# The $p$-Weil–Petersson Teichmüller space
and the quasiconformal extension of curves

Huaying Wei  Department of Mathematics and Statistics, Jiangsu Normal
University Xuzhou 221116, PR China Department of Mathematics, School of
Education, Waseda University Shinjuku, Tokyo 169-8050, Japan<EMAIL_ADDRESS>and  Katsuhiko Matsuzaki  Department of Mathematics, School of Education,
Waseda University Shinjuku, Tokyo 169-8050, Japan<EMAIL_ADDRESS>
###### Abstract.

We consider the correspondence between the space of $p$-Weil–Petersson curves
$\gamma$ on the plane and the $p$-Besov space of $u=\log\gamma^{\prime}$ on
the real line for $p>1$. We prove that the variant of the Beurling–Ahlfors
extension defined by using the heat kernel yields a holomorphic map for $u$ on
a domain of the $p$-Besov space to the space of $p$-integrable Beltrami
coefficients. This in particular gives a global real-analytic section for the
Teichmüller projection from the space of $p$-integrable Beltrami coefficients
to the $p$-Weil–Petersson Teichmüller space.

###### Key words and phrases:

Weil–Petersson Teichmüller space, Beurling–Ahlfors extension, integrable
Beltrami coefficients, global section of Teichmüller projection,
$A_{\infty}$-weights, BMO functions, Besov space

###### 2020 Mathematics Subject Classification:

Primary 32G15, 30C62, 30H25, 30H35; Secondary 42A45, 26A46, 46G20

Research supported by Japan Society for the Promotion of Science (KAKENHI
18H01125 and 21F20027).

##  1\. Introduction

###  1.1. Background on the Weil–Petersson class and its generalization

An increasing homeomorphism $h$ of the real line $\mathbb{R}$ onto itself
belongs to the $2$-Weil–Petersson class on $\mathbb{R}$ (nowadays this is
usually called the Weil–Petersson class in the literature, but we add the
index $2$ here for its generalization) if, by definition, it can be extended
to a quasiconformal homeomorphism of the upper half-plane $\mathbb{U}$ onto
itself whose Beltrami coefficient is $2$-integrable in the hyperbolic metric
(the index $2$ actually comes from here). Let $W_{2}(\mathbb{R})$ be the set
of all normalized $2$-Weil–Petersson class homeomorphisms on $\mathbb{R}$
which keeps $0$, $1$ and $\infty$ fixed. This is the real model of the
$2$-Weil–Petersson Teichmüller space.

A study of the $2$-Weil–Petersson Teichmüller space was initiated by Cui [6]
where he gave some characterizations of the $2$-Weil–Petersson class and
showed that this is the completion of the set of all normalized
$C^{\infty}$-diffeomorphisms under the Weil–Petersson metric. Later, Takhtajan
and Teo [27] studied systematically the $2$-Weil–Petersson Teichmüller space.
They proved that it is the connected component of the identity in the
universal Teichmüller space viewed as a complex Hilbert manifold and
established many other equivalent characterizations of the $2$-Weil–Petersson
Teichmüller space. Moreover, they proposed a problem for characterizing
intrinsically the elements in the $2$-Weil–Petersson class without using
quasiconformal extension. Then, Shen and his coauthors [22, 23, 24] did among
other work solve this problem by characterizing the $2$-Weil–Petersson class
directly in terms of the fractional dimensional Sobolev space
$H_{\mathbb{R}}^{1/2}$ of real-valued functions. Recently, Bishop [3] gave
lots of new characterizations of the $2$-Weil–Petersson class which link to
various concepts in geometric measure theory and hyperbolic geometry. In
addition, this work has motivations from string theory and SLE theory, and in
reverse, it has applications to these theories (see [3, 29] and references
therein).

For $p>1$, the normalized $p$-Weil–Petersson class $W_{p}(\mathbb{R})$ can be
defined similarly by just changing $2$-integrability into $p$-integrability.
The generalization apart from the case of $p=2$ is natural, and several works
have been done in this direction (see [13, 16, 17, 26, 33]). These
generalizations are usually straightforward, but there are really a few
crucial differences in the arguments between the cases of $p=2$ and $p\neq 2$.

In the present paper, we study the Weil–Petersson theory of the universal
Teichmüller space, and mainly consider a $p$-Weil–Petersson curve from
$\mathbb{R}$ into the whole plane $\mathbb{C}$, which is the generalization of
a $p$-Weil–Petersson class homeomorphism. Here, by saying a curve, we include
its parametrization, which has more information than just the image of a
curve. We will prove the existence of a canonical quasiconformal extension of
a Weil–Petersson curve to $\mathbb{C}$ using the variant of the
Beurling–Ahlfors extension by the heat kernel introduced in Fefferman, Kenig
and Pipher [10]. Its detailed exposition is in our previous paper [30]. Then,
by the restriction of this quasiconformal extension operator to the
$p$-Weil–Petersson class $W_{p}(\mathbb{R})$, we can obtain novel results and
also the reformation of the existing results on the $p$-Weil–Petersson
Teichmüller space. It is worthwhile to mention that our new approach is
natural for the investigation of absolutely continuous curves on $\mathbb{R}$
induced by quasiconformal mappings of $\mathbb{C}$, and can be used for other
problems.

###  1.2. Parametrization of Weil–Petersson curves

Taking the space ${\mathcal{M}}_{p}(\mathbb{U})$ of the Beltrami coefficients
that are $p$-integrable with respect to the hyperbolic metric on $\mathbb{U}$,
the $p$-Weil–Petersson Teichmüller space $T_{p}(\mathbb{U})$ is given by the
Teichmüller projection $\pi:{\mathcal{M}}_{p}(\mathbb{U})\to
T_{p}(\mathbb{U})$. It is known that $T_{p}(\mathbb{U})$ has a unique complex
Banach manifold structure via the Bers embedding through the Schwarzian
derivative (or via the logarithmic derivative embedding) such that the
Teichmüller projection $\pi$ is holomorphic with local holomorphic inverse for
$p\geq 2$ (see [26, 31]). This can be considered on the lower half-plane
$\mathbb{L}$ in the same way.

A $p$-Weil–Petersson curve $\gamma:\mathbb{R}\to\mathbb{C}$ is the restriction
of a quasiconformal homeomorphism of $\mathbb{C}$ whose complex dilatation on
$\mathbb{U}$ and $\mathbb{L}$ belongs to ${\mathcal{M}}_{p}(\mathbb{U})$ and
${\mathcal{M}}_{p}(\mathbb{L})$, respectively. We impose the normalization
$\gamma(0)=0$ and $\gamma(1)=1$ and $\gamma(\infty)=\infty$ on every
$p$-Weil–Petersson curve $\gamma$. Let ${\rm WPC}_{p}$ be the set of all
normalized $p$-Weil–Petersson curves. Hence, the space ${\rm WPC}_{p}$ can be
understood in the spirit of the Bers simultaneous uniformization so that ${\rm
WPC}_{p}$ is identified with the product of the $p$-Weil–Petersson Teichmüller
spaces $T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$, which endows ${\rm
WPC}_{p}$ with the product complex Banach manifold structure. In our recent
paper, we have proved the following.

######  Theorem 1.1 ([31]).

For any normalized $p$-Weil–Petersson curve $\gamma\in{\rm WPC}_{p}$, the
logarithm of the derivative $\log\gamma^{\prime}$ belongs to the $p$-Besov
space $B_{p}(\mathbb{R})$. Moreover, this correspondence $L:{\rm WPC}_{p}\to
B_{p}(\mathbb{R})$ is a biholomorphic homeomorphism onto its image.

In the present paper, we will derive the inverse of $L$ in Theorem 1.1 by
constructing the quasiconformal mappings explicitly. Precisely, we show that
if $\log\gamma^{\prime}$ is in some neighborhood
$U(B_{p}^{\mathbb{R}}(\mathbb{R}))$ of the real Banach subspace
$B_{p}^{\mathbb{R}}(\mathbb{R})$ consisting of all real-valued $p$-Besov
functions, then the variant of the Beurling–Ahlfors extension by the heat
kernel of $\gamma$ to both $\mathbb{U}$ and $\mathbb{L}$ has complex
dilatations in $\mathcal{M}_{p}(\mathbb{U})$ and
$\mathcal{M}_{p}(\mathbb{L})$. Moreover, this correspondence
$\widetilde{\Lambda}$ is holomorphic. Then, if we further take the composition
with the product of the Teichmüller projections
$\widetilde{\pi}:\mathcal{M}_{p}(\mathbb{U})\times\mathcal{M}_{p}(\mathbb{L})\to
T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$, this gives the inverse of $L:{\rm
WPC}_{p}\cong T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})\to B_{p}(\mathbb{R})$
on the neighborhood $U(B_{p}^{\mathbb{R}}(\mathbb{R}))$.

######  Theorem 1.2 (see Theorem 4.5).

There is a holomorphic map

$\widetilde{\Lambda}:U(B_{p}^{\mathbb{R}}(\mathbb{R}))\to{\mathcal{M}}_{p}(\mathbb{U})\times{\mathcal{M}}_{p}(\mathbb{L})$

defined on a neighborhood $U(B_{p}^{\mathbb{R}}(\mathbb{R}))\subset
B_{p}(\mathbb{R})$ of $B_{p}^{\mathbb{R}}(\mathbb{R})$ such that
$L\circ\widetilde{\pi}\circ\widetilde{\Lambda}$ is the identity on
$U(B_{p}^{\mathbb{R}}(\mathbb{R}))$.

