{1,p}(\Omega))}\lesssim\mathscr{O}\left[\left\|{O}\right\|_{\mathcal{B}_{p+\varepsilon}}+\mathscr{O}^{3}\left(1+\|\mu\|_{W^{1,p}(\Omega)}^{6}\right)\right]$

The implicit constant depends on the same parameters of Theorem B, as well as
$\varepsilon^{-1}$.

Let us provide a more specific description of the relation between Theorem B,
as well as Corollary B.1, and the results of [prats19]. In particular,
[prats19]*Theorem 1.1 tells us that if $\Omega\in\mathcal{B}_{p}$ and $\mu\in
W^{1,p}(\Omega)$, then the principal solution $f$ lies in $W^{2,p}(\Omega)$.
Standard trace results [gagliardo, triebel83] then entail that $O=f(\Omega)$
is a $\mathcal{B}_{p}$ domain as well, which in turn is qualitatively
equivalent, see Lemma 4.2 below, to $\mathcal{S}_{O}:W^{1,p}(O,\omega)\to
W^{1,p}(O,\omega)$, where $\omega$ is as in Theorem B. Thus, Theorem B holds
under the same assumptions as [prats19]*Theorem 1.1, $n=1$, and may be viewed
as a strict quantification of that result. Furthermore, Corollary B.1 replaces
the analytic condition on $\mathcal{S}_{O}$ with a fully geometric testing
condition on $O$, namely its membership to $\mathcal{B}_{q}$ for some $q>p$,
thus providing an explicit dependence on the data $\mu,\Omega$ and $O$.

We close this introduction with a circle of questions motivated by Corollary
B.1. First of all, a crude version of Corollary B.1 with $\varepsilon=0$ can
be obtained without weighted estimates at the price of exponential dependence
on the data $\left\|{\Omega}\right\|_{\mathcal{B}_{p}}$ and
$\left\|{O}\right\|_{\mathcal{B}_{p}}$; cf. Remark 4.4 below. It is thus
natural to ask whether a version of Corollary B.1 holds with $\varepsilon=0$
and uniform polynomial estimates in the sharp Besov norms on $\Omega$ and $O$.
This would hold if the Jacobian power $\omega$ were an $\mathrm{A}_{1}(O)$
weight polynomially in $\mathscr{O}$ and $\|\mu\|_{W^{1,p}(\Omega)}$, and
$\|\mathcal{S}_{O}\mathbf{1}\|_{W^{1,p}(v,O)}$ were controlled by a constant
depending only on $[v]_{\mathrm{A}_{1}(O)}$ and on the $\mathcal{B}_{p}$
character of $O$. The latter statement for $v=\mathbf{1}$ is the content of
[cruz-tolsa]*Theorem 1.1, whence it is legitimate to ask whether Lebesgue
measure can be replaced with a generic $\mathrm{A}_{1}$ weight therein, and
whether a full analogue of Corollary B.1 holds for $\varepsilon=0$.

Furthermore, let us propose a strategy for removing the exponential dependence
on the auxiliary $W^{1,r}$-norm of $\mu$ in both Theorem B and Corollary B.1.
The space $W^{1,r}(\Omega)$ can actually be replaced by any Sobolev-type space
$X(\Omega)$ enjoying both properties

  * a.

$X(\Omega)$ continuously embeds into both $W^{1,2}(\Omega)$ and
$L^{\infty}(\Omega)$;

  * b.

For some $Y\in\\{X(\mathbb{C}),W^{1,2}(\mathbb{C})\\}$, there exists $M>0$
such that

$\left\|{(I-\mu\mathcal{S})^{-1}}\right\|_{Y}\lesssim
1+\left\|{\mu}\right\|_{X(\mathbb{C})}^{M},$

By Theorem A, $X=W^{1,r}$ for $r>2$ satisfies these conditions. A candidate
for a space $X$ larger than $W^{1,r}$ is the Lorentz-Sobolev space consisting
of $L^{2}$ functions with derivatives in the Lorentz space $L^{2,1}$. While it
is known that $I-\mu\mathcal{S}$ is invertible on this space
[cruz2013beltrami], no norm estimates are known. This leads to ask whether
there is a version of Theorem A for $D\mu\in L^{2,1}(\mathbb{C})$.

### Structure of the article

In Section 2, after a few preliminaries, we deduce quantitative estimates for
the Beltrami resolvent associated to dilatations $\mu\in W^{1,2}(\mathbb{C})$
from a precise $\mathrm{A}_{p}$-class embedding for powers of Jacobians of the
corresponding principal solutions, see Lemma 2.3 and 2.2 respectively. Section
3 contains the proofs of Theorem A and Corollary A.1. In Section 4, we provide
proofs of Theorem B and Corollary B.1. As an intermediate step, in Proposition
4.8, we establish a quantitative version of a recent result of Astala, Prats
and Saksman of [astala-prats-saksman]*Theorem 1.1 on the regularity of
quasiconformal solutions to Beltrami equations on $\mathcal{B}_{p}$ domains.
Furthermore, we explain why methods that do not treat Jacobians of principal
solutions as Muckenhoupt weights lead to exponential type estimates in the
data $\mu$, $\Omega$, and $O$, so that Theorem B is not within their reach,
cf. Remark 4.4 below. Finally, Section 5 deals with the technical proof of
Lemma 2.2.

##  2\. The Beltrami resolvent when $\mu\in W^{1,2}(\mathbb{C})$

To prove Theorem A, we will need a few preliminaries. The facts that we will
need from the classical theory of quasiconformal maps will be recalled
throughout from the monograph [astala-book]. Recall the definition of the
Beurling-Ahlfors transform from (1.3). It is of particular use because it
intertwines the derivatives $\partial:=\frac{\partial}{\partial z}$ and
$\overline{\partial}:=\frac{\partial}{\partial{\bar{z}}}$, which means

(2.1) $\mathcal{S}(\overline{\partial}f)=\partial f,\qquad f\in
W^{1,2}(\mathbb{C}).$

This property can be established by appealing to the Fourier transform, or
through the Cauchy transform $\mathcal{K}$ defined by

(2.2) $\mathcal{K}f(z)=\frac{1}{\pi}\lim_{\varepsilon\to
0}\int_{|z-w|>\varepsilon}\frac{f(w)}{z-w}\,\mathrm{d}w.$

The Cauchy transform is the inverse of the $\overline{\partial}$ operator and
$\partial\mathcal{K}=\mathcal{S}$, so that $\partial
f=\partial\mathcal{K}(\overline{\partial}f)=\mathcal{S}(\overline{\partial}f)$.

Use $Df$ to denote the gradient $(\partial f,\overline{\partial}f)$ and for
each integer $n\geq 2$, let $D^{n}f$ denote the vector function consisting of
all combinations of $n$-th order partial derivatives in $z$ and $\overline{z}$
of $f$. $D^{1}f=Df$ and $D^{0}f=f$. We will use $|D^{n}f|$ to denote the
$\ell^{1}$ norm of this vector.

Given an open set $E\subset\mathbb{C}$, an a.e. positive element of
$L^{1}_{\mathrm{loc}}(E)$ is called a weight on $E$. For $\omega$ a weight on
$E$, $n$ a nonnegative integer, and $0<p<\infty$, define the homogeneous and
inhomogeneous weighted Sobolev norms by

$\left\|{f}\right\|_{\dot{W}^{n,p}(E,\omega)}=\sum_{|\alpha|=n}\left\|{\left(\partial^{\alpha_{1}}\overline{\partial}^{\alpha_{2}}f\right)\omega^{\frac{1}{p}}}\right\|_{L^{p}(E)},\qquad\left\|{f}\right\|_{{W}^{n,p}(E,\omega)}=\sum_{j=0}^{n}\left\|{f}\right\|_{\dot{W}^{j,p}(E,\omega)},$

where $\alpha=(\alpha_{1},\alpha_{2})\in\mathbb{N}^{2}$ and
$|\alpha|=\alpha_{1}+\alpha_{2}$. We also use the local average notation for a
cube $Q\subset\mathbb{C}$,

$\left\langle{f}\right\rangle_{p,Q}=\left(|Q|^{-1}\int_{Q}\left|{f(z)}\right|^{p}\,\mathrm{d}z\right)^{\frac{1}{p}},$

with the simplification
$\left\langle{f}\right\rangle_{Q}=\left\langle{f}\right\rangle_{1,Q}$ when
$p=1$. We say a weight $\omega$ on $\mathbb{C}$ belongs to the Muckenhoupt
class $\mathrm{A}_{p}(\mathbb{C})$ if the associated characteristic,

$[\omega]_{\mathrm{A}_{p}(\mathbb{C})}=\sup_{Q\,\mathrm{cube}\,\mathrm{in}\,\mathbb{C}}\left\langle{\omega}\right\rangle_{Q}\left\langle{\omega^{-1}}\right\rangle_{\frac{1}{p-1},Q}$

is finite. To apply the strategy of [astala01], we will need now weighted
Sobolev estimates for $\mathcal{S}$, which were recently obtained for smooth
Calderón-Zygmund operators in sharp quantitative form in [diplinio22wrt]. The
estimates we require are summarized in the following proposition.

######  Proposition 2.1.

Let $n\in\mathbb{N}$, $1<p<\infty$. There exists $C_{p,n}>0$ such that for any
$\omega\in\mathrm{A}_{p}(\mathbb{C})$,

(2.3) $\left\|{\mathcal{S}f}\right\|_{\dot{W}^{n,p}(\mathbb{C},\omega)}\leq
C_{p,n}[\omega]^{\max\\{1,\frac{1}{p-1}\\}}_{\mathrm{A}_{p}(\mathbb{C})}\left\|{f}\right\|_{\dot{W}^{n,p}(\mathbb{C},\omega)}.$

###### Proof.

Since $\mathcal{S}$ is of convolution type, (2.3) more or less follows from
the case $n=0$, which is well-known, [petermichl02], though some care must be
taken with the principal value integral. So, one can consult
[diplinio22wrt]*Corollary A.1 for a complete proof of (2.3). ∎

Introduce the notation

(2.4) $\left|{Jf}\right|=\left|{\partial
f}\right|^{2}-\left|{\overline{\partial}f}\right|^{2},$

which is equal to the determinant of the Jacobian of $f$ as a mapping from
$\mathbb{R}^{2}$ to itself. A characterization of $K$-quasiconformal mappings
equivalent to (B) is the distortion inequality

(2.5) $\left|{Df}\right|^{2}\leq K\left|{Jf}\right|.$

The main lemma concerning $\left|{Jf}\right|$ for $\mu\in W^{1,2}(\mathbb{C})$
is a consequence of the critical Moser-Trudinger Sobolev embedding, and is
proved in §5.1 and 5.3.

######  Lemma 2.2.

Suppose $\mu$ satisfies (1.1) for some $K\geq 1$, and in addition that $\mu\in
W^{1,2}(\mathbb{C})$ with $\|\mu\|_{W^{1,2}C}\leq L$. Let $f$ be the principal
solution to (B), $a\in\mathbb{R}$, and $1<p<\infty$. Then, the Jacobians
$\left|{Jf}\right|^{a}$ and $\left|{Jf^{-1}}\right|^{a}$ are both
$\mathrm{A}_{p}(\mathbb{C})$ weights. In particular, there exists a constant
$C=C(K)>0$ such that for any $1<p<\infty$,

(2.6)
$\displaystyle\left[\left|{Jf^{-1}}\right|^{1-\frac{p}{2}}\right]_{\mathrm{A}_{p}(\mathbb{C})}^{\max\left\\{1,\frac{1}{p-1}\right\\}}$
$\displaystyle\leq
C\exp\left(C\max\left\\{p,\tfrac{1}{p-1}\right\\}^{2}L^{2}\right);$ (2.7)
$\displaystyle\left[\left|{Jf^{-1}}\right|^{1-p}\right]_{\mathrm{A}_{p}(\mathbb{C})}^{\max\left\\{1,\frac{1}{p-1}\right\\}}$
$\displaystyle\leq
C\exp\left(C\max\left\\{p^{2},\tfrac{1}{p-1}\right\\}L^{2}\right).$

The second lemma we will use follows from Lemma 2.2 and the strategy of
[astala01]. The estimate (2.8) below is known qualitatively since
$W^{1,2}(\mathbb{C})$ embeds into $\mathrm{VMO}(\mathbb{C})$, the functions
with vanishing mean oscillation. And it is well-known that $I-\mu\mathcal{S}$
is invertible on all $L^{p}(\mathbb{C})$ for $1<p<\infty$ and
$\mu\in\mathrm{VMO}(\mathbb{C})$, a result that be found in [astala01]*Theorem
5.

######  Lemma 2.3.

Suppose $\mu$ satisfies (1.1) for some $K\geq 1$, and in addition that $\mu\in
W^{1,2}(\mathbb{C})$ with $\|\mu\|_{W^{1,2}C}\leq L$. Then, there exists
$C=C(K)>0$ such that for all $1<p<\infty$,

(2.8)
$\left\|{(I-\mu\mathcal{S})^{-1}}\right\|_{\mathcal{L}(L^{p}(\mathbb{C}))}\leq
C\exp\left(C\max\left\\{p,\tfrac{1}{p-1}\right\\}^{2}L^{2}\right).$

###### Proof of Lemma 2.3.

The Astala-Iwaniec-Saksman strategy from [astala01] shows that

(2.9)
$\|(I-\mu\mathcal{S})^{-1}\|_{\mathcal{L}(L^{p}(\mathbb{C}))}\lesssim_{K}\|\mathcal{S}\|_{\mathcal{L}(L^{p}(\mathbb{C},\omega))},$

where $\omega=\left|{J{f^{-1}}}\right|^{1-\frac{p}{2}}$ and $f$ is the
principal solution to (B). See [astala-book]*§14.2 (14.25) for this exact
statement, or refer to the proof of Proposition 4.6 below. Estimating the
right hand side of (2.9) by (2.3) in Proposition 2.1 with $n=0$, and (2.6) in
Lemma 2.2 concludes the proof. ∎

##  3\. Proof of Theorem A and Corollary A.1

We will prove the critical (1.4) and supercritical (1.5) estimates in Theorem
A at the same time. To this end, let $2\leq p<\infty$ and introduce

(3.1)
$r\in\left\\{\begin{array}[]{ll}\left\\{\tfrac{pp^{\prime}}{2},p\right\\}&p>2,\\\
(1,2)&p=2,\end{array}\right.\quad
q\coloneqq\left\\{\begin{array}[]{ll}p,&p>2,\\\ r,&p=2.\end{array}\right.$

The key relationship among these exponents is that

(3.2)
$\left\|{D(g_{1}g_{2})}\right\|_{L^{r}(\mathbb{C})}\lesssim\left(\|g_{2}\|_{L^{\infty}(\mathbb{C})}+\|g_{2}\|_{W^{1,p}(\mathbb{C})}\right)\|g_{1}\|_{W^{1,q}(\mathbb{C})},$

whenever the right hand side is finite. When $p=2$ or $r=p$, (3.2) is a
consequence of the product rule, Hölder’s inequality, and Sobolev embedding.
When $p>2$ and $r=\frac{pp^{\prime}}{2}$, Hölder’s inequality shows that

$\left\|{(Dg_{1})g_{2}}\right\|_{L^{\frac{pp^{\prime}}{2}}(\mathbb{C})}\leq\left\|{Dg_{1}}\right\|_{L^{p}(\mathbb{C})}\left\|{g_{2}}\right\|_{L^{\frac{pp^{\prime}}{2-p^{\prime}}}(\mathbb{C})}.$

The second term is then handled by interpolation between $L^{\infty}$ and
$L^{p}$, and subsequently bounded by
$\left\|{g_{2}}\right\|_{W^{1,p}(\mathbb{C})}$ due to Sobolev embedding. This
same argument with $g_{1}$ and $g_{2}$ exchanged establishes (3.2) in all
cases.

With this notation, to prove Theorem A it suffices to estimate
$\left\|{(I-\mu\mathcal{S})^{-1}}\right\|_{W^{1,r}(\mathbb{C})}$ with the
prescribed dependence on $\mu$, $p$, $q$, and $r$. Since the final estimate
will have exponential blow-up at the endpoints of our ranges ($r$ approaching
$1$ or $2$ in the case $p=2$, or in the other case as $p$ approaches $2$ or
$\infty$), we use $A\lesssim B$ to denote $A\leq CB$ for some $C$ depending
polynomially on $k$, $p$, $r$, $q$, and $L$. Most importantly, the Sobolev
embedding theorems have polynomial blow up at these endpoints, hence (3.2)
holds with this prescribed convention for $\lesssim$.

We make the further reduction following [astala01]*pp. 39-40 to consider
$w,h\in\mathcal{C}^{\infty}(\mathbb{C})$ satisfying the following
inhomogeneous Beltrami equation

$\overline{\partial}w=\mu\partial w+h.$

Normalize so that $\left\|{h}\right\|_{W^{1,q}(\mathbb{C})}\leq 1$ and let
$L\coloneqq\|\mu\|_{W^{1,2}(\mathbb{C})}$Then, Theorem A amounts to the a
priori estimate

(3.3)
$\|D\overline{\partial}w\|_{L^{r}(\mathbb{C})}\lesssim\left\\{\begin{array}[]{cc}\exp\left(C\max\left\\{\tfrac{1}{r-1},\tfrac{1}{(2-r)^{2}}\right\\}L^{2}\right)&p=2;\\\
\exp\left(C\max\left\\{p,\tfrac{1}{p-2}\right\\}^{2}L^{2}\right)\left(1+\|\mu\|_{W^{1,p}(\mathbb{C})}\right)^{2}&p>2.\end{array}\right.$

Indeed, the estimates in Theorem A are now special cases of (3.3). The
critical case (1.4) is recovered when $1<r<2$ and $p=2$; (1.5) is recovered by
$r=p>2$.

###  3.1. Main line of proof of Theorem A

Let $f$ be the principal solution to (B). By [astala-book]*Theorem 5.2.3 for
$\mu\in W^{1,2}(\mathbb{C})$, $\partial f=\operatorname{{e}}^{\sigma}$ where
$\sigma$ satisfies

(3.4) $\overline{\partial}\sigma=\mu\partial\sigma+\partial\mu.$

Applying Lemma 2.3 shows that

(3.5)
$\|\sigma\|_{W^{1,p}(\mathbb{C})}\lesssim\exp(Cp^{2}L^{2})\|\mu\|_{W^{1,p}(\mathbb{C})}.$

Next, introduce $u=w\circ f^{-1}$ so by the chain rule,

(3.6) $\displaystyle\overline{\partial}w$ $\displaystyle=\left(\partial u\circ
f\right)\overline{\partial}f+\left(\overline{\partial}u\circ
f\right)\overline{\partial f}$ $\displaystyle\mu\partial w+h$
$\displaystyle=\mu\left[\left(\partial u\circ f\right)\partial
f+\left(\overline{\partial}u\circ
f\right)\overline{\overline{\partial}f}\right]+h.$

####  3.1.1. _A priori_ bounds on the antiholomorphic part

Using the equations for $f$ and $w$, we obtain from (3.6) that

(3.7) $\left(\overline{\partial}u\circ f\right)\overline{\partial
f}=\frac{h}{1-|\mu|^{2}}\eqqcolon H.$

It follows from (3.2) that

(3.8) $\left\|{DH}\right\|_{L^{r}(\mathbb{C})}\lesssim
1+\|\mu\|_{W^{1,p}(\mathbb{C})}.$

Referring to (3.7), observe the crucial equality

$\displaystyle DH$ $\displaystyle=D\left[\left(\overline{\partial}u\circ
f\right)\operatorname{{e}}^{\overline{\sigma}}\right]=\left(\overline{\partial}u\circ
f\right)\overline{\partial fD\sigma}+D\left(\overline{\partial}u\circ
f\right)\overline{\partial f}$
$\displaystyle=H\overline{D\sigma}+D\left(\overline{\partial}u\circ
f\right)\overline{\partial f}\equiv H\overline{D\sigma}+G.$

Using the regularity of $\sigma$ from (3.5) and of $H$ from (3.8), together
with Sobolev embedding gives

(3.9)
$\|G\|_{L^{r}(\mathbb{C})}\lesssim\|DH\|_{L^{r}(\mathbb{C})}+\|hD\sigma\|_{L^{r}(\mathbb{C})}\lesssim\exp\left(Cp^{2}L^{2}\right)\left(1+\|\mu\|_{W^{1,p}(\mathbb{C})}\right).$

Another calculation relying on the chain rule yields the equality

(3.10) $\left(\overline{\partial f}\right)^{2}\left(D\overline{\partial}u\circ
f\right)=\mathbb{A}\overline{\partial f}D\left(\overline{\partial}u\circ
f\right),\qquad\mathbb{A}\coloneqq\frac{1}{\left|{Jf}\right|}\left[\begin{array}[]{cc}\overline{\partial
f}^{2}&-\overline{\partial f\overline{\partial}f}\\\ -\overline{\partial
f}\overline{\partial}f&\left|\partial f\right|^{2}\end{array}\right],$

and (3.9) together with $K$-quasiconformality of $f$ from (2.5) entail the
estimate

(3.11) $\left\|\left(\overline{\partial
f}\right)^{2}\left(D\overline{\partial}u\circ
f\right)\right\|_{L^{r}(\mathbb{C})}\lesssim\|\mathbb{A}\|_{\infty}\|G\|_{L^{r}(\mathbb{C})}\lesssim\exp\left(Cp^{2}L^{2}\right)\left(1+\|\mu\|_{W^{1,p}(\mathbb{C})}\right).$

####  3.1.2. Estimating the norm $\|\overline{\partial}w\|_{W^{1,r}}$

Differentiating the first line of (3.6), we see that

(3.12) $D\overline{\partial}w=D\left(\partial u\circ
f\right)\overline{\partial}f+\left(\partial u\circ
f\right)D\overline{\partial}f+DH.$