The point of this consequence is that a single formula of the the variant of
the Beurling–Ahlfors extension by the heat kernel can be applied to all
complex-valued functions in some neighborhood of the real-valued $p$-Besov
functions. Other quasiconformal extensions used in the literature are not
known to have this property.

###  1.3. Applications to the Weil–Petersson class

We apply Theorem 1.2 restricted to $B_{p}^{\mathbb{R}}(\mathbb{R})$ itself in
$U(B_{p}^{\mathbb{R}}(\mathbb{R}))$ (or restricted to $W_{p}(\mathbb{R})$ in
${\rm WPC}_{p}$). This produces new assertions on the $p$-Weil–Petersson
Teichmüller space.

The first implication of Theorem 1.2 is the following, which is just a special
case of this theorem.

######  Corollary 1.3.

The holomorphic map $\widetilde{\Lambda}$ sends $u\in
B_{p}^{\mathbb{R}}(\mathbb{R})$ to a symmetric pair of Beltrami coefficients
$\left(\mu_{u}(z),\,\overline{\mu_{u}(\bar{z})}\right)\in{\mathcal{M}}_{p}(\mathbb{U})\times{\mathcal{M}}_{p}(\mathbb{L})$,
and hence the normalized curve
$\gamma_{u}(x)=\left(\int_{0}^{1}e^{u(t)}dt\right)^{-1}\int_{0}^{x}e^{u(t)}dt$
belongs to the $p$-Weil–Petersson class $W_{p}(\mathbb{R})$. Moreover, the
correspondence $B_{p}^{\mathbb{R}}(\mathbb{R})\to W_{p}(\mathbb{R})$ given by
$u\mapsto\gamma_{u}$ and its inverse are real-analytic homeomorphisms.

The statements that $\gamma_{u}$ belongs to $W_{2}(\mathbb{R})$ if $u\in
B_{2}^{\mathbb{R}}(\mathbb{R})$ and that $B_{2}^{\mathbb{R}}(\mathbb{R})\to
W_{2}(\mathbb{R})$ (and its inverse) is real-analytic were proved in Shen and
Tang [23] by using a modified Beurling–Ahlfors extension due to Semmes [21]
which can work only for such $u$ with small norm. In this case, the
quasiconformal extensions after decomposing $\gamma_{u}$ into small norm
pieces and then the composition of such extensions are required. On the
contrary, our extension has a better property (no norm assumption on $u$ is
needed) and can be applied for any $p>1$. Moreover, as the advantage of the
one-time extension by a single formula, it holds several desirable properties
of its complex dilatation, and also it yields the next.

The second implication of Theorem 1.2 is that the variant of the
Beurling–Ahlfors extension by the heat kernel yields a global real-analytic
section $\Lambda\circ L$ for the $p$-Weil–Petersson Teichmüller space
$T_{p}\cong W_{p}(\mathbb{R})\subset{\rm WPC}_{p}$, where $\Lambda$ is the
diagonal reduction of $\widetilde{\Lambda}$.

######  Corollary 1.4.

Under the identification of $W_{p}(\mathbb{R})$ with $T_{p}$, the map

$\Lambda\circ L|_{W_{p}(\mathbb{R})}:T_{p}\to{\mathcal{M}}_{p}$

is a global real-analytic section for the Teichmüller projection
$\pi:{\mathcal{M}}_{p}\to T_{p}$.

For the universal Teichmüller space $T$ and its subspaces invariant under
Fuchsian groups, the Douady–Earle extension in [8] gives a global real-
analytic section for the Teichmüller projection. Here, the result of Corollary
1.4 is a counterpart for $T_{p}$. From Corollary 1.4, we also see that $T_{p}$
is contractible since ${\mathcal{M}}_{p}$ is contractible. A contraction
$\phi:T_{p}\times[0,1]\to T_{p}$ is given explicitly by
$\phi(h,t)=\pi\left((1-t)\Lambda\circ L|_{W_{p}(\mathbb{R})}(h)\right)$. The
holomorphic contractibility of $T_{2}$, which means that the contraction
$\phi(\cdot,t)$ is holomorphic for each fixed $t\in[0,1]$, was obtained by Fan
and Hu [9] though this does not imply the existence of a global holomorphic
section for $\pi$.

###  1.4. Plan of this paper

We end this introduction section with the organization of the paper. In
Section 2, we recall definitions and properties of the BMO space, Muckenhoupt
weights, and the Besov space, and prepare several basic results for later use.
In Section 3, we give a detailed exposition on the variant of the Beurling-
Ahlfors extension by the heat kernel for complex-valued BMO functions. This
extension plays an important role in the proof of our main theorem (Theorem
4.5). Section 4 is devoted to this proof and its consequences as we described
above. In Section 5 as an appendix, we show that our extension translated to
the setting of the unit circle also yields the desired quasiconformal
extension to the unit disk.

##  2\. Preliminaries on BMO, ${\rm A}_{\infty}$-weights, and the Besov space

A locally integrable complex-valued function $u$ on $\mathbb{R}$ is of BMO if

$\|u\|_{*}=\sup_{I\subset\mathbb{R}}\frac{1}{|I|}\int_{I}|u(x)-u_{I}|dx<\infty,$

where the supremum is taken over all bounded intervals $I$ on $\mathbb{R}$ and
$u_{I}$ denotes the integral mean of $u$ over $I$. The set of all BMO
functions on $\mathbb{R}$ is denoted by ${\rm BMO}(\mathbb{R})$. This is
regarded as a Banach space with norm $\|\cdot\|_{*}$ by ignoring the
difference of constant functions. It is said that $u\in{\rm BMO}(\mathbb{R})$
is of VMO if

$\lim_{|I|\to 0}\frac{1}{|I|}\int_{I}|u(x)-u_{I}|dx=0,$

and the set of all such functions is denoted by ${\rm VMO}(\mathbb{R})$. This
is a closed subspace of ${\rm BMO}(\mathbb{R})$. The John–Nirenberg inequality
for BMO functions (see [11, VI.2], [25, IV.1.3]) asserts that there exists two
universal positive constants $C_{0}$ and $C_{JN}$ such that for any BMO
function $u$, any bounded interval $I$ of $\mathbb{R}$, and any $\lambda>0$,
it holds that

$\frac{1}{|I|}|\\{t\in I:|u(t)-u_{I}|\geq\lambda\\}|\leq
C_{0}\exp\left(\frac{-C_{JN}\lambda}{\|u\|_{*}}\right).$ (1)

A locally integrable non-negative measurable function $\omega\geq 0$ on
$\mathbb{R}$ is called a weight. We say that $\omega$ is an $A_{p}$-weight of
Muckenhoupt [18] for $p>1$ if there exists a constant $C_{p}(\omega)\geq 1$
such that

$\left(\frac{1}{|I|}\int_{I}\omega(x)dx\right)\left(\frac{1}{|I|}\int_{I}\left(\frac{1}{\omega(x)}\right)^{\frac{1}{p-1}}dx\right)^{p-1}\leq
C_{p}(\omega)$ (2)

for any bounded interval $I\subset\mathbb{R}$. We call the optimal value of
such $C_{p}(\omega)$ the $A_{p}$-constant for $\omega$. We define $\omega$ to
be an $A_{\infty}$-weight if $\omega$ is an $A_{p}$-weight for some $p>1$,
that is, $A_{\infty}=\bigcup_{p>1}A_{p}$. It is known that $\omega$ is an
$A_{\infty}$-weight if and only if there are positive constants
$\alpha(\omega)$, $K(\omega)>0$ such that

$\frac{\int_{E}\omega(x)dx}{\int_{I}\omega(x)dx}\leq
K(\omega)\left(\frac{|E|}{|I|}\right)^{\alpha(\omega)}$ (3)

for any bounded interval $I\subset\mathbb{R}$ and for any measurable subset
$E\subset I$ (see [5, Theorem V] and [11, Lemma VI.6.11]).

The Jensen inequality implies that

$\exp\left(\frac{1}{|I|}\int_{I}\log\omega(x)dx\right)\leq\frac{1}{|I|}\int_{I}\omega(x)dx.$
(4)

Another characterization of $A_{\infty}$-weights can be given by the inverse
Jensen inequality. Namely, $\omega\geq 0$ belongs to the class of
$A_{\infty}$-weights if and only if there exists a constant
$C_{\infty}(\omega)\geq 1$ such that

$\frac{1}{|I|}\int_{I}\omega(x)dx\leq
C_{\infty}(\omega)\exp\left(\frac{1}{|I|}\int_{I}\log\omega(x)dx\right)$ (5)

for every bounded interval $I\subset\mathbb{R}$ (see [14]). We call the
optimal value of such $C_{\infty}(\omega)$ the $A_{\infty}$-constant for
$\omega$. If $\omega$ is an $A_{p}$-weight, then $C_{\infty}(\omega)\leq
C_{p}(\omega)$ by the Jensen inequality. If $\omega$ is an
$A_{\infty}$-weight, the constants $\alpha(\omega)$ and $K(\omega)$ in (3) are
estimated by $C_{\infty}(\omega)$ as is shown in [14, Theorem 1], and
$C_{p}(\omega)$ and $p$ are estimated by $\alpha(\omega)$ and $K(\omega)$ (see
[5, Section 3]). One can also refer to [25, p.218] for these implications.