By virtue of (3.8), we are left with estimating the first two terms on the
right hand side. By (2.1), the chain rule, (2.5), and a change of variables,
the first term in (3.12) is estimated as

(3.13) $\displaystyle\left\|D\left(\partial u\circ
f\right)\overline{\partial}f\right\|_{L^{r}(\mathbb{C})}^{r}$
$\displaystyle=\int_{\mathbb{C}}\big{|}D[\mathcal{S}(\overline{\partial}u)\circ
f]\big{|}^{r}\left|{\overline{\partial}f}\right|^{r}$
$\displaystyle\lesssim\int_{\mathbb{C}}\big{|}D[\mathcal{S}(\overline{\partial}u)]\circ
f\big{|}^{r}|Jf|^{r}$
$\displaystyle\lesssim\int_{\mathbb{C}}|D\mathcal{S}(\overline{\partial}u)|^{r}|Jf^{-1}|^{1-r}.$

By Lemma 2.2, $|Jf^{-1}|^{1-r}\in\mathrm{A}_{r}(\mathbb{C})$ with the
dependence (2.7) so the weighted Sobolev estimate for $\mathcal{S}$
(Proposition (2.1) with $n=1$ and $p=r$), (3.13) implies

(3.14) $\displaystyle\left\|{D(\partial u\circ
f)\overline{\partial}f}\right\|_{L^{r}(\mathbb{C})}$
$\displaystyle\lesssim\exp\left(C\max\left\\{r^{2},\tfrac{1}{r-1}\right\\}L^{2}\right)\left(\int_{\mathbb{C}}|D\overline{\partial}u|^{r}|J{f^{-1}}|^{1-r}\right)^{\frac{1}{r}}$
$\displaystyle\lesssim\exp\left(C\max\left\\{p^{2},\tfrac{1}{r-1}\right\\}L^{2}\right)\left(1+\|\mu\|_{W^{1,p}(\mathbb{C})}\right),$

where the last line followed from changing variables back, applying (3.11),
and the fact that $r\leq p$. Combining (3.12), (3.14), and (3.8), we have

(3.15)
$\left\|{D\overline{\partial}w}\right\|_{L^{r}(\mathbb{C})}\lesssim\exp\left(C\max\left\\{p^{2},\tfrac{1}{r-1}\right\\}L^{2}\right)\left(1+\|\mu\|_{W^{1,p}(\mathbb{C})}\right)+\left\|{(\partial
u\circ f)D\overline{\partial}f}\right\|_{L^{r}(\mathbb{C})}.$

It remains to estimate the final term in (3.15). Notice that
$D\overline{\partial}f=(D\mu+\mu D\sigma)\partial f\eqqcolon\lambda\partial f$
and $\lambda\in L^{p}(\mathbb{C})$ by (3.5). With (3.15) in hand, we now
complete the proof of (3.3) in the case $r<p$. Let $p<s<\infty$ be determined
by $\frac{1}{r}=\frac{1}{p}+\frac{1}{s}$. Then, by Hölder’s inequality,

(3.16) $\|(\partial u\circ
f)D\overline{\partial}f\|_{L^{r}(\mathbb{C})}\leq\|(\partial u\circ f)\partial
f\|_{L^{s}(\mathbb{C})}\|\lambda\|_{L^{p}(\mathbb{C})}.$

As in (3.13) above, by (2.1), a change of variable, (2.5), and the weighted
estimate for $\mathcal{S}$ on $L^{s}(\mathbb{C})$,

(3.17) $\int_{\mathbb{C}}|\partial u\circ f|^{s}|\partial
f|^{s}\lesssim\int_{\mathbb{C}}|\mathcal{S}(\overline{\partial}u)|^{s}|Jf^{-1}|^{1-\frac{s}{2}}\lesssim\exp\left(Cs^{3}L^{2}\right)\int_{\mathbb{C}}|\overline{\partial}u|^{s}|Jf^{-1}|^{1-\frac{s}{2}}.$

The same logic leading to (3.2), after changing variables, facilitates the
final estimate

(3.18)
$\left(\int_{\mathbb{C}}|\overline{\partial}u|^{s}|Jf^{-1}|^{1-\frac{s}{2}}\right)^{\frac{1}{s}}\lesssim\|H\|_{L^{s}(\mathbb{C})}\lesssim\|h\|_{L^{s}(\mathbb{C})}\lesssim
1.$

Stringing together (3.16), (3.17), and (3.18), using (3.5) to estimate
$\lambda$, and recalling $s>p$, the final term in (3.15) is estimated for
$r<p$ by

(3.19) $\|(\partial u\circ
f)D\overline{\partial}f\|_{L^{r}(\mathbb{C})}\lesssim\exp\left(Cs^{2}L^{2}\right)\left(1+\left\|{\mu}\right\|_{W^{1,p}(\mathbb{C})}\right).$

In the case $p=2$, $s=\frac{2r}{2-r}\sim\frac{1}{2-r}$, so by (3.15) and
(3.19), (3.3) is established for $p=2$. For $p>2$, $r=\frac{pp^{\prime}}{2}$
and one has $s=\frac{p^{2}}{p-2}\sim\max\left\\{p,\frac{1}{p-2}\right\\}$. In
this case (3.15) and (3.19) imply

(3.20)
$\left\|{\overline{\partial}w}\right\|_{W^{1,\frac{pp^{\prime}}{2}}(\mathbb{C})}\lesssim\exp\left(C\max\left\\{p,\tfrac{1}{p-2}\right\\}^{2}L^{2}\right)\left(1+\|\mu\|_{W^{1,p}(\mathbb{C})}\right).$

Finally, to achieve (3.3) when $r=p$, we notice that since
$\frac{pp^{\prime}}{2}>2$, by (3.6), Sobolev embedding, and (3.20),

(3.21) $\left\|(\partial u\circ f)\partial
f\right\|_{L^{\infty}(\mathbb{C})}\lesssim\left\|\partial
w\right\|_{L^{\infty}(\mathbb{C})}+\|\mu
H\|_{L^{\infty}(\mathbb{C})}\lesssim\exp\left(C\max\left\\{p,\tfrac{1}{p-2}\right\\}^{2}L^{2}\right)\left(1+\|\mu\|_{W^{1,p}(\mathbb{C})}\right).$

Apply (3.21) to estimate the final term in (3.15),

$\|(\partial u\circ
f)D\overline{\partial}f\|_{L^{p}(\mathbb{C})}\lesssim\left\|(\partial u\circ
f)\partial
f\right\|_{L^{\infty}(\mathbb{C})}\|\lambda\|_{L^{p}(\mathbb{C})}\lesssim\exp\left(C\max\left\\{p,\tfrac{1}{p-2}\right\\}^{2}L^{2}\right)\left(1+\|\mu\|_{W^{1,p}(\mathbb{C})}\right)^{2},$

and (3.3) immediately follows for $r=p$.

###  3.2. Proof of Corollary A.1

We will again give a unified proof of the critical and supercritical cases. To
this end, let $p\geq 2$ and consider

$q\in\left\\{\begin{array}[]{cc}(2,\infty),&p=2,\\\
\left[\frac{p}{p-1},\infty\right)&p>2,\end{array}\right.\qquad
r=\frac{q}{q-1}.$

Then, (1.7) and (1.9) will follow from

(3.22)
$\|\eta(Df)\|_{L^{q}(\mathbb{C})}\lesssim\|(D\eta)f\|_{L^{q}(\mathbb{C})},$

while (1.8) and (1.10) will follow from

(3.23)
$\|\eta(D^{2}f)\|_{L^{r}(\mathbb{C})}\lesssim\|(D\eta)f\|_{L^{r}(\mathbb{C})}+\|(D\eta)(Df)\|_{L^{r}(\mathbb{C})}+\|(D^{2}\eta)f\|_{L^{r}(\mathbb{C})}.$

In this proof, we will not track the constants implicit in (3.22) and (3.23)
but they are an absolute constant multiple of the corresponding estimates for
$(I-\mu\mathcal{S})^{-1}$ in Lemma 2.3 and Theorem A. First we will prove
(3.22). Since $f\in L^{q}_{\mathrm{loc}}$, we claim that we can extend (1.6)
to all $\psi$ of the form $\eta\phi$ for $\eta\in\mathcal{C}_{0}^{\infty}(E)$
and $\phi\in W^{1,r}(\mathbb{C})$. Indeed, for such $\eta$ and $\phi$, by
Hölder’s inequality and Sobolev embedding,

$\|(\partial-\partial\mu)(\eta\phi)\|_{L^{r}(\mathbb{C})}\lesssim\left\|{\eta\phi}\right\|_{W^{1,r}(\mathbb{C})}+\left(\left\|{\eta\mu}\right\|_{L^{\infty}(\mathbb{C})}+\left\|{\eta\mu}\right\|_{W^{1,p}(\mathbb{C})}\right)\left\|{\phi}\right\|_{W^{1,r}(\mathbb{C})}.$

Therefore, (1.6) holds for all such $\psi=\eta\phi$ by density. Now, fix
$\eta\in\mathcal{C}_{0}^{\infty}(E)$, let $\chi\in\mathcal{C}_{0}^{\infty}(E)$
with $\chi\equiv 1$ on $\operatorname{{supp}}\eta$, and set $\nu=\mu\chi$. Let
$g\in C^{\infty}_{0}(E)$ and let $v$ satisfy

(3.24) $\overline{\partial}v-\nu\partial v=g.$

Then, Lemma 2.3 and Theorem A respectively imply

(3.25)
$\|v\|_{W^{1,r}(\mathbb{C})}\lesssim\|g\|_{L^{r}(\mathbb{C})},\qquad\|v\|_{W^{2,r}(\mathbb{C})}\lesssim\|g\|_{W^{1,r}(\mathbb{C})}.$

Furthermore, applying $\partial$ to (3.24) we obtain

$\partial g=(\overline{\partial}-\partial\nu)(\partial v).$

Pair the above display with $F=\eta f$ to obtain

$\left\langle F,\partial g\right\rangle=\left\langle\eta
f,\left(\overline{\partial}-\partial\nu\right)\partial
v\right\rangle=\left\langle
f,\left(\overline{\partial}-\partial\nu\right)(\eta\partial
v)\right\rangle-\left\langle
f,\left(\overline{\partial}\eta-\nu\partial\eta\right)\partial
v\right\rangle.$

The precise form of $\chi$ shows that $\nu\eta=\mu\chi\eta=\mu\eta$ so that,
since $\partial v\in W^{1,r}(\mathbb{C})$ by (3.25), the final term in the
above display vanishes by (1.6). On the other hand, the first estimate in
(3.25) shows that

$\left|{\left\langle
f,\left(\overline{\partial}\eta-\nu\partial\eta\right)\partial
v\right\rangle}\right|\lesssim\|(D\eta)f\|_{L^{q}(\mathbb{C})}\|g\|_{L^{r}(\mathbb{C})}.$