For a convenience of reference later, we verify inequality (5) showing the
dependence of $C_{\infty}(\omega)$ on $\omega$ when $\|\log\omega\|_{*}$ is
sufficiently small. In particular, $\omega$ is an $A_{\infty}$-weight in this
case. Conversely, for any $A_{\infty}$-weight $\omega$, we have
$\log\omega\in\rm BMO(\mathbb{R})$ (see [11, Lemma VI.6.5]).

######  Proposition 2.1.

Suppose that a weight $\omega\geq 0$ satisfies $\log\omega\in{\rm
BMO}(\mathbb{R})$. If the BMO norm $\|\log\omega\|_{*}$ is less than the
constant $C_{JN}$, then $\omega$ is in $A_{2}\subset A_{\infty}$ and the
$A_{2}$\- and $A_{\infty}$-constants depend only on $\|\log\omega\|_{*}$ and
tend to $1$ as $\|\log\omega\|_{*}\to 0$.

###### Proof.

Let $u=\log\omega\in{\rm BMO}(\mathbb{R})$. For any bounded interval
$I\subset\mathbb{R}$, the John–Nirenberg inequality (1) yields that

$\begin{split}\frac{1}{|I|}\int_{I}e^{|u(x)-u_{I}|}dx&=\int_{0}^{\infty}\frac{1}{|I|}|\\{x\in
I:|u(x)-u_{I}|>\lambda\\}|e^{\lambda}d\lambda+1\\\ &\leq
C_{0}\int_{0}^{\infty}\exp\left(\frac{-C_{JN}\lambda}{\|u\|_{*}}\right)e^{\lambda}d\lambda+1\\\
&=\frac{C_{0}\|u\|_{*}}{C_{JN}-\|u\|_{*}}+1\end{split}$ (6)

when $\|u\|_{*}<C_{JN}$. We set the right side of the above inequality as
$C(\omega)^{1/2}\geq 1$, which tends to $1$ as $\|u\|_{*}\to 0$. From this
inequality, we have

$\frac{1}{|I|}\int_{I}e^{u(x)-u_{I}}dx\leq
C(\omega)^{1/2};\quad\frac{1}{|I|}\int_{I}e^{u_{I}-u(x)}dx\leq
C(\omega)^{1/2}.$

Hence, we see that $\omega$ is an $A_{2}$-weight by

$\begin{split}\left(\frac{1}{|I|}\int_{I}\omega(x)dx\right)\left(\frac{1}{|I|}\int_{I}\frac{1}{\omega(x)}dx\right)&=\left(\frac{1}{|I|}\int_{I}e^{u(x)}dx\right)\left(\frac{1}{|I|}\int_{I}e^{-u(x)}dx\right)\\\
&=\left(\frac{1}{|I|}\int_{I}e^{u(x)-u_{I}}dx\right)\left(\frac{1}{|I|}\int_{I}e^{u_{I}-u(x)}dx\right)\leq
C(\omega).\end{split}$

Moreover, the Jensen inequality (4) implies that

$\frac{1}{|I|}\int_{I}\frac{1}{\omega(x)}dx\geq\exp\left(\frac{-1}{|I|}\int_{I}\log\omega(x)dx\right),$

which shows that inequality (5) is satisfied for the constant $C(\omega)$. ∎

The $p$-Besov space $B_{p}(\mathbb{R})$ for $p>1$ is the set of all measurable
complex-valued functions $u$ on $\mathbb{R}$ that satisfy

$\|u\|^{p}_{B_{p}}=\int_{-\infty}^{\infty}\\!\int_{-\infty}^{\infty}\frac{|u(t)-u(s)|^{p}}{|t-s|^{2}}dsdt<\infty.$

For the case of $p=2$, $B_{2}(\mathbb{R})$ coincides with the Sobolev space
$H^{1/2}(\mathbb{R})$. It is easy to see that if $u\in B_{p}(\mathbb{R})$ then
$|u|,{\rm Re}\,u,{\rm Im}\,u\in B_{p}^{\mathbb{R}}(\mathbb{R})$. Here,
$B_{p}^{\mathbb{R}}(\mathbb{R})$ denotes the set of all real-valued $p$-Besov
functions. As in the case of BMO functions, $B_{p}(\mathbb{R})$ can be
regarded as a Banach space with norm $\|\cdot\|_{B_{p}}$ by modulo of constant
functions. In other words, we regard $B_{p}(\mathbb{R})$ as a homogeneous
Besov space, which is often denoted by $\dot{B}_{p}(\mathbb{R})$ in the
literature.

The following relation between $B_{p}(\mathbb{R})$ and ${\rm VMO}(\mathbb{R})$
is important throughout this paper. We can also find this for $p=2$ in [24,
Section 3].

######  Proposition 2.2.

The inclusion $B_{p}(\mathbb{R})\subset{\rm VMO}(\mathbb{R})$ holds. Moreover,
$\|u\|_{*}\leq\|u\|_{B_{p}}$ for every $u\in B_{p}(\mathbb{R})$, and in
particular, the inclusion map is continuous.

###### Proof.

Let $I\subset\mathbb{R}$ be any bounded interval. Then,

$\displaystyle\frac{1}{|I|}\int_{I}|u(t)-u_{I}|dt$
$\displaystyle=\frac{1}{|I|}\int_{I}\left|u(t)-\frac{1}{|I|}\int_{I}u(s)ds\right|dt$
$\displaystyle\leq\frac{1}{|I|^{2}}\int_{I}\\!\int_{I}|u(t)-u(s)|dsdt$
$\displaystyle\leq\left(\frac{1}{|I|^{2}}\int_{I}\\!\int_{I}|u(t)-u(s)|^{p}dsdt\right)^{1/p}\leq\left(\int_{I}\\!\int_{I}\frac{|u(t)-u(s)|^{p}}{|t-s|^{2}}dsdt\right)^{1/p}.$

This implies that $\|u\|_{*}\leq\|u\|_{B_{p}}$. If $u\in B_{p}(\mathbb{R})$,
then $|u(t)-u(s)|^{p}/|t-s|^{2}$ is integrable on $\mathbb{R}^{2}$. Hence, for
any $\varepsilon>0$, there is some $\delta>0$ such that if $|I|<\delta$ then
its integral over $I\times I$ is less than $\varepsilon$. This shows that
$u\in{\rm VMO}(\mathbb{R})$. ∎

Moreover, we see that each element of $B_{p}^{\mathbb{R}}(\mathbb{R})$
corresponds to an $A_{\infty}$-weight.

######  Proposition 2.3.

If a weight $\omega\geq 0$ satisfies that $\log\omega\in
B_{p}^{\mathbb{R}}(\mathbb{R})$, then $\log\omega$ is in the closure of
$L^{\infty}(\mathbb{R})$ in the BMO norm. In particular, $\omega\in
A_{2}\subset A_{\infty}$.

###### Proof.

Let $u=\log\omega$. For any $N>0$, we set
$u_{N}(x)=\max\\{\min\\{u(x),N\\},-N\\}$, which belongs to
$L^{\infty}(\mathbb{R})$. We can easily check that

$|(u-u_{N})(s)-(u-u_{N})(t)|\leq|u(s)-u(t)|$

for almost all $(s,t)\in\mathbb{R}^{2}$. Then, the dominated convergence
theorem implies that

$\lim_{N\to\infty}\|u-u_{N}\|^{p}_{B_{p}}=\lim_{N\to\infty}\int_{-\infty}^{\infty}\\!\int_{-\infty}^{\infty}\frac{|(u-u_{N})(t)-(u-u_{N})(s)|^{p}}{|t-s|^{2}}dsdt=0.$

Since $\|u-u_{N}\|_{*}\leq\|u-u_{N}\|_{B_{p}}$ by Proposition 2.2, we see that
$u_{N}\in L^{\infty}(\mathbb{R})$ converges to $u$ in the BMO norm. In this
case, we have $\omega\in A_{2}$. See [12, Chap.IV, Theorem 5.14]. ∎

We prepare the following three claims on $\rm BMO(\mathbb{R})$ and
$B_{p}(\mathbb{R})$, which play key roles in the next two sections. Let
$I(x,y)\subset\mathbb{R}$ be the interval $(x-y,x+y)$ for any $x\in\mathbb{R}$
and $y>0$.

######  Proposition 2.4.

Let $u$ and $\varphi$ be complex-valued functions on $\mathbb{R}$ such that
$u\in\rm BMO(\mathbb{R})$ and $|\varphi(x)|\leq Ce^{-|x|}$ for some constant
$C>0$. Let $k>0$. Then,

$\int_{\mathbb{R}}|\varphi_{y}(x-t)||u(t)-u_{I(x,y)}|^{k}dt\leq
C(k)\|u\|_{*}^{k}$

for some constant $C(k)>0$.