Therefore, combining the previous two displays and using compact support,

$|\langle\partial
F,g\rangle|\lesssim\|(D\eta)f\|_{L^{q}(\mathbb{C})}\|g\|_{L^{r}(\mathbb{C})},$

for all $g\in C^{\infty}_{0}$. This establishes that $\left\|{\eta\partial
f}\right\|_{L^{q}}$ is bounded by the right hand side of (3.22). To prove the
same for to $\eta\overline{\partial}f$, simply notice that by (1.6), there
holds for any $h\in\mathcal{C}^{\infty}_{0}(\mathbb{C})$,

$\left|\left\langle\eta\overline{\partial}f,h\right\rangle\right|\leq\left|\left\langle\mu\eta\partial
f,h\right\rangle\right|.$

We proceed to prove (3.23). Notice that (3.22), which we just proved, together
with the assumption $f\in L^{q}_{\mathrm{loc}}$ establishes that $F=\eta f$
indeed belongs to $W^{1,s}(\mathbb{C})$ for every $s\leq q$. Therefore,

$\overline{\partial}F-\nu\partial
F=(\overline{\partial}\eta-\nu\partial\eta)f.$

Using Sobolev embedding, the right hand side of the above display belongs to
$W^{1,r}$ with norm

(3.26)
$\|(\overline{\partial}\eta-\nu\partial\eta)f\|_{W^{1,r}(\mathbb{C})}\lesssim\|(D\eta)f\|_{L^{r}(\mathbb{C})}+\|(D\eta)(Df)\|_{L^{r}(\mathbb{C})}+\|(D^{2}\eta)f\|_{L^{r}(\mathbb{C})}.$

On the other hand, by (2.1) and Proposition 2.1,

$\|D^{2}F\|_{L^{r}(\mathbb{C})}\leq\left\|\overline{\partial}F\right\|_{W^{1,r}(\mathbb{C})}+\|\partial
F\|_{W^{1,r}(\mathbb{C})}\lesssim\left\|\overline{\partial}F\right\|_{W^{1,r}(\mathbb{C})}.$

Furthermore, Theorem A applies to give

$\left\|{\overline{\partial}F}\right\|_{W^{1,r}(\mathbb{C})}\lesssim\|(\overline{\partial}\eta-\nu\partial\eta)f\|_{W^{1,r}(\mathbb{C})}$

and the proof of (3.23) is concluded by (3.26).

Finally, we establish the concluding statement that $f\in
W^{2,r}_{\mathrm{loc}}(E)$. To this end, we iterate the first Caccioppoli
inequality (3.22) to show that $f\in W^{1,s}_{\mathrm{loc}}(E)$ for every
$1<s<\infty$. Indeed, let $f\in L^{q}_{\mathrm{loc}}(E)$ and suppose $s>q>1$.
By (3.22), $f\in W^{1,q}_{\mathrm{loc}}(E)$ so by Sobolev embedding, $f\in
L^{q_{1}}_{\mathrm{loc}}(E)$ for every $q_{1}$ satisfying

$q_{1}<\left\\{\begin{array}[]{cc}\frac{2q}{2-q},&q\leq 2;\\\
\infty,&q>2.\end{array}\right.$

In particular, we can choose $q_{1}>2$ so that a second application of (3.22)
implies $f\in W^{1,q_{1}}_{\mathrm{loc}}(E)$ and by Sobolev embedding $f\in
L^{q_{2}}_{\mathrm{loc}}(E)$ for every $q_{2}<\infty$. One final application
of (3.22) proves the claim. In particular, $f\in W^{1,r}_{\mathrm{loc}}(E)$ so
the right hand side of (3.23) is finite hence $f\in
W^{2,r}_{\mathrm{loc}}(E)$.

##  4\. Proof of Theorem B and Corollary B.1

Throughout, let $2<p<\infty$ and $\Omega$ be a simply connected bounded domain
in $\mathbb{C}$. The restriction to simply connected domains is for
convenience, but can be lifted to finitely connected domains. In this section
we still consider global solutions to the Beltrami equation, but we assume
$\mu$ to be of a special form, linked to $\Omega$ in the following way. Let
$\mu$ satisfy

(4.1)
$\operatorname{{supp}}\mu\subset\overline{\Omega},\qquad\left\|\mu\right\|_{\infty}=k=\frac{K-1}{K+1}<1,\qquad\mu\in
W^{1,p}(\Omega).$

In proving Theorem B, the Beurling-Ahlfors operator $\mathcal{S}$ is replaced
by its compression to a domain $O$, defined by

$\mathcal{S}_{O}=\mathbf{1}_{\overline{O}}\mathcal{S}(\cdot\mathbf{1}_{\overline{O}}).$

Lebesgue space estimates for $\mathcal{S}_{O}$ follow from those established
for $\mathcal{S}$ in Proposition 2.1, by simply considered functions supported
in $\overline{O}$. However, the Sobolev estimates must be approached
differently. The systematic study of unweighted Sobolev estimates for
compressions of Calderón-Zygmund operators was taken up by Prats and Tolsa in
[prats-tolsa]. A completely different approach by the authors has led to
general $T(1)$ theorems and weighted estimates in Sobolev spaces in
[diplinio23domains]. The necessary ingredients from these papers are extracted
in Lemma 4.3 below.

######  Definition 4.1.

We first define two norms for functions $f:\Gamma\to\mathbb{C}$ where $\Gamma$
is a piecewise continuous curve. Say $f$ is Dini continuous if the norm

$\left\|{f}\right\|_{\mathrm{Dini}}=\int_{0}^{1}\sup_{|x-y|\leq
t}\left|{f(x)-f(y)}\right|\,\frac{\mathrm{d}t}{t}$

is finite. Define the homogeneous Besov norm on $\Gamma$ by

$\left\|{f}\right\|_{\dot{B}^{1-\frac{1}{q}}_{q,q}(\Gamma)}=\left(\int_{\Gamma}\int_{\Gamma}\frac{|f(x)-f(y)|^{q}}{|x-y|^{q}}\,\mathrm{d}s(x)\,\mathrm{d}s(y)\right)^{\frac{1}{q}}<\infty,$

where $\mathrm{d}s$ is the surface measure on $\Gamma$. Say a bounded domain
$O$ is a Dini-smooth domain if there exists $\delta,R>0$ such that for each
$z\in\partial O$ there exists a function $A:\mathbb{R}\to\mathbb{R}$ with
$\left\|{A^{\prime}}\right\|_{\mathrm{Dini}}<\delta$ and an angle $\theta$
such that

$\Omega\cap B(z,r)=\left\\{\operatorname{{e}}^{\mathrm{i}\theta}(x+iy)\in
B(z,r):y>A(x)\right\\}.$

Dini-smooth domains are the natural setting for higher order conformal
estimates as the conformal map onto a Dini-smooth domain is bi-Lipschitz. We
denote $[O]_{\mathrm{Dini}}=(\delta,R)$. Given $q>2$, we say a Dini-smooth
domain $O$ is a $\mathcal{B}_{q}$ domain if there exists $C>0$ such that each
parameterization $A$ described above also satisfies

$\left\|{A^{\prime}}\right\|_{\dot{B}^{1-\frac{1}{q}}_{q,q}(\mathbb{R})}\leq
C,$

Covering the boundary of $\partial\Omega$ by balls $B_{j}$ of radius $R$, and
letting $A_{j}$ denote the associated parameterizations, one has

(4.2)
$\sum_{j}\left\|{A_{j}^{\prime}}\right\|_{\dot{B}^{1-\frac{1}{q}}_{q,q}(\mathbb{R})}\sim\left\|{N_{O}}\right\|_{\dot{B}^{1-\frac{1}{q}}_{q,q}(\partial
O)},$

where $N_{O}$ is the normal vector to the boundary $\partial O$ and the
implicit constants in (4.2) depend only on $[O]_{\mathrm{Dini}}$, see [cruz-
tolsa]*Lemmata 3.1 and 3.3. In light of (4.2), we define

$\left\|{O}\right\|_{\mathcal{B}_{q}}=\left\|{N_{O}}\right\|_{\dot{B}^{1-\frac{1}{q}}_{q,q}(\partial
O)}.$

###  4.1. Standing assumptions and implicit constants

Let us fix some parameters and establish some notational conventions for the
remainder of this section. Henceforth, let $2<r\leq p$, $\Omega$ be a bounded
simply connected $\mathcal{B}_{p}$ domain, and $\mu$ satisfy (4.1).
Furthermore, let $f$ be the principal solution of (B), set $O=f(\Omega)$.

The shorthand $\lesssim$ and $\sim$ will denote one or two-sided inequalities
with implicit dependence on $[\Omega]_{\mathrm{Dini}}$, $[O]_{\mathrm{Dini}}$,
$\operatorname{{diam}}\Omega$, $\operatorname{{diam}}O$, $p$, $K$, and
$L=\left\|{\mu}\right\|_{W^{1,2}(\Omega)}$. In Theorem B, the dependence will
be double exponential on these quantities, so we do not track them precisely.
Furthermore, $\mathcal{G}$ and $\mathcal{E}$ will be generic functions such
that $\mathcal{G}$ and $\log\mathcal{E}$ have polynomial growth, which is
implicitly determined by these parameters.

###  4.2. Weighted estimates for compressions

To apply the same strategy as in the proof of Theorem A, we will need
estimates for $\mathcal{S}_{O}$ on a certain weighted Sobolev space
$W^{1,p}(O,\omega)$. It is known that the Besov norm of the boundary normal
introduced above is precisely connected to unweighted Sobolev estimates for
$\mathcal{S}_{O}$; consult the following references [cruz-tolsa,
tolsa2013regularity, prats17]. In particular,

(4.3) $\left\|\mathcal{S}_{O}\right\|_{\mathcal{L}(W^{1,p}(O))}\sim
1+\left\|O\right\|_{\mathcal{B}_{p}},\qquad p>2.$

An analog of the quantitative geometric characterization (4.3) for
$\left\|{\mathcal{S}_{O}}\right\|_{\mathcal{L}(W^{1,p}(O,\omega))}$ is not
currently available, though qualitatively they are equivalent. In fact, the
following Lemma demonstrates the sharpness of our assumption on $O=f(\Omega)$
in Theorem B.

######  Lemma 4.2.

If $I-\mu\mathcal{S}_{\Omega}$ is invertible on $W^{1,p}(\Omega)$, then

$\mathcal{S}_{O}:W^{1,p}(O,\omega)\to
W^{1,p}(O,\omega),\qquad\omega=\left|{Jf^{-1}}\right|^{1-p}.$

The proof is postponed until §4.4 below. Concerning quantitative geometric
conditions on $O$, using the $T(1)$ theorems developed by the authors in
[diplinio23domains], we can give some alternatives of varying of degrees of
sharpness.

######  Lemma 4.3.

Set $\omega=\left|Jf^{-1}\right|^{1-p}$. Then,

  * i.