###### Proof.

For any integer $n\geq 0$, we consider the average $u_{I_{n}}$ of $u$ over the
interval $I_{n}=(x-2^{n}y,x+2^{n}y)$, where $I_{0}=I(x,y)$. Using an
inequality

$|u_{I_{n}}-u_{I_{n-1}}|\leq\frac{1}{|I_{n-1}|}\int_{I_{n-1}}|u-u_{I_{n}}|dt\leq\frac{2}{|I_{n}|}\int_{I_{n}}|u-u_{I_{n}}|dt\leq
2\|u\|_{*}$

repeatedly, we have

$|u_{I_{n}}-u_{I_{0}}|\leq 2n\|u\|_{*}.$ (7)

Dividing the integral over $\mathbb{R}$ into those on dyadic intervals, we
obtain

$\begin{split}&\quad\int_{\mathbb{R}}|\varphi_{y}(x-t)||u(t)-u_{I(x,y)}|^{k}dt\\\
&=\int_{|x-t|<y}|\varphi_{y}(x-t)||u(t)-u_{I_{0}}|^{k}dt+\sum_{n=0}^{\infty}\int_{2^{n}y\leq|x-t|<2^{n+1}y}|\varphi_{y}(x-t)||u(t)-u_{I_{0}}|^{k}dt\\\
&\leq\frac{C}{y}\int_{|x-t|<y}e^{-\frac{|x-t|}{y}}|u(t)-u_{I_{0}}|^{k}dt+\sum_{n=0}^{\infty}\frac{C}{y}\int_{2^{n}y\leq|x-t|<2^{n+1}y}e^{-\frac{|x-t|}{y}}|u(t)-u_{I_{0}}|^{k}dt\\\
&\leq\frac{2C}{|I_{0}|}\int_{I_{0}}|u(t)-u_{I_{0}}|^{k}dt+\sum_{n=0}^{\infty}\frac{2^{n+2}C}{e^{2^{n}}|I_{n+1}|}\int_{I_{n+1}}|u(t)-u_{I_{0}}|^{k}dt\\\
&\leq
C\sum_{n=0}^{\infty}\frac{2^{n+2}}{e^{2^{n-1}}|I_{n}|}\int_{I_{n}}(|u(t)-u_{I_{n}}|+|u_{I_{n}}-u_{I_{0}}|)^{k}dt\\\
&\leq
C\sum_{n=0}^{\infty}\frac{2^{n+2+k}}{e^{2^{n-1}}}\left(\frac{1}{|I_{n}|}\int_{I_{n}}|u(t)-u_{I_{n}}|^{k}dt+\frac{1}{|I_{n}|}\int_{I_{n}}|u_{I_{n}}-u_{I_{0}}|^{k}dt\right).\end{split}$
(8)

For the last inequality above, we have used $(a+b)^{k}\leq 2^{k}(a^{k}+b^{k})$
for $a,b\geq 0$.

Here, by the John–Nirenberg inequality (1), we have

$\begin{split}&\quad\frac{1}{|I_{n}|}\int_{I_{n}}|u(t)-u_{I_{n}}|^{k}dt\\\
&=k\int_{0}^{\infty}\frac{1}{|I_{n}|}|\\{t\in
I_{n}\mid|u(t)-u_{I_{n}}|>\lambda\\}|\,\lambda^{k-1}d\lambda\\\ &\leq
k\int_{0}^{\infty}C_{0}\exp\left(\frac{-C_{JN}\lambda}{\|u\|_{*}}\right)\lambda^{k-1}d\lambda=\frac{C_{0}\Gamma(k+1)}{C_{JN}^{k}}\|u\|_{*}^{k}.\end{split}$
(9)

Moreover, (7) yields

$\begin{split}\frac{1}{|I_{n}|}\int_{I_{n}}|u_{I_{n}}-u_{I_{0}}|^{k}dt\leq(2n)^{k}\|u\|_{*}^{k}.\end{split}$
(10)

Plugging these inequalities into the last line of (8), we obtain the required
estimate. ∎

######  Proposition 2.5.

Let $u$ and $\varphi$ be complex-valued functions on $\mathbb{R}$ such that
$u\in\rm BMO(\mathbb{R})$ and $|\varphi(x)|\leq Ce^{-|x|}$ for some constant
$C>0$. If $\|u\|_{*}<C_{JN}$, then

$\int_{\mathbb{R}}|\varphi_{y}(x-t)|e^{|u(t)-u_{I(x,y)}|}dt\leq C(u)$

for a constant $C(u)$ given in terms of $\|u\|_{*}$.

###### Proof.

As in the proof of Proposition 2.4, we have

$\begin{split}&\quad\int_{\mathbb{R}}|\varphi_{y}(x-t)|e^{|u(t)-u_{I(x,y)}|}dt\\\
&\leq\frac{C}{y}\int_{|x-t|<y}e^{-\frac{|x-t|}{y}}e^{|u(t)-u_{I_{0}}|}dt+\sum_{n=0}^{\infty}\frac{C}{y}\int_{2^{n}y\leq|x-t|<2^{n+1}y}e^{-\frac{|x-t|}{y}}e^{|u(t)-u_{I_{0}}|}dt\\\
&\leq
C\sum_{n=0}^{\infty}\frac{2^{n+2}e^{2n\|u\|_{*}}}{e^{2^{n-1}}|I_{n}|}\int_{I_{n}}e^{|u(t)-u_{I_{n}}|}dt.\end{split}$
(11)

If $\|u\|_{*}<C_{JN}$, then the John–Nirenberg inequality as in (6) yields

$\begin{split}\frac{1}{|I_{n}|}\int_{I_{n}}e^{|u(t)-u_{I_{n}}|}dt&\leq\frac{C_{0}\|u\|_{*}}{C_{JN}-\|u\|_{*}}+1.\end{split}$

Thus, we obtain the statement of the proposition. ∎

######  Lemma 2.6.

For each $u_{0}\in B_{p}^{\mathbb{R}}(\mathbb{R})$, there exists a constant
$C(u_{0})>0$ such that every $u\in B_{p}(\mathbb{R})$ with
$\|u-u_{0}\|_{B_{p}}\leq C_{JN}/4$ satisfies

$\frac{1}{|I|}\int_{I}e^{|u(t)-u_{I}|}dt\leq C(u_{0})$

for all bounded intervals $I\subset\mathbb{R}$. Likewise,

$\int_{\mathbb{R}}|\varphi_{y}(x-t)|e^{|u(t)-u_{I(x,y)}|}dt\leq
C^{\prime}(u_{0})$

is satisfied for another constant $C^{\prime}(u_{0})>0$.

###### Proof.

We fix any $u_{0}\in B_{p}^{\mathbb{R}}(\mathbb{R})$. By Proposition 2.3,
there exists some $b\in L^{\infty}(\mathbb{R})$ such that $\|u_{0}-b\|_{*}\leq
C_{JN}/4$. Then, $\|u-b\|_{*}\leq C_{JN}/2$ for every $u\in B_{p}(\mathbb{R})$
with $\|u-u_{0}\|_{*}\leq\|u-u_{0}\|_{B_{p}}\leq C_{JN}/4$. Applying the
John–Nirenberg inequality as in (6), we have

$\begin{split}\frac{1}{|I|}\int_{I}e^{|u(t)-b(t)-u_{I}+b_{I}|}dt&\leq\frac{C_{0}\|u-b\|_{*}}{C_{JN}-\|u-b\|_{*}}+1\leq
2C_{0}+1.\end{split}$

This implies that

$\frac{1}{|I|}\int_{I}e^{|u(t)-u_{I}|}dt\leq(2C_{0}+1)e^{2\|b\|_{\infty}}.$

Then by (11), the latter statement also follows. ∎

##  3\. The variant of the Beurling–Ahlfors extension for BMO

Hereafter, we use the following convention. The notation $A(u)\asymp B(u)$
concerning formulas of $u$ means that there is a constant $C\geq 1$
independent of $u$ such that $A(u)/C\leq B(u)\leq CA(u)$. The notation
$A(u)\lesssim B(u)$ means that there is such a constant $C$ that satisfies
$A(u)\leq CB(u)$. We also use a variant of this convention in the following
case even if $C$ depends on $u$ in general. When a function $u$ is in ${\rm
BMO}(\mathbb{R})$, the notation $A(u)\asymp B(u)$ means that there is a
constant $C(u)\geq 1$ such that $A(u)/C(u)\leq B(u)\leq C(u)A(u)$, where
$C(u)$ is bounded whenever $\|u\|_{*}$ is bounded.

Beurling and Ahlfors [1] characterized the boundary value of a quasiconformal
homeomorphism of the upper half-plane $\mathbb{U}$ onto itself as a
quasisymmetric homeomorphism $f$ of the real line $\mathbb{R}$. Here, an
increasing homeomorphism $f$ of $\mathbb{R}$ onto itself is quasisymmetric if
there is a constant $\rho>1$, which is called the doubling constant, such that
$|f(2I)|\leq\rho|f(I)|$ for any bounded interval $I\subset\mathbb{R}$, where
$|\cdot|$ is the Lebesgue measure and $2I$ denotes the interval of the same
center as $I$ with $|2I|=2|I|$. Let $\phi(x)=\frac{1}{2}1_{[-1,1]}(x)$ and
$\psi(x)=\frac{r}{2}1_{[-1,0]}(x)+\frac{-r}{2}1_{[0,1]}(x)$ for some $r>0$,
where $1_{E}$ denotes the characteristic function of $E\subset\mathbb{R}$. For
any function $\varphi(x)$ on $\mathbb{R}$ and for $y>0$, we set
$\varphi_{y}(x)=y^{-1}\varphi(y^{-1}x)$. Then, for a quasisymmetric
homeomorphism $f$, the Beurling–Ahlfors extension $F(x,y)=(U(x,y),V(x,y))$ for
$(x,y)\in\mathbb{U}$ is defined by the convolutions
$U(x,y)=(f\ast\phi_{y})(x)$ and $V(x,y)=(f\ast\psi_{y})(x)$. Modification and
variation to the Beurling–Ahlfors extension have been made by changing the
functions $\phi$ and $\psi$.