$\left\|{\mathcal{S}_{O}}\right\|_{\mathcal{L}(W^{1,p}(O,\omega))}\lesssim\left(\|Jf\|_{\infty}\|Jf^{-1}\|_{\infty}\right)^{1-\frac{1}{p}}\left(1+\left\|O\right\|_{\mathcal{B}_{p}}\right)$.

  * ii.

$\left\|{\mathcal{S}_{O}}\right\|_{\mathcal{L}(W^{1,p}(O,\omega))}\lesssim\left(1+\frac{\left\|\mathcal{S}_{O}(\mathbf{1})\right\|_{W^{1,p}(O,\omega)}}{\left\|{\mathbf{1}_{O}}\right\|_{L^{p}(O,\omega)}}\right)\mathcal{E}\left(\left\|{\mu}\right\|_{W^{1,r}(\Omega)}\right)$.

  * iii.

For any $\varepsilon>0$,
$\left\|{\mathcal{S}_{O}}\right\|_{\mathcal{L}(W^{1,p}(O,\omega))}\lesssim\left(1+\left\|O\right\|_{\mathcal{B}_{p+\varepsilon}}\right)\mathcal{E}\left(\frac{1}{\varepsilon},\left\|{\mu}\right\|_{W^{1,r}(\Omega)}\right)$.

######  Remark 4.4.

Lemma 4.3.i is achieved by appealing to the unweighted estimate (4.3), and
crude bounds on $\left\|{\left|{Jf}\right|^{-1}}\right\|_{\infty}$ and
$\left\|{Jf}\right\|_{\infty}$ can be derived from (4.7) of Proposition 4.8.
In this way, if the reader is interested in a quick bound for
$(I-\mu\mathcal{S}_{\Omega})^{-1}$ which bypasses the more difficult weighted
Sobolev theory of CZOs that ii. and iii. rely on, one can appeal to the
logarithmic Sobolev inequality

$\left\|{g}\right\|_{\infty}\lesssim\frac{1}{q-2}\left(1+\left\|{g}\right\|_{W^{1,2}(\Omega)}\right)\log\left(\operatorname{{e}}+\left\|{g}\right\|_{W^{1,q}(\Omega)}\right),\qquad
q>2,$

to obtain

$\left\|{\mathcal{S}_{O}}\right\|_{\mathcal{L}(W^{1,p}(O,\omega))}\lesssim\left(1+\left\|{O}\right\|_{\mathcal{B}_{q}}+\left\|{\Omega}\right\|_{\mathcal{B}_{q}}+\left\|{\mu}\right\|_{W^{1,q}(\Omega)}\right)^{\frac{C}{q-2}}\left\|O\right\|_{\mathcal{B}_{p}}.$

Notice that the middle factor blows up exponentially as $q\to 2$, while
weighted estimates in e.g. Lemma 4.3.iii and Corollary B.1 produce an absolute
polynomial bound in terms of the boundary data
$\left\|{\Omega}\right\|_{\mathcal{B}_{p}}$ and
$\left\|{O}\right\|_{\mathcal{B}_{p}}$.

To prove Lemma 4.3, we need to verify that the weights
$\omega=\left|{Jf^{-1}}\right|^{1-p}$ belong to the appropriate Muckenhoupt
$\mathrm{A}_{p}$ class for the results from [diplinio23domains] to apply. The
following adaptation of Lemma 2.2 is proved in §5.2.

######  Lemma 4.5.

Let $F\in W^{1,2}_{\mathrm{loc}}(\Omega)$ be a homeomorphism satisfying

$\overline{\partial}F(z)=\mu(z)\partial F(z),\quad z\in\Omega,$

and further assume that $F(\Omega)$ is a bounded $\mathcal{B}_{r}$ domain.
Then, for each $a\in\mathbb{R}$ and $1<q<\infty$, there exists
$\upsilon:\mathbb{C}\to[0,\infty]$ such that

$\upsilon=\left|JF^{-1}\right|^{a}\mbox{ on }F(\Omega),$

and

(4.4)
$\left[\upsilon\right]_{\mathrm{A}_{q}(\mathbb{C})}\lesssim\mathcal{E}\left(a,q,\tfrac{1}{q-1},\left\|{\mu}\right\|_{W^{1,r}(\Omega)}\right).$

###### Proof of Lemma 4.3.

i. follows by pulling the weight $\omega$ outside, applying the unweighted
estimate, using (4.3), and then reinserting the weight. To prove ii. and iii.,
we rely on the weighted Sobolev estimates for Calderón-Zygmund operators on
domains established in [diplinio23domains]. According to Lemma 4.5 below,
$\omega$ does belong to the relevant weight classes. The main result for
compressions of Calderón-Zygmund operators from [diplinio23domains]*Corollary
B.1 and Lemma 4.22 states that if the weight $\omega$ possesses an extension
which belongs to $\mathrm{A}_{s}(\mathbb{C})$ for some $1<s<\frac{p}{2}$, then
it suffices to test $\mathcal{S}_{O}$ on the constant function $\mathbf{1}$.
Letting $\upsilon$ be the extension of $\omega$ provided by Lemma 4.5,

$[\upsilon]_{\mathrm{A}_{\frac{p+2}{4}}(\mathbb{C})}\lesssim\mathcal{E}\left(\left\|{\mu}\right\|_{W^{1,r}(\Omega)}\right),$

from which ii. follows. To prove iii., let $\varepsilon>0$, set
$s=\frac{p+\varepsilon}{\varepsilon}$, and let $\tilde{\upsilon}$ be the
extension of $\omega^{s}$ from Lemma 4.5. Let $Q$ be a large cube so that
$O\subset Q$ and $|O|\sim|Q|$. Standard considerations show that

$\left(\int_{O}\omega^{s}\right)^{\frac{1}{ps}}\lesssim\mathcal{G}\left([\tilde{\upsilon}]_{\mathrm{A}_{2}(\mathbb{C})}\right)\left(\int_{O}\omega\right)^{\frac{1}{p}}.$

Therefore, by Hölder’s inequality, and (4.3)

$\left\|{\mathcal{S}_{O}(\mathbf{1})}\right\|_{W^{1,p}(O,\omega)}\leq\left\|{\mathcal{S}_{O}(\mathbf{1})}\right\|_{W^{1,p+\varepsilon}(O)}\left\|{\omega^{\frac{1}{p}}}\right\|_{L^{pr}(O)}\lesssim\left(1+\left\|{O}\right\|_{B_{p+\varepsilon}}\right)\mathcal{G}\left([\tilde{\upsilon}]_{\mathrm{A}_{2}(\mathbb{C})}\right)\left\|{\mathbf{1}}\right\|_{L^{p}(O,\omega)},$

and iii. is now a consequence of ii. and (4.4) ∎

The second consequence of Lemma 4.5 is the following analogue of Lemma 2.3.

######  Proposition 4.6.

For each $1<s<\infty$,

$\left\|(I-\mu\mathcal{S}_{\Omega})^{-1}\right\|_{\mathcal{L}(L^{s}(\Omega))}\lesssim\mathcal{E}\left(s,\tfrac{1}{s-1},\left\|{\Omega}\right\|_{\mathcal{B}_{r}},\left\|{\mu}\right\|_{W^{1,r}(\Omega)},\left\|{f(\Omega)}\right\|_{B_{2}}\right).$

###### Proof.

The proof consists of one small modification to the argument leading to (2.9)
which we outline. Let $g\in\mathcal{C}^{\infty}(\mathbb{C})$ and set
$w=\mathcal{K}(\mathbf{1}_{\overline{\Omega}}g)$ where $\mathcal{K}$ is the
Cauchy transform defined in (2.2). Setting
$h=(I-\mu\mathcal{S}_{\Omega})(\mathbf{1}_{\overline{\Omega}}g)$, one can
verify

$\overline{\partial}w=\mu\partial w+h$

and the desired estimate will follow from
$\|\overline{\partial}w\|_{L^{s}(\Omega)}\lesssim\|h\|_{L^{s}(\Omega)}$. Let
$f$ be $\mu$-quasiconformal and set $u=w\circ f^{-1}$. The chain rule shows
that $\overline{\partial}w=(\partial u\circ
f)\overline{\partial}f+(\overline{\partial}u\circ f)\overline{\partial f}$ and
the equations for $w$ and $f$ imply
$(1-|\mu|^{2})^{-1}h=(\overline{\partial}u\circ f)\overline{\partial f}$.
Therefore it remains to estimate $(\partial u\circ f)\overline{\partial}f$ by
$h$ in $L^{s}(\Omega)$-norm. Changing variables, using (2.5) and (2.1),

$\int_{\Omega}\left|(\partial u\circ
f)\bar{\partial}f\right|^{s}\lesssim\int_{f(\Omega)}\left|\mathcal{S}(\overline{\partial}u)\right|^{s}\left|Jf^{-1}\right|^{1-s/2}.$

Letting $\upsilon$ be the extension of
$\left|{Jf^{-1}}\right|^{1-\frac{s}{2}}$ from Lemma 4.5, and applying
Proposition 2.1,

$\int_{f(\Omega)}\left|\mathcal{S}(\overline{\partial}u)\right|^{s}\left|Jf^{-1}\right|^{1-s/2}\leq\left\|\mathcal{S}(\overline{\partial}u)\right\|_{L^{s}(\mathbb{C},\upsilon)}^{s}\leq\mathcal{G}\left([\upsilon]_{\mathrm{A}_{s}(\mathbb{C})}\right)\left\|\overline{\partial}u\right\|_{L^{s}(\mathbb{C},\upsilon)}^{s}.$

The proof is concluded by (4.4) and recalling that $\overline{\partial}u$ is
supported on $f(\Omega)$ so that by a change of variable,
$\left\|\overline{\partial}u\right\|_{L^{s}(\mathbb{C},\upsilon)}=\|h\|_{L^{s}(\Omega)}$.
∎

###  4.3. Conformal estimates

The final ingredients for the proof of Theorem B are the following conformal
estimates which are qualitatively contained in [astala-prats-saksman]. The
first one is a quantification of [astala-prats-saksman]*Theorem 1.2.

######  Lemma 4.7.

Let $O_{1}$ and $O_{2}$ be simply connected $\mathcal{B}_{p}$ domains, and $g$
conformally map $O_{1}$ onto $O_{2}$. Then $\log g^{\prime}\in W^{1,p}(O_{1})$
with

(4.5) $\displaystyle\left\|{\left[\log
g^{\prime}\right]^{\prime}}\right\|_{L^{q}(O_{1})}\lesssim
1+\left\|{O_{1}}\right\|_{\mathcal{B}_{q}}+\left\|{O_{2}}\right\|_{\mathcal{B}_{q}}\qquad
1<q\leq p.$

###### Proof.

It is shown in [astala-prats-saksman]*Theorem 1.2 that if $h_{j}$ is a
conformal mapping from $\mathbb{D}$ onto $O_{j}$, then $h_{j}^{\prime},\log
h_{j}^{\prime}\in W^{1,p}(\mathbb{D})$ for a suitable branch of the logarithm.
To track the dependence on $\left\|{O_{j}}\right\|_{\mathcal{B}_{p}}$, let us
outline their argument. For $\operatorname{{e}}^{\mathrm{i}t}\in\mathbb{T}$,
it is not hard to compute that

$\arg N_{O_{j}}(h_{j}(\operatorname{{e}}^{\mathrm{i}t}))=-\arg
h_{j}^{\prime}(\operatorname{{e}}^{\mathrm{i}t})+t$

so that $1+\left\|{O_{j}}\right\|_{\mathcal{B}_{p}}\sim\left\|{\arg
h_{j}^{\prime}}\right\|_{B^{1-\frac{1}{q}}_{q,q}(\mathbb{T})}$ for any $q>1$.
Furthermore, the Herglotz extension maps
$B^{1-\frac{1}{q}}_{q,q}(\mathbb{T})\to W^{1,q}(\mathbb{D})$
[triebel83]*Theorem 4.3.3, which together with the well-known Herglotz
representation [pommerenke-book]*Theorem 3.3.2, yields

(4.6) $\log
h_{j}^{\prime}=\log\left|{h_{j}^{\prime}(0)}\right|+H_{j},\qquad\left\|{H_{j}}\right\|_{W^{1,q}(\mathbb{D})}\lesssim
1+\left\|{O_{j}}\right\|_{\mathcal{B}_{q}}.$

Now $g$ can be factored as $h_{2}\circ h_{1}^{-1}$. By the chain rule and
inverse function theorem,

$\left[\log g^{\prime}\right]^{\prime}=\frac{\tilde{H}}{h_{1}^{\prime}}\circ
h_{1}^{-1},\qquad\tilde{H}=\left[\log
h_{2}^{\prime}\right]^{\prime}-\left[\log h_{1}^{\prime}\right]^{\prime}.$

So, changing variables

$\int_{O}\left|{\left[\log
g^{\prime}\right]^{\prime}}\right|^{q}=\int_{\mathbb{D}}\left|{\tilde{H}}\right|^{q}\left|{h_{1}^{\prime}}\right|^{2-q}.$

Because $O_{1}$ is Dini-smooth, $h_{1}^{\prime}$ is bounded above and below
[pommerenke-book]*Theorem 3.3.5, so the proof is concluded by appealing to
(4.6). ∎

Immediately from Lemma 4.7 and Theorem A, we obtain a quantitative version of
[astala-prats-saksman]*Theorem 1.1 in the Sobolev case.

######  Proposition 4.8.

Let $F\in W^{1,2}_{\mathrm{loc}}(\Omega)$ be a homeomorphism satisfying

$\overline{\partial}F(z)=\mu(z)\partial F(z),\qquad z\in\Omega,$

such that $F(\Omega)$ is also a bounded $\mathcal{B}_{p}$ domain. Then,

(4.7) $\displaystyle\left\|{D\log\partial
F}\right\|_{L^{p}(\Omega)}\leq\left[\left\|{F(\Omega)}\right\|_{\mathcal{B}_{p}}+\left(1+\left\|{\Omega}\right\|_{\mathcal{B}_{p}}\right)\left(1+\left\|{\mu}\right\|_{W^{1,p}(\Omega)}^{3}\right)\right]\left(1+\left\|{\mu}\right\|_{W^{1,r}(\Omega)}\right)^{1-\frac{2}{p}};$
(4.8) $\displaystyle\left\|{D\log\partial F}\right\|_{L^{2}(\Omega)}\lesssim
1+\left\|{F(\Omega)}\right\|_{\mathcal{B}_{2}}+\left\|{\Omega}\right\|_{\mathcal{B}_{2}}\left(1+\left\|{\mu}\right\|_{W^{1,r}(\Omega)}^{3}\right).$

###### Proof.

Let $E\mu$ be an extension of $\mu$ from $\Omega$ to $\mathbb{C}$ satisfying

$\left\|{E\mu}\right\|_{\infty}\leq\varkappa(k)<1,\qquad\left\|{E\mu}\right\|_{W^{1,q}(\mathbb{C})}\lesssim\left\|{\mu}\right\|_{W^{1,q}(\Omega)},\quad
2\leq q\leq p.$

The difficulty in constructing $E\mu$ is to guarantee the first property. To
do so, one needs only to modify the parameters in the usual first order
extension [evans]*Theorem 5.4.1. To demonstrate, extending over the line
$x_{n}=0$, one can use

$Eu(x_{1},\ldots,x_{n})=-\varepsilon
u\left(x_{1},\ldots,x_{n-1},-\tfrac{3}{2\varepsilon}x_{n}\right)+(1+\varepsilon)u\left(x_{1},\ldots,x_{n-1},-\tfrac{1}{2(1+\varepsilon)}x_{n}\right),$

where $\varepsilon>0$ is chosen small enough that
$k(1+2\varepsilon)=\varkappa(k)<1$. Composing with $\mathcal{C}^{1}$ boundary
parameterizations does not affect the $L^{\infty}$ norm of the extension and
will only add a constant to the relevant Sobolev norms.

Let $G$ be the principal solution to $\overline{\partial}G=(E\mu)\partial G$,
which has the well-known solution formula $G(z)=z+\mathcal{K}(\rho)$ where
$(I-(E\mu)\mathcal{S})\rho=E\mu$. By Theorem A,

(4.9) $\|G\|_{W^{2,s}(\mathbb{C})}\lesssim
1+\|E\mu\|_{W^{1,s}(\mathbb{C})}^{3}\lesssim
1+\|\mu\|_{W^{1,s}(\Omega)}^{3},\qquad s\in\\{r,p\\}.$

Since $\Omega$ is a $\mathcal{B}_{p}$ domain and $G\in W^{2,p}(\mathbb{C})$,
standard trace results [gagliardo, triebel83] together with (4.9) show that
$G(\Omega)$ is also a $\mathcal{B}_{p}$ domain and moreover the estimates

(4.10)
$\displaystyle\left\|{G(\Omega)}\right\|_{\mathcal{B}_{2}}\lesssim\left\|{\Omega}\right\|_{\mathcal{B}_{2}}\left(1+\left\|{\mu}\right\|_{W^{1,r}(\Omega)}^{3}\right),\qquad\left\|{G(\Omega)}\right\|_{\mathcal{B}_{p}}\lesssim\left\|{\Omega}\right\|_{\mathcal{B}_{p}}\left(1+\left\|{\mu}\right\|_{W^{1,p}(\Omega)}^{3}\right)$

hold. Introducing $g=F\circ G^{-1}$, $g$ is conformal on $G(\Omega)$ with
$g(G(\Omega))=F(\Omega)$. Furthermore,

(4.11) $\partial\log\partial F=\frac{\partial^{2}F}{\partial
F}=\frac{(g^{\prime}\circ G)\partial^{2}G}{(g^{\prime}\circ G)\partial
G}+\frac{(g^{\prime\prime}\circ G)(\partial G)^{2}}{(g^{\prime}\circ
G)\partial G}\eqqcolon F_{1}+F_{2}.$

Recall that $\partial G=\operatorname{{e}}^{\sigma}$, for $\sigma$ satisfying
$\overline{\partial}\sigma=(E\mu)\partial\sigma+\partial(E\mu)$. Noticing that
$F_{1}=\partial\sigma$ and applying (3.5), we have

(4.12)
$\left\|{F_{1}}\right\|_{L^{q}(\mathbb{C})}\leq\left\|{\sigma}\right\|_{W^{1,q}(\mathbb{C})}\lesssim\|E\mu\|_{W^{1,q}(\mathbb{C})}\lesssim\|\mu\|_{W^{1,q}(\Omega)},\qquad
2\leq q\leq p,$

which obeys the estimates (4.7) and (4.8). To estimate $F_{2}$, change
variables and use the quasiconformality of $G$ in (2.5) to obtain for any
$2\leq q\leq p$,

(4.13)
$\int_{\Omega}\left|F_{2}\right|^{q}\lesssim\int_{G(\Omega)}\left|\frac{g^{\prime\prime}}{g^{\prime}}\right|^{q}\left(|JG|\circ
G^{-1}\right)^{\frac{q}{2}-1}\leq\left\|{\left[\log
g^{\prime}\right]^{\prime}}\right\|_{L^{q}(G(\Omega))}^{q}\|JG\|_{\infty}^{\frac{q}{2}-1}.$

For $q\in\\{2,p\\}$, estimate the first factor by (4.5) from Lemma 4.7 and
(4.10). When $q=2$, the $JG$ term vanishes in (4.13), so (4.8) is established.
When $q=p$, estimate $JG$ by the Sobolev embedding and (4.9) with $s=r$. ∎

###  4.4. Proof of Lemma 4.2

By (1.11) and (4.3), the assumptions of this lemma imply $f\in
W^{2,p}(\Omega)$. This implies first that $\left|{Jf}\right|\in
L^{\infty}(\Omega)$ and second that $f(\Omega)$ is a $\mathcal{B}_{p}$ domain.
Therefore, by Lemma 4.3.i, the conclusion will follow if we can show
$\left|{Jf}\right|^{-1}\sim\left|{\partial f}\right|^{-1}\in L^{\infty}$. To
this end, by (4.7) in Proposition 4.8, $\log\partial f\in W^{1,p}(\Omega)$ and
hence

$\left|{\partial f}\right|^{-1}=\left|{\exp(-\log\partial
f)}\right|\lesssim\exp\left(\left|{\log\partial f}\right|\right)$

belongs to $L^{\infty}$ by Sobolev embedding. ∎

###  4.5. Proof of Theorem B

Let $g\in\mathcal{C}_{0}^{\infty}(\mathbb{C})$ and set
$w=\mathcal{K}(\mathbf{1}_{\overline{\Omega}}g)$ so that $w$ satisfies the
inhomogeneous Beltrami equation $\overline{\partial}w=\mu\partial w+h$ on
$\mathbb{C}$ with
$h=(I-\mu\mathcal{S}_{\Omega})(\mathbf{1}_{\overline{\Omega}}g)$. Notice that
$\overline{\partial}w=\mathbf{1}_{\overline{\Omega}}g\in\mathcal{C}^{\infty}(\overline{\Omega})$
and since $\Omega$ is a $\mathcal{B}_{p}$ domain, $\partial w$ and $h$ both
belong to $W^{1,p}(\Omega)$. The precise estimate we will prove is

(4.14)
$\displaystyle\left\|\overline{\partial}w\right\|_{W^{1,p}(\Omega)}\lesssim\mathrm{R}\,\mathrm{P}\left[\left\|{\mathcal{S}_{O}}\right\|_{\mathcal{L}(W^{1,p}(O,\omega))}+\mathrm{P}\left(1+\left\|{\Omega}\right\|_{\mathcal{B}_{p}}\right)\left\|{O}\right\|_{\mathcal{B}_{p}}\right]\|h\|_{W^{1,p}(\Omega)},$
(4.15)
$\displaystyle\mathrm{R}=\mathcal{E}\left(\left\|{\mu}\right\|_{W^{1,r}(\Omega)}\right),\qquad\mathrm{P}=\left\|{O}\right\|_{\mathcal{B}_{p}}+\left(1+\left\|{\Omega}\right\|_{\mathcal{B}_{p}}\right)\left(1+\left\|{\mu}\right\|_{W^{1,p}}^{3}\right).$