For a complex-valued function $u$ on $\mathbb{R}$ such that $u\in\rm
BMO(\mathbb{R})$, we consider a curve
$\gamma_{u}=\gamma:\mathbb{R}\to\mathbb{C}$ given by

$\gamma(x)=\gamma(0)+\int_{0}^{x}e^{u(t)}dx.$ (12)

Let $\phi(x)=\frac{1}{\sqrt{\pi}}e^{-x^{2}}$ and
$\psi(x)=\phi^{\prime}(x)=-2x\phi(x)$. Then, we extend $\gamma$ to the upper
half-plane $\mathbb{U}$ setting a differentiable map
$F_{u}=F:\mathbb{U}\to\mathbb{C}$ by

$\begin{split}&F(x,y)=U(x,y)+iV(x,y);\\\ U(x,y)&=(\gamma\ast\phi_{y})(x),\
V(x,y)=(\gamma\ast\psi_{y})(x).\end{split}$ (13)

The partial derivatives of $U$ and $V$ can be represented as follows:

$\displaystyle U_{x}(x,y)$ $\displaystyle=\frac{\partial U}{\partial
x}=(e^{u}\ast\phi_{y})(x);$ $\displaystyle V_{x}(x,y)$
$\displaystyle=\frac{\partial V}{\partial x}=(e^{u}\ast\psi_{y})(x);$
$\displaystyle U_{y}(x,y)$ $\displaystyle=\frac{\partial U}{\partial
y}=(\gamma\ast\frac{\partial\phi_{y}}{\partial
y})(x)=\frac{1}{2}(e^{u}\ast\psi_{y})(x)=\frac{1}{2}V_{x}(x,y);$
$\displaystyle V_{y}(x,y)$ $\displaystyle=\frac{\partial V}{\partial
y}=(\gamma\ast\frac{\partial\psi_{y}}{\partial
y})(x)=U_{x}(x,y)+\frac{y^{2}}{2}(e^{u}\ast(\phi_{y})^{\prime\prime})(x),$

where we have used

$\displaystyle\frac{\partial\phi_{y}}{\partial y}$
$\displaystyle=\frac{y}{2}\frac{\partial^{2}\phi_{y}}{\partial
x^{2}}=\frac{1}{2}(\psi_{y})^{\prime};$
$\displaystyle\frac{\partial\psi_{y}}{\partial y}$
$\displaystyle=\frac{\partial\phi_{y}}{\partial
x}+\frac{y^{2}}{2}\frac{\partial^{3}\phi_{y}}{\partial
x^{3}}=(\phi_{y})^{\prime}+\frac{y^{2}}{2}(\phi_{y})^{\prime\prime\prime}.$

In particular, each of $U_{y}(x,y)$, $V_{x}(x,y)$, and $(U_{x}-V_{y})(x,y)$
can be represented by the convolution $(e^{u}\ast a_{y})(x)$ explicitly for a
certain real-valued function $a\in C^{\infty}(\mathbb{R})$ such that
$\int_{\mathbb{R}}a(x)dx=0$, $|a(x)|$ is an even function, and
$a(x)=O(x^{2}e^{-x^{2}})$ $(|x|\to\infty)$. For instance,
$V_{x}(x,y)=(e^{u}\ast\psi_{y})(x)$ for
$\psi(x)=-\frac{2}{\sqrt{\pi}}xe^{-x^{2}}$.

Next, we consider the complex derivatives

$\displaystyle F_{\bar{z}}(x,y)$
$\displaystyle=\frac{1}{2}(F_{x}+iF_{y})=\frac{1}{2}\left((U_{x}-V_{y})+iU_{y}+iV_{x}\right);$
$\displaystyle F_{z}(x,y)$
$\displaystyle=\frac{1}{2}(F_{x}-iF_{y})=U_{x}+\frac{1}{2}\left(-(U_{x}-V_{y})-iU_{y}+iV_{x}\right).$

From these expressions, we can find two complex-valued functions
$\alpha,\beta\in C^{\infty}(\mathbb{R})$ independent of $u$ such that

$F_{\bar{z}}=e^{u}\ast\alpha_{y}(x),\quad F_{z}=e^{u}\ast\beta_{y}(x),$ (14)

and $\int_{\mathbb{R}}\alpha(x)dx=0$, $\alpha(x)=O(x^{2}e^{-x^{2}})$
$(|x|\to\infty)$, $\int_{\mathbb{R}}\beta(x)dx=1$,
$\beta(x)=O(x^{2}e^{-x^{2}})$ $(|x|\to\infty)$. In particular, we can assume
that

$|\alpha(x)|\leq Ce^{-|x|},\quad|\beta(x)|\leq Ce^{-|x|}$

for some constant $C>0$. We set $\mu_{u}(x,y)=F_{\bar{z}}/F_{z}$, and call it
the complex dilatation of $F$ even though the map $F=F_{u}$ by $(\ref{F})$ is
not necessarily quasiconformal.

In the case where $u$ is a real-valued function such that $e^{u}$ is an
$A_{\infty}$-weight, the situation becomes simpler. In this case, the
extension $F:\mathbb{U}\to\mathbb{U}$ of $\gamma:\mathbb{R}\to\mathbb{R}$ is
the variant of the Beurling–Ahlfors extension by the heat kernel introduced in
[10]. The following result is obtained in [30, Theorems 3.4, 4.1].

######  Theorem 3.1.

For a real-valued $u\in{\rm BMO}(\mathbb{R})$ such that $e^{u}$ is an $\rm
A_{\infty}$-weight, the map $F_{u}$ induced by $u$ is a quasiconformal
diffeomorphism of $\mathbb{U}$ onto itself. Moreover, for the complex
dilatation $\mu_{u}$ of $F_{u}$, $\frac{1}{y}|\mu_{u}(x,y)|^{2}dxdy$ is a
Carleson measure on $\mathbb{U}$. If $u$ belongs to ${\rm VMO}(\mathbb{R})$ in
addition, then this is a vanishing Carleson measure.

We say that a measure $\lambda(x,y)dxdy$ is a Carleson measure on $\mathbb{U}$
if

$\sup_{I\subset\mathbb{R}}\frac{1}{|I|}\int_{0}^{|I|}\\!\\!\int_{I}\lambda(x,y)dxdy$

is bounded, where the supremum is taken over all bounded intervals $I$ in
$\mathbb{R}$. Moreover, a Carleson measure is vanishing if

$\lim_{|I|\to 0}\frac{1}{|I|}\int_{0}^{|I|}\\!\\!\int_{I}\lambda(x,y)dxdy=0.$

Now, we add one more property on $F_{u}$ to this theorem. We say that a
diffeomorphism $F(x,y)$ of $\mathbb{U}$ onto itself is bi-Lipschitz with
respect to the hyperbolic metric with constant $L\geq 1$ if

$\frac{1}{Ly}\leq\frac{|d_{(x,y)}F(v)|}{{\rm Im}\,F(x,y)}\leq\frac{L}{y}$

for any $(x,y)\in\mathbb{U}$ and any unit tangent vector $v$ at $(x,y)$. The
derivative satisfies

$\frac{1}{K}|F_{z}(x,y)|\leq|d_{(x,y)}F(v)|\leq K|F_{z}(x,y)|$ (15)

for the maximal dilatation $K=(1+\|\mu\|_{\infty})/(1-\|\mu\|_{\infty})$ with
$\mu(x,y)=F_{\bar{z}}(x,y)/F_{z}(x,y)$.

######  Proposition 3.2.

The quasiconformal diffeomorphism $F_{u}$ in Theorem 3.1 is bi-Lipschitz with
respect to the hyperbolic metric on $\mathbb{U}$. The maximal dilatation $K$
and the bi-Lipschitz constant $L$ of $F_{u}$ depend only on the doubling
constant $\rho$ of $e^{u}$. If $\|u\|_{*}<C_{JN}$, then $\rho$ depends only on
$\|u\|_{*}$.