We will repeatedly use the facts that $\mathrm{R}^{2}\sim\mathrm{R}$ and that
$\left\|{O}\right\|_{\mathcal{B}_{2}},\left\|{\Omega}\right\|_{\mathcal{B}_{2}}\lesssim
1$ by virtue of the standing assumption that the domains are Dini-smooth. To
establish (4.14), let us normalize so that $\|h\|_{W^{1,p}(\Omega)}=1$. By
Sobolev embedding and interpolation, since $p>2$,

(4.16) $\|h\|_{L^{s}(\Omega)}\lesssim_{p,s}1,\qquad p\leq s\leq\infty.$

Defining $u=w\circ f^{-1}$, one can check

(4.17) $\begin{array}[]{l}\overline{\partial}w=(\partial u\circ
f)\overline{\partial}f+H,\\\\[5.69054pt] \partial w=(\partial u\circ
f)\partial f+\overline{\mu}H,\end{array}\qquad
H\coloneqq\frac{h}{1-|\mu|^{2}}=\left(\overline{\partial}u\circ
f\right)\overline{\partial f}.$

Let $q\in\left\\{\tfrac{pp^{\prime}}{2},p\right\\}$. Setting
$\sigma=\log\partial f$, by (4.7) we have the analogous estimates to (3.8),
and (3.11):

(4.18) $\displaystyle\left\|H\right\|_{W^{1,q}(\Omega)}$
$\displaystyle\lesssim 1+\|\mu\|_{W^{1,p}(\Omega)};$ (4.19)
$\displaystyle\left\|\left(\overline{\partial
f}\right)^{2}[D\overline{\partial}u\circ f]\right\|_{L^{q}(\Omega)}$
$\displaystyle\lesssim
1+\left\|\mu\right\|_{W^{1,p}(\Omega)}+\left\|\partial\sigma\right\|_{L^{p}(\Omega)}\lesssim\mathrm{R}\,\mathrm{P}.$

In light of the identity

(4.20) $D\overline{\partial}w=D(\partial u\circ
f)\overline{\partial}f+(\partial u\circ f)D\overline{\partial}f+DH,$

we want to obtain $L^{q}(\Omega)$ bounds for the first two terms in (4.20). To
handle the first term, notice that since $h$ is supported in
$\overline{\Omega}$, (4.17) implies that $\overline{\partial}u$ is supported
in $\overline{O}$. Therefore, $\mathbf{1}_{\overline{O}}\partial
u=\mathcal{S}_{O}\overline{\partial}u$ whence, following the same steps as
(3.13),

$\int_{\Omega}\left|D(\partial u\circ
f)\overline{\partial}f\right|^{q}\lesssim\int_{O}\left|D\mathcal{S}_{O}(\overline{\partial}u)\right|^{q}\left|Jf^{-1}\right|^{1-q}.$

Setting $\omega_{q}=\left|{Jf^{-1}}\right|^{1-q}$,

$\left(\int_{O}\left|D\mathcal{S}_{O}(\overline{\partial}u)\right|^{q}\left|Jf^{-1}\right|^{1-q}\right)^{1/q}\lesssim\left\|\mathcal{S}_{O}\right\|_{W^{1,q}(O,\omega_{q})}\left[\left\|D{\overline{\partial}}u\right\|_{L^{p}(O,\omega)}+\left\|{\overline{\partial}}u\right\|_{L^{p}(O,\omega)}\right].$

The first summand within the bracket is bounded by (4.19). Referring to
(4.17), the second summand is bounded by $\|h\partial f\|_{L^{p}(\Omega)}$.
However, $\partial f=1+\mathcal{S}(I-\mu\mathcal{S}_{\Omega})^{-1}\mu\in
L^{p}(\Omega)$ by Proposition 4.6 and the $L^{\infty}$-norm of $h$ is
controlled by (4.16). We will now have to consider the two cases
$q=\frac{pp^{\prime}}{2}$ and $q=p$ separately, so we summarize the proof thus
far by

(4.21)
$\|\overline{\partial}w\|_{W^{1,q}(\Omega)}\lesssim\mathrm{R}\,\mathrm{P}\left\|\mathcal{S}_{O}\right\|_{W^{1,r}(O,\omega_{r})}+\|(\partial
u\circ f)D\overline{\partial}f\|_{L^{q}(\Omega)}.$

Let us first consider $q=\frac{pp^{\prime}}{2}<p$. By Lemma 4.3.iii,

(4.22)
$\left\|\mathcal{S}_{O}\right\|_{W^{1,q}(O,\omega_{q})}\lesssim\mathrm{R}\left(1+\left\|{O}\right\|_{\mathcal{B}_{p}}\right),\qquad
q=\frac{pp^{\prime}}{2}.$

To estimate the final term in (4.21) in this case, define $\lambda\coloneqq
D\mu+\mu D\sigma$ which belongs to $L^{p}(\Omega)$ by (4.19) and satisfies
$D\overline{\partial}f=\lambda\partial f$. Taking
$s=\frac{pp^{\prime}}{2-p^{\prime}}<\infty$, applying Hölder’s inequality,
arguing as in the proof of Proposition 4.6 with (4.4), and recalling (4.7),

$\displaystyle\left\|(\partial u\circ f)\partial
f\cdot\lambda\right\|_{L^{q}(\Omega)}\lesssim\left\|(\partial u\circ
f)\partial
f\right\|_{L^{s}(\Omega)}\left\|\lambda\right\|_{L^{p}(\Omega)}\lesssim\mathrm{R}\|h\|_{L^{s}(\Omega)}\|\lambda\|_{L^{p}(\Omega)}\lesssim\mathrm{R}\,\mathrm{P}.$

Inserting this estimate into (4.21), using (4.22) and (4.3), we obtain

(4.23) $\|\partial
w\|_{W^{1,q}(\Omega)}\lesssim\left(1+\left\|{\Omega}\right\|_{\mathcal{B}_{q}}\right)\|\overline{\partial}w\|_{W^{1,q}(\Omega)}\lesssim\mathrm{R}\,\mathrm{P}\left(1+\left\|{\Omega}\right\|_{\mathcal{B}_{p}}\right)\left(1+\left\|{O}\right\|_{\mathcal{B}_{p}}\right),\qquad
q=\frac{pp^{\prime}}{2}.$

Since $\frac{pp^{\prime}}{2}>2$, by Sobolev embedding, (4.23) establishes an
upper bound for the $L^{\infty}(\Omega)$ norm of $\overline{\partial}w$.
Referring back to (4.17) and (4.16), (4.23) further implies

(4.24) $\|(\partial u\circ f)\partial
f\|_{L^{\infty}(\Omega)}\lesssim\mathrm{R}\,\mathrm{P}\left(1+\left\|{\Omega}\right\|_{\mathcal{B}_{p}}\right)\left(1+\left\|{O}\right\|_{\mathcal{B}_{p}}\right).$

Now we can pick up at (4.21) with $r=p$ and immediately estimate

$\|(\partial u\circ f)D\partial f\|_{L^{p}(\Omega)}\lesssim\|(\partial u\circ
f)\partial
f\|_{L^{\infty}(\Omega)}\|\lambda\|_{L^{p}(\Omega)}\lesssim\mathrm{R}\,\mathrm{P}^{2}\left(1+\left\|{\Omega}\right\|_{\mathcal{B}_{p}}\right)\left(1+\left\|{O}\right\|_{\mathcal{B}_{p}}\right)$

which clearly implies (4.14), concluding the proof.

##  5\. Muckenhoupt weights and Jacobians of Quasiconformal maps

In this section we prove Lemmata 2.2 and 4.5 concerning the Muckenhoupt weight
properties of $\left|{Jf}\right|$ and $\left|{Jf^{-1}}\right|$. The main tool
is the following critical Sobolev embedding Lemma of Moser-Trudinger type.

######  Lemma 5.1.

There exists an absolute constant $C$ such that for every
$\sigma:\mathbb{C}\to\mathbb{C}$ with $D\sigma\in L^{2}(\Omega)$,
$a\in\mathbb{R}$, and $1<p<\infty$,

$\sup_{Q\subset\mathbb{C}}\left\langle{\left|{\operatorname{{e}}^{a\sigma}}\right|}\right\rangle_{Q}\left\langle{\left|{\operatorname{{e}}^{\frac{-a\sigma}{p-1}}}\right|}\right\rangle_{\frac{1}{p-1},Q}\leq
C\exp\left(C\frac{|a|p}{p-1}\left\|{D\sigma}\right\|_{L^{2}}\right).$

###### Proof.

Using the Taylor formula, we can write, for any cube $Q\subset\mathbb{C}$ and
any $z\in Q$,

$|\sigma(z)-\sigma_{Q}|\lesssim\int_{Q}\frac{|D\sigma(w)|}{|w-z|}\,\mathrm{d}w,\quad\sigma_{Q}\coloneqq\frac{1}{|Q|}\int_{Q}\sigma(w)\,\mathrm{d}w.$

For any $p>2$, by Young’s inequality

$\int_{Q}\left|\int_{Q}\frac{|D\sigma(w)|}{|w-z|}\,\mathrm{d}w\right|^{p}\,dz\leq\|D\sigma\|_{L^{2}(Q)}^{p}\left(\int_{[-\ell(Q),\ell(Q)]^{2}}|z|^{-r}\,\mathrm{d}z\right)^{p/r},\quad\frac{1}{2}+\frac{1}{r}=\frac{1}{p}+1.$

Notice that by definition, $r<2$ so

$\int_{[-\ell(Q),\ell(Q)]^{2}}|z|^{-r}\,\mathrm{d}z\sim\frac{\ell(Q)^{2-r}}{2-r}.$

However, some calculations show that $2-r=\frac{4}{p+2}$ and $p/r=p/2+1$ so
that

$\frac{1}{|Q|}\int_{Q}|\sigma(z)-\sigma_{Q}|^{p}\,\mathrm{d}z\lesssim\|D\sigma\|_{L^{2}(Q)}^{p}\left(\frac{p+2}{4}\right)^{p/2+1}.$

For each $a>0$, we can now compute

(5.1)
$\displaystyle\frac{1}{|Q|}\int_{Q}\operatorname{{e}}^{a|\sigma(z)-\sigma_{Q}|}\,\mathrm{d}z$
$\displaystyle=\sum_{k=0}^{\infty}\frac{a^{k}}{k!}\frac{1}{|Q|}\int_{Q}|\sigma(z)-\sigma_{Q}|^{k}\,\mathrm{d}z\lesssim\sum_{k=0}^{\infty}(a\|D\sigma\|_{L^{2}})^{k}\frac{k^{k/2}}{k!}.$