###### Proof.

We assume that $F=F_{u}$ is $K$-quasiconformal, and consider

$\displaystyle|F_{z}(x,y)|^{2}$
$\displaystyle=\frac{1}{4}(U_{x}^{2}+U_{y}^{2}+V_{x}^{2}+V_{y}^{2})+\frac{1}{2}(U_{x}V_{y}-U_{y}V_{x})$
$\displaystyle=\frac{1}{2}(U_{x}^{2}+U_{y}^{2}+V_{x}^{2}+V_{y}^{2})-|F_{\bar{z}}(x,y)|^{2}$
$\displaystyle\leq\frac{1}{2}(U_{x}^{2}+U_{y}^{2}+V_{x}^{2}+V_{y}^{2}).$

In the proof of [30, Theorem 3.3], we see that

$U_{x}^{2}+U_{y}^{2}+V_{x}^{2}+V_{y}^{2}\asymp U_{x}^{2};\quad
U_{x}V_{y}-U_{y}V_{x}\gtrsim U_{x}^{2},$

where the comparability is given in terms of the doubling constant $\rho$ of
$e^{u}$. Therefore, we have

$\begin{split}|F_{z}(x,y)|\asymp
U_{x}(x,y)&=(e^{u}\ast\phi_{y})(x)=(\gamma\ast(\phi_{y})^{\prime})(x)\\\
&=\frac{1}{y}(\gamma\ast\psi_{y})(x)=\frac{1}{y}V(x,y)=\frac{1}{y}{\rm
Im}\,F(x,y).\end{split}$ (16)

Thus, the bi-Lipschitz constant $L$ of $F$ depends only on $\rho$ and $K$ by
(15) and (16). Moreover, again in the proof of [30, Theorem 3.3], we see that
$K$ also depends only on $\rho$.

If $\|u\|_{*}<C_{JN}$, then $e^{u}\in A_{2}$ and the $A_{2}$-constant is given
in terms of $\|u\|_{*}$ as is shown in Proposition 2.1. Since the doubling
constant for an $A_{2}$-weight depends only on the $A_{2}$-constant (see [5,
Section 3]), we see that $\rho$ depends only on $\|u\|_{*}$. ∎

We return to the general case where $u\in{\rm BMO}(\mathbb{R})$ is complex-
valued. Parallel to (13), the curve
$\gamma_{u}=\gamma:\mathbb{R}\to\mathbb{C}$ given by (12) is extendable also
to the lower half-plane $\mathbb{L}$ setting $F_{u}=F:\mathbb{L}\to\mathbb{C}$
by

$\begin{split}&F(x,y)=U(x,-y)-iV(x,-y);\\\ U(x,y)&=(\gamma\ast\phi_{y})(x),\
V(x,y)=(\gamma\ast\psi_{y})(x).\end{split}$ (17)

In the rest of this section, we will show that under a small norm condition on
$u\in{\rm BMO}(\mathbb{R})$, the variant of the Beurling–Ahlfors extension
$F_{u}$ by the heat kernel yields a quasiconformal homeomorphism of
$\mathbb{C}$ (see Theorem 3.7 below). This is an analogous result to that by
Semmes [21] who considered the case that the kernels $\phi$ and $\psi$ of the
convolution are compactly supported. In fact, the following two claims toward
Theorem 3.7 are based on those in [21, p.251].

######  Proposition 3.3.

Let $u$ and $\varphi$ be complex-valued functions on $\mathbb{R}$ such that
$u\in\rm BMO(\mathbb{R})$, $|\varphi(x)|\leq Ce^{-|x|}$ for some constant
$C>0$, and $\int_{\mathbb{R}}\varphi(x)dx=1$. Let $I(x,y)\subset\mathbb{R}$ be
the interval $(x-y,x+y)$ for any $x\in\mathbb{R}$ and $y>0$. Then,

$|e^{\varphi_{y}\ast u(x)}|\asymp|e^{u_{I(x,y)}}|$

is satisfied.

###### Proof.

We apply Proposition 2.4 for $k=1$. Then, by setting $C_{1}=C(1)$, we see that

$|\varphi_{y}\ast
u(x)-u_{I(x,y)}|\leq\int_{\mathbb{R}}|\varphi_{y}(x-t)||u(t)-u_{I(x,y)}|dt\leq
C_{1}\|u\|_{*}.$

This implies that

$e^{-C_{1}\|u\|_{*}}\leq\left|e^{\varphi_{y}\ast u(x)-u_{I(x,y)}}\right|\leq
e^{C_{1}\|u\|_{*}},$

from which we have $\left|e^{\varphi_{y}\ast u(x)-u_{I(x,y)}}\right|\asymp 1$,
that is, $|e^{\varphi_{y}\ast u(x)}|\asymp|e^{u_{I(x,y)}}|$. ∎

######  Lemma 3.4.

Let $u$ and $\varphi$ be complex-valued functions on $\mathbb{R}$ such that
$u\in\rm BMO(\mathbb{R})$, $|\varphi(x)|\leq Ce^{-|x|}$ for some constant
$C>0$, and $\int_{\mathbb{R}}\varphi(x)dx=1$. If $\|u\|_{*}$ is sufficiently
small, then

$|e^{\varphi_{y}\ast u(x)}|\asymp|(\varphi_{y}\ast e^{u})(x)|$

is satisfied.

###### Proof.

By Proposition 3.3, it suffices to show that
$|e^{u_{I(x,y)}}|\asymp|(\varphi_{y}\ast e^{u})(x)|$. We will estimate

$\left|\frac{(\varphi_{y}\ast
e^{u})(x)}{e^{u_{I(x,y)}}}-1\right|=|\varphi_{y}\ast(e^{u(\cdot)-u_{I(x,y)}}-1)(x)|$

from above when $\|u\|_{*}<C_{JN}/2$. This can be done by

$\begin{split}&\quad|\varphi_{y}\ast(e^{u(\cdot)-u_{I(x,y)}}-1)(x)|\\\
&\leq\int_{\mathbb{R}}|\varphi_{y}(x-t)||e^{u(t)-u_{I(x,y)}}-1|dt\\\
&\leq\int_{\mathbb{R}}\left|\varphi_{y}(x-t)\right||u(t)-u_{I(x,y)}|e^{|u(t)-u_{I(x,y)}|}dt\\\
&\leq\left(\int_{\mathbb{R}}|\varphi_{y}(x-t)||u(t)-u_{I(x,y)}|^{2}dt\right)^{1/2}\left(\int_{\mathbb{R}}|\varphi_{y}(x-t)|e^{2|u(t)-u_{I(x,y)}|}dt\right)^{1/2},\end{split}$
(18)

where we have used an inequality $|e^{z}-1|\leq|z|e^{|z|}$.

Here, Proposition 2.4 implies that the first factor of the last line in (18)
is bounded by a multiple of $\|u\|_{*}$, and Proposition 2.5 implies that the
second factor is bounded if $\|2u\|_{*}<C_{JN}$. Therefore, we have

$\left|\frac{(\varphi_{y}\ast
e^{u})(x)}{e^{u_{I(x,y)}}}-1\right|\lesssim\|u\|_{*}$

in this case, and thus $|e^{\varphi_{y}\ast u(x)}|\asymp|(\varphi_{y}\ast
e^{u})(x)|$ when $\|u\|_{*}$ is sufficiently small. ∎

From these two claims, we see that the supremum norm of the complex dilatation
is dominated by the BMO norm. In the following Proposition 3.5, we only
consider the upper half-plane case. The lower half-plane case can be treated
similarly.

######  Proposition 3.5.

For a complex-valued function $u$ on $\mathbb{R}$ with $u\in\rm
BMO(\mathbb{R})$, let $\mu_{u}$ be the complex dilatation of the map $F_{u}$
on $\mathbb{U}$ induced by $u$ as above. If $\|u\|_{*}$ is sufficiently small,
then $\|\mu_{u}\|_{\infty}\lesssim\|u\|_{*}$.

###### Proof.

It follows from Proposition 3.3 and Lemma 3.4 that

$|\mu_{u}(x,y)|=\frac{|\alpha_{y}\ast e^{u}(x)|}{|\beta_{y}\ast
e^{u}(x)|}\asymp\frac{|\alpha_{y}\ast e^{u}(x)|}{|e^{\beta_{y}\ast u(x)}|}\\\
\asymp\frac{|\alpha_{y}\ast e^{u}(x)|}{|e^{u_{I(x,y)}}|}\\\ =|\alpha_{y}\ast
e^{u-u_{I(x,y)}}(x)|.$

Then, by $\int_{\mathbb{R}}\alpha(x)dx=0$, which implies
$\int_{\mathbb{R}}\alpha_{y}(x)dx=0$, and by $|e^{z}-1|\leq|z|e^{|z|}$, we
have

$\begin{split}|\mu_{u}(x,y)|&\asymp|\alpha_{y}\ast(e^{u-u_{I(x,y)}}-1)(x)|\\\
&\leq\int_{\mathbb{R}}|\alpha_{y}(x-t)||u(t)-u_{I(x,y)}|e^{|u(t)-u_{I(x,y)}|}dt.\end{split}$
(19)

By (18) applying to $\alpha=\varphi$, the last line of (19) is bounded by some
multiple of $\|u\|_{*}$. This implies that
$\|\mu_{u}\|_{\infty}\lesssim\|u\|_{*}$. ∎

Finally, aiming at the goal of this section, we prove a certain approximation
of BMO functions. This was stated in [21, p.251] without proof in order to
prove that $F_{u}$ is quasiconformal in its situation.

######  Lemma 3.6.

For any $u\in{\rm BMO}(\mathbb{R})$, there exist a sequence
$\\{u_{j}\\}\subset{\rm BMO}(\mathbb{R})$ and constants $a\in\mathbb{C}$ and
$C>0$ such that $u_{j}-a$ is continuous and compactly supported with
$\|u_{j}\|_{\ast}\leq C\|u\|_{\ast}$ for all $j\in\mathbb{N}$ and $u_{j}$
converges to $u$ locally in $L^{p}$ for $1\leq p<\infty$, i.e., $u_{j}\to u$
in $L^{p}(I)$ for any bounded interval $I\subset\mathbb{R}$. Moreover, if
$\|u\|_{\ast}$ is sufficiently small in addition, then $e^{u_{j}}$ converges
to $e^{u}$ locally in $L^{1}$.

###### Proof.

Any function $u\in{\rm BMO}(\mathbb{R})$ can be written as
$u=\varphi+H(\psi)+a$, where $\varphi,\psi\in L^{\infty}(\mathbb{R})$ satisfy
$\|\varphi\|_{\infty}\lesssim\|u\|_{\ast}$,
$\|\psi\|_{\infty}\lesssim\|u\|_{\ast}$, $a$ is a constant, and $H$ is the
Hilbert transformation for $L^{\infty}(\mathbb{R})$ defined by

$H(\psi)(x)=\lim_{\varepsilon\to
0}\frac{1}{\pi}\int_{|x-t|>\varepsilon}\left(\frac{1}{x-t}+\frac{t}{1+t^{2}}\right)\psi(t)dt.$

See [11, Corollary VI.4.5]. Moreover, let

$b(\psi)(x)=\frac{1}{\pi}\int_{|x-t|\geq
1}\left(\frac{1}{x-t}+\frac{t}{1+t^{2}}\right)\psi(t)dt,$

which satisfies $b(\psi)\in L^{\infty}(\mathbb{R})$ with
$\|b(\psi)\|_{\infty}\lesssim\|\psi\|_{\infty}$. Then,
$u(x)=\varphi(x)+b(\psi)(x)+H^{\prime}(\psi)(x)+a$ for

$H^{\prime}(\psi)(x)=\lim_{\varepsilon\to
0}\frac{1}{\pi}\int_{1>|x-t|>\varepsilon}\left(\frac{1}{x-t}+\frac{t}{1+t^{2}}\right)\psi(t)dt.$

We define a function

$\eta(x)=\begin{cases}c\exp(\frac{1}{x^{2}-1})&(-1<x<1)\\\ 0&{\rm
otherwise},\end{cases}$ (20)

where $c$ is a constant satisfying $\int_{\mathbb{R}}\eta(x)dx=1$. Then, we
consider the mollifier $\eta_{1/j}(x)=j\eta(jx)$ for every $j\in\mathbb{N}$,
and define

$\begin{split}u_{j}(x)&=\eta_{1/j}\ast\left(\varphi
1_{[-j,j]}+b(\psi)1_{[-j,j]}+H^{\prime}(\psi 1_{[-j,j]})\right)(x)+a\\\
&=\eta_{1/j}\ast\left(\varphi 1_{[-j,j]}+b(\psi)1_{[-j,j]}-b(\psi
1_{[-j,j]})+H(\psi 1_{[-j,j]})\right)(x)+a\\\ &=\eta_{1/j}\ast\left(\varphi
1_{[-j,j]}+b(\psi)1_{[-j,j]}-b(\psi
1_{[-j,j]})\right)(x)+H(\eta_{1/j}\ast(\psi 1_{[-j,j]}))(x)+a.\end{split}$

For the third equality above, we used a fact that the convolution by the
mollifier and the Hilbert transformation commute (see [28, Lemma 6]). Then,
each $u_{j}-a$ is smooth and compactly supported.

By a property of the mollifier, we have

$\begin{split}\|\eta_{1/j}\ast(\varphi
1_{[-j,j]})\|_{*}&\leq\|\eta_{1/j}\ast(\varphi
1_{[-j,j]})\|_{\infty}\leq\|\varphi\|_{\infty}\lesssim\|u\|_{\ast};\\\
\|\eta_{1/j}\ast(b(\psi)1_{[-j,j]})\|_{*}&\leq\|\eta_{1/j}\ast(b(\psi)1_{[-j,j]})\|_{\infty}\leq\|b(\psi)1_{[-j,j]}\|_{\infty}\lesssim\|\psi\|_{\infty}\lesssim\|u\|_{\ast};\\\
\|\eta_{1/j}\ast(b(\psi 1_{[-j,j]}))\|_{*}&\leq\|\eta_{1/j}\ast(b(\psi
1_{[-j,j]}))\|_{\infty}\leq\|b(\psi
1_{[-j,j]})\|_{\infty}\lesssim\|\psi\|_{\infty}\lesssim\|u\|_{\ast}.\end{split}$

Moreover, for the Hilbert transformation $H$, we know that
$\|H(\psi)\|_{*}\lesssim\|\psi\|_{\infty}$ (see [11, Theorem VI.1.5]). Hence,

$\begin{split}\|H(\eta_{1/j}\ast(\psi
1_{[-j,j]}))\|_{*}&\lesssim\|\eta_{1/j}\ast(\psi
1_{[-j,j]})\|_{\infty}\leq\|\psi\|_{\infty}\lesssim\|u\|_{\ast}.\end{split}$

Thus, we can conclude that there is some constant $C>0$ such that
$\|u_{j}\|_{*}\leq C\|u\|_{*}$ for all $j\in\mathbb{N}$.

Let $I\subset\mathbb{R}$ be an arbitrary bounded interval. Then, for all
sufficiently large $j$, we have $\varphi 1_{[-j,j]}=\varphi$, $b1_{[-j,j]}=b$,
and $H^{\prime}(\psi 1_{[-j,j]})=H^{\prime}(\psi)$ on $I$. Therefore, by
another property of the mollifier, we see that $u_{j}$ converges to $u$ in
$L^{p}(I)$ for $1\leq p<\infty$.

Finally, we show that $e^{u_{j}}\to e^{u}$ in $L^{1}(I)$ when $\|u\|_{*}$ is
sufficiently small. For simplicity, we may assume that $u_{I}=0$, for
otherwise, we only have to put the constant $u_{I}$ at appropriate places.
Since $u_{j}\to u$ in $L^{1}(I)$, we have $(u_{j})_{I}\to 0$ as $j\to\infty$.
Hence, $|(u_{j})_{I}|\leq\delta$ for some $\delta>0$. As before, we have

$\begin{split}&\quad\
\frac{1}{|I|}\int_{I}\left|e^{u(t)}-e^{u_{j}(t)}\right|dt\leq\frac{1}{|I|}\int_{I}e^{|u(t)|}e^{|u(t)-u_{j}(t)|}|u(t)-u_{j}(t)|dt\\\
&\leq\frac{e^{\delta}}{|I|}\int_{I}e^{2|u(t)-u_{I}|}e^{|u_{j}(t)-(u_{j})_{I}|}|u(t)-u_{j}(t)|dt\\\
&\leq
e^{\delta}\left(\frac{1}{|I|}\int_{I}e^{4|u(t)-u_{I}|}dt\right)^{1/2}\left(\frac{1}{|I|}\int_{I}e^{4|u_{j}(t)-(u_{j})_{I}|}dt\right)^{1/4}\left(\frac{1}{|I|}\int_{I}|u(t)-u_{j}(t)|^{4}dt\right)^{1/4}.\end{split}$

By the John–Nirenberg inequality as used in (6), the first integral factor in
the last line is bounded in terms of $\|u\|_{*}$ and the second integral
factor is bounded in terms of $\|u_{j}\|_{*}\leq C\|u\|_{*}$ if $\|u\|_{*}$ is
sufficiently small. Since $u_{j}\to u$ in $L^{p}(I)$, this proves that
$e^{u_{j}}\to e^{u}$ in $L^{1}(I)$. ∎

After these preparations, we obtain the following theorem on the variant of
the Beurling–Ahlfors extension $F_{u}$ by the heat kernel for BMO functions
$u$. This asserts that $F_{u}$ is quasiconformal if $\|u\|_{\ast}$ is small.
Combining this with Theorem 3.1, we see that $F_{u}$ is quasiconformal if $u$
is either real-valued with $e^{u}\in A_{\infty}$ or of small BMO norm. In the
next section, we consider this extension in the case where $u$ is in the
$p$-Besov space.

######  Theorem 3.7.

The map $F_{u}$ extends continuously to a quasiconformal homeomorphism of
$\mathbb{C}$ onto itself with $F_{u}|_{\mathbb{R}}=\gamma_{u}$ if
$\|u\|_{\ast}$ is sufficiently small.

###### Proof.

By Proposition 3.5, we may assume that $\|\mu_{u}\|_{\infty}$ is also
sufficiently small, and in particular $\|\mu_{u}\|_{\infty}<1$. By the
property of heat kernel, we see that $U(x,y)\to\gamma_{u}(x)$ and $V(x,y)\to
0$ as $y\to 0$. This shows that $F=F_{u}$ extends continuously to $\gamma_{u}$
on $\mathbb{R}$, and then $F$ is continuous on $\mathbb{C}$.