A crude estimate for the power series is given by

$\sum_{k=0}^{\infty}A^{k}\frac{k^{k/2}}{k!}\lesssim\exp(CA^{2}),$

for some absolute constant $C$, by splitting into even and odd integers, using
Stirling’s approximation, and the crude estimate $(2k)!\geq(k!)^{2}$. So, for
any $a\in\mathbb{R}$ and $p>1$, by (5.1),

$\displaystyle\frac{1}{|Q|}\int_{Q}\left|{\operatorname{{e}}^{a\sigma}}\right|$
$\displaystyle\left(\frac{1}{|Q|}\int_{Q}\left|{\operatorname{{e}}^{-\frac{a}{p-1}\sigma}}\right|\right)^{p-1}$
$\displaystyle=\frac{1}{|Q|}\int_{Q}\left|{\operatorname{{e}}^{a(\sigma-\sigma_{Q})}}\right|\left(\frac{1}{|Q|}\int_{Q}\left|{\operatorname{{e}}^{-\frac{a}{p-1}(\sigma-\sigma_{Q})}}\right|\right)^{p-1}$
$\displaystyle\leq\frac{1}{|Q|}\int_{Q}\operatorname{{e}}^{|a|\cdot|\sigma-\sigma_{Q}|}\left(\frac{1}{|Q|}\int_{Q}\operatorname{{e}}^{\frac{|a|}{p-1}|\sigma-\sigma_{Q}|}\right)^{p-1}$
$\displaystyle\leq
C\exp\left(C|a|^{2}L^{2}\right)\exp\left(\frac{C|a|^{2}L^{2}}{p-1}\right).$

∎

###  5.1. Proof of Lemma 2.2

As in the proof of Theorem A, $\partial f=\operatorname{{e}}^{\sigma}$ where
$\sigma$ satisfies

$\overline{\partial}\sigma=\mu\partial v+\partial\mu,\qquad\mbox{on
}\mathbb{C}.$

Therefore,

$(I-\mu\mathcal{S}_{\Omega})\overline{\partial}\sigma=\partial\mu,\qquad\mbox{on
}\Omega.$

However, $\left\|{\mathcal{S}}\right\|_{\mathcal{L}(L^{2}(\mathbb{C}))}=1$
hence

$\left\|{\partial\mu}\right\|_{L^{2}(\mathbb{C})}=\left\|{(I-\mu\mathcal{S})\overline{\partial}\sigma}\right\|_{L^{2}(\mathbb{C})}\geq(1-k)\left\|{\overline{\partial}\sigma}\right\|_{L^{2}(\mathbb{C})}.$

Therefore, $\left\|{D\sigma}\right\|_{L^{2}(\mathbb{C})}\lesssim L$. Now, the
Jacobian takes the form

$|J{f}|=(1-|\mu|^{2})|\operatorname{{e}}^{2\sigma}|\sim\left|\operatorname{{e}}^{2\sigma}\right|,$

so Lemma 5.1 establishes that $\left|{Jf}\right|^{a}$ belongs to
$\mathrm{A}_{p}(\mathbb{C})$ with the estimate

(5.2) $\left[\left|{Jf}\right|^{a}\right]_{\mathrm{A}_{p}(\mathbb{C})}\leq
C\exp\left(\frac{Cp|a|^{2}L^{2}}{p-1}\right).$

We next demonstrate that (5.2) implies $\left|{Jf}\right|^{a}$ belongs to
every $\mathrm{RH}_{s}(\mathbb{C})$. Indeed, for each $1<s<\infty$, by
Hölder’s inequality, for each cube $Q\subset\mathbb{C}$, setting
$p=\frac{s+1}{s}$,

$|Q|^{p}=\left(\int_{Q}\left|{Jf}\right|^{\frac{a}{p}}\left|{Jf}\right|^{-\frac{a}{p}}\right)^{p}\leq\left(\int\left|{Jf}\right|^{a}\right)\left(\int\left|{Jf}\right|^{-as}\right)^{p-1}$
$\leq\left(\int\left|{Jf}\right|^{a}\right)\left(\left[\left|{Jf}\right|^{as}\right]_{\mathrm{A}_{2}(\mathbb{C})}|Q|^{2}\left(\int\left|{Jf}\right|^{as}\right)\right)^{p-1}.$

Therefore, applying (5.2) and rearranging the above display,

(5.3)
$\left[\left|{Jf}\right|^{a}\right]_{\mathrm{RH}_{s}(\mathbb{C})}\coloneqq\sup_{Q\
\mathrm{cube}}\left\langle{\left|{Jf}\right|^{a}}\right\rangle_{s,Q}\left\langle{\left|{Jf}\right|^{a}}\right\rangle_{Q}^{-1}\leq
C\exp\left(C|a|^{2}sL^{2}\right).$

Now, to handle $\left|{J{f^{-1}}}\right|^{a}$ we use the identity, for any
$t\in\mathbb{R}$,

(5.4) $\int_{Q}\left|{J{f^{-1}}}\right|^{t}=\int_{f^{-1}(Q)}|J{f^{-1}}\circ
f|^{t}|Jf|=\int_{f^{-1}(Q)}|Jf|^{1-t}.$

Since $f$ is quasiconformal, $f$ and $f^{-1}$ are quasisymmetric (see e.g.
[astala-book]*Corollary 3.10.4). Therefore, there exist cubes $R,P$ such that
$R\subset f^{-1}(Q)\subset P$ with $|R|\sim|P|$. So, we claim that for every
$t\in\mathbb{R}$,

(5.5)
$\frac{1}{|P|^{t}}\left(\int_{P}|Jf|\right)^{t-1}\int_{P}|Jf|^{1-t}\lesssim\left\\{\begin{array}[]{cc}\exp\left(C(1-t)^{2}L^{2}\right),&1-t>1;\\\
1,&0\leq 1-t\leq 1;\\\
\exp\left(Ct(t-1)L^{2}\right),&1-t<0.\end{array}\right.$

When $1-t\geq 1$, (5.5) is simply the $\mathrm{RH}_{1-t}$ property of
$\left|{Jf}\right|$ from (5.3). If $0\leq 1-t\leq 1$, then (5.5) follows by
Hölder’s inequality. Finally, if $t>1$, apply the $\mathrm{A}_{p}(\Omega)$
condition for $|Jf|$ with $1-t=\frac{-1}{p-1}$ from (5.2). Thus (5.5) is
established. Finally, for any $a\in\mathbb{R}$ and $1<p<\infty$, applying
(5.4) followed by (5.5) with $t=a$ and $t=-\frac{a}{p-1}$,

$\left(\frac{1}{|Q|}\int_{Q}\left|{J{f^{-1}}}\right|^{a}\right)\left(\frac{1}{|Q|}\int_{Q}\left|{J{f^{-1}}}\right|^{-\frac{a}{p-1}}\right)^{p-1}\lesssim\frac{|P|^{a}}{|Q|^{a}}\left(\frac{|P|^{-\frac{a}{p-1}}}{|Q|^{-\frac{a}{p-1}}}\right)^{p-1}=1,$

with the appropriate implicit constant according to (5.5).

###  5.2. Proof of Lemma 4.5

By Proposition 4.8 $\log\partial F\in W^{1,2}(\Omega)$. Since $\Omega$ is
$\mathcal{B}_{p}$ it is also $\mathcal{C}^{1}$ so there exists an extension
$\rho\in W^{1,2}(\mathbb{C})$. By Lemma 5.1
$\left|{\operatorname{{e}}^{a\rho}}\right|\in\mathrm{A}_{q}(\mathbb{C})$. The
same argument used to extend to $JF^{-1}$ in the proof of Lemma 2.2 in §5.1
shows that $\left|{JF^{-1}}\right|^{a}$ belongs to the following
$\mathrm{A}_{p}(O)$ class for $O=F(\Omega)$, defined by finiteness of the
characteristic

(5.6)
$\left[\left|{JF^{-1}}\right|^{a}\right]_{\mathrm{A}_{p}(O)}=\sup_{Q\subset\mathbb{C}}\left\langle{\mathbf{1}_{O}\left|{JF^{-1}}\right|^{a}}\right\rangle_{Q}\left\langle{\mathbf{1}_{O}\left|{JF^{-1}}\right|^{-a}}\right\rangle_{\frac{1}{p-1},Q}.$

An unpublished result of Wolff [wolff-Ap] states that for any measurable set
$O$, if $\omega^{1+\varepsilon}\in\mathrm{A}_{p}(O)$ then there exists
$\upsilon\in\mathrm{A}_{p}(\mathbb{C})$ such that $\upsilon=\omega$ on $O$.
See [gc-book]*Theorem IV.5.5 for a proof, [diplinio23domains]*Lemma 3.8 for a
quantitative version, and [holden, kurki22] for two related results. Because
(5.6) holds for arbitrary $a$, we can apply Wolff’s result to obtain the
promised $\upsilon$.

###  5.3. Explicit dependence in special cases

Let us compute this precise dependence in the relevant special cases to
establish (2.6) and (2.7).

Let $1<p<\infty$ and denote by $p^{\prime}=\frac{p}{p-1}$ the Hölder
conjugate. The computation
$(1-\frac{p}{2})\frac{-1}{p-1}=1-\frac{p^{\prime}}{2}$ reveals the symmetry

(5.7)
$\left[\left|{Jf^{-1}}\right|^{1-\frac{p}{2}}\right]_{\mathrm{A}_{p}(\mathbb{C})}^{\max\\{1,\frac{1}{p-1}\\}}=\left[\left|{Jf^{-1}}\right|^{1-\frac{p^{\prime}}{2}}\right]_{\mathrm{A}_{p^{\prime}}(\mathbb{C})}^{\max\\{1,\frac{1}{p^{\prime}-1}\\}}.$

Therefore, we can assume $p>2$ so that $\max\\{1,\frac{1}{p-1}\\}=1$,
$1-(1-\frac{p}{2})>1$, and $\frac{1}{2}\leq 1-(1-\frac{p^{\prime}}{2})\leq 1$
in order to compute

$\left[\left|{Jf^{-1}}\right|^{1-\frac{p}{2}}\right]_{\mathrm{A}_{p}(\mathbb{C})}^{\max\\{1,\frac{1}{p-1}\\}}\leq
C\exp\left(Cp^{2}L^{2}\right).$

Therefore, (2.6) follows by the symmetry (5.7). To prove (2.7), let
$1<p<\infty$, so that $1-(1-r)>1$, and hence

$\left[\left|{Jf^{-1}}\right|^{1-r}\right]_{\mathrm{A}_{r}(\mathbb{C})}^{\max\left\\{1,\frac{1}{r-1}\right\\}}\leq
C\exp\left(Cr^{2}\max\left\\{1,\tfrac{1}{r-1}\right\\}L^{2}\right)\leq
C\exp\left(C_{1}\max\left\\{r^{2},\tfrac{1}{r-1}\right\\}L^{2}\right).$

## References