Suppose first that $u-a$ is continuous and has a compact support for some
$a\in\mathbb{C}$. Then, $F_{\bar{z}}$ and $F_{z}$ are continuous off
$\mathbb{R}$. Noting that $F_{\bar{z}}=e^{u}\ast\alpha_{y}(x)$,
$F_{z}=e^{u}\ast\beta_{y}(x)$ and $\int_{\mathbb{R}}\alpha(x)dx=0$,
$\int_{\mathbb{R}}\beta(x)dx=1$, we conclude by the Lebesgue dominated
convergence theorem that $F_{\bar{z}}(x,y)\to 0$ and
$F_{z}(x,y)\to\gamma^{\prime}(x)$ as $y\to 0$. Thus, $F$ is continuously
differentiable on $\mathbb{C}$. Since $u-a$ has a compact support,
$\gamma(x)=e^{a}x+O(1)$ at $\infty$, and then $F(z)=e^{a}z+O(1)$ at $\infty$,
which implies that $F(z)\to\infty$ as $z\to\infty$. Moreover, since
$|F_{z}(x,y)|\asymp|e^{\beta_{y}\ast u(x)}|$ by Lemma 3.4, which is never $0$,
the Jacobian determinant of $F$ is positive everywhere. This implies that $F$
is locally homeomorphic on $\mathbb{U}$. The same is true for the map $F$
defined on $\mathbb{L}$. Then, a topological argument deduces that $F$ extends
continuously to an orientation-preserving global homeomorphism of $\mathbb{C}$
onto itself. By $\|\mu_{u}\|_{\infty}<1$, we see that $F$ is quasiconformal.

Consider now the general case. Given $u\in\rm BMO(\mathbb{R})$ with a small
norm, Lemma 3.6 implies that there is a continuous and compactly supported
sequence $\\{u_{j}-a\\}\subset{\rm BMO}(\mathbb{R})$ such that
$\|u_{j}\|_{*}\leq C\|u\|_{*}$, $u_{j}\to u$ locally in $L^{1}$, and
$e^{u_{j}}\to e^{u}$ locally in $L^{1}$. We assume that
$\gamma_{j}(0)=\gamma(0)$ for all $j$. The corresponding $F_{u_{j}}$ for
$u_{j}$ is a quasiconformal homeomorphism of $\mathbb{C}$ by the above
arguments. Since $\gamma_{u_{j}}\to\gamma$ uniformly on compact sets by the
condition that $e^{u_{j}}\to e^{u}$ locally in $L^{1}$, so does $F_{u_{j}}\to
F$. Since $\|u_{j}\|_{*}$ is also sufficiently small by $\|u_{j}\|_{*}\leq
C\|u\|_{*}$, we can regard that all $\|\mu_{u_{j}}\|_{\infty}$ are uniformly
bounded by a constant less than $1$. Then, passing to a subsequence if
necessary, $F_{u_{j}}$ converges to a quasiconformal homeomorphism of
$\mathbb{C}$, which coincides with $F_{u}$. Thus, we see that $F_{u}$ is
quasiconformal. ∎

##  4\. Quasiconformal extension of curves

In this section, we establish the main result of this paper (Theorem 4.5,
which is the precise version of Theorem 1.2). The key point is to consider the
quasiconformal extension of a curve $\gamma_{u}:\mathbb{R}\to\mathbb{C}$
produced by $u\in B_{p}(\mathbb{R})$ by applying the variant of the
Beurling–Ahlfors extension by the heat kernel. Then, several results
concerning the $p$-Weil–Petersson Teichmüller space follow this as mentioned
in Section 1.

Let us start with some notations. Let $M(\mathbb{U})$ denote the open unit
ball of the Banach space $L^{\infty}(\mathbb{U})$ of all essentially bounded
measurable functions on $\mathbb{U}$. An element in $M(\mathbb{U})$ is called
a Beltrami coefficient. Let $L_{0}(\mathbb{U})$ denote a subspace of
$L^{\infty}(\mathbb{U})$ consisting of all elements $\mu$ vanishing at the
boundary, that is,

$\lim_{t\to 0}\,{\rm
ess.}\\!\\!\\!\\!\\!\\!\\!\\!\sup_{0<y<t\qquad}\\!\\!\\!\\!\\!\\!\\!\\!|\mu(x,y)|=0.$

Then, we define the subset of all Beltrami coefficients vanishing at the
boundary by $M_{0}(\mathbb{U})=M(\mathbb{U})\cap L_{0}(\mathbb{U})$. For
$\mu\in L^{\infty}(\mathbb{U})$, we define the $p$-integrable norm with
respect to the hyperbolic metric by

$\|\mu\|_{p}=\left(\iint_{\mathbb{U}}|\mu(x,y)|^{p}\frac{dxdy}{y^{2}}\right)^{\frac{1}{p}}.$

Then, we introduce a new norm $\|\mu\|_{\infty}+\|\mu\|_{p}$ for $\mu$. Let
$\mathcal{L}_{p}(\mathbb{U})$ denote a subspace of $L^{\infty}(\mathbb{U})$
consisting of all elements $\mu$ with $\|\mu\|_{\infty}+\|\mu\|_{p}<\infty$,
which is a Banach space. Moreover, we set
$\mathcal{M}_{p}(\mathbb{U})=M(\mathbb{U})\cap\mathcal{L}_{p}(\mathbb{U})$.

As before, $\mu_{u}$ denotes the complex dilatation of $F_{u}$ given by (13)
and (14), but we take $u\in B_{p}(\mathbb{R})$ in the present setting. The
following three claims are concerning the boundedness of the norms of
$\mu_{u}$.

######  Proposition 4.1.

For any $u_{0}\in B^{\mathbb{R}}_{p}(\mathbb{R})$, there are $\delta>0$ and
$M>0$ such that if $\tilde{u}\in B_{p}(\mathbb{R})$ satisfies
$\|\tilde{u}-u_{0}\|_{B_{p}}<\delta$, then
$\|\mu_{\tilde{u}}\|_{\infty}+\|\mu_{\tilde{u}}\|_{p}\leq M$.

###### Proof.

For any $\tilde{u}\in B_{p}(\mathbb{R})$, we set $\tilde{u}=u+iv$ where $u$
and $v$ are real-valued. Fixing $I(x,y)=(x-y,x+y)\subset\mathbb{R}$ for
$(x,y)\in\mathbb{U}$, we consider

$|\mu_{\tilde{u}}(x,y)|=\frac{|\alpha_{y}\ast
e^{\tilde{u}}(x)|}{|\beta_{y}\ast e^{\tilde{u}}(x)|}=\frac{|\alpha_{y}\ast
e^{\tilde{u}-\tilde{u}_{I(x,y)}}(x)|}{|\beta_{y}\ast
e^{\tilde{u}-\tilde{u}_{I(x,y)}}(x)|}.$ (21)

The denominator is estimated from below as

$\begin{split}|\beta_{y}\ast
e^{\tilde{u}-\tilde{u}_{I(x,y)}}(x)|&=|\beta_{y}\ast(e^{u-u_{I(x,y)}}(e^{i(v-v_{I(x,y)})}-1))+\beta_{y}\ast
e^{u-u_{I(x,y)}}|\\\ &\geq|\beta_{y}\ast
e^{u-u_{I(x,y)}}|-|\beta_{y}\ast(e^{u-u_{I(x,y)}}(e^{i(v-v_{I(x,y)})}-1))|.\end{split}$

Here, since $u$ is real-valued, we see as in the proof of Proposition 3.2 that

$|\beta_{y}\ast e^{u-u_{I(x,y)}}|\asymp|\phi_{y}\ast e^{u-u_{I(x,y)}}|$

for $\phi(x)=\frac{1}{\sqrt{\pi}}e^{-x^{2}}$. The Jensen inequality implies
that

$\begin{split}\left|(\phi_{y}\ast
e^{u-u_{I(x,y)}})(x)\right|&\geq\int_{\mathbb{R}}\phi_{y}(x-t)e^{-|u(t)-u_{I(x,y)}|}dt\\\
&\geq\frac{2}{\sqrt{\pi}e}\left(\frac{1}{2y}\int_{|x-t|<y}e^{-|u(t)-u_{I(x,y)}|}dt\right)\\\
&\geq\frac{2}{\sqrt{\pi}e}\exp(-\|u\|_{*})\gtrsim 1.\end{split}$ (22)

On the contrary, the Cauchy–Schwarz inequality and $|e^{ix}-1|\leq|x|$ yield
that

$\begin{split}&\quad|\beta_{y}\ast(e^{u-u_{I(x,y)}}(e^{i(v-v_{I(x,y)})}-1))(x)|\\\
&\leq\int_{\mathbb{R}}|\beta_{y}(x-t)|e^{u(t)-u_{I(x,y)}}|e^{i(v(t)-v_{I(x,y)})}-1|dt\\\
&\leq\left(\int_{\mathbb{R}}|\beta_{y}(x-t)|e^{2(u(t)-u_{I(x,y)})}dt\right)^{1/2}\left(\int_{\mathbb{R}}|\beta_{y}(x-t)||v(t)-v_{I(x,y)}|^{2}dt\right)^{1/2}.\end{split}$
(23)

By Lemma 2.6, the first factor of the last line of (23) is locally bounded. By
Proposition 2.4, the second factor is bounded by a multiple of
$\|v\|_{*}\leq\|v\|_{B_{p}}$. Thus, the denominator in the fraction of (21) is
bounded away from $0$ if $\|v\|_{B_{p}}$ is sufficiently small depending on
where $u$ moves. In particular, there is some $\delta>0$ with $\delta\leq
C_{JN}/8$ such that if $\|u-u_{0}\|_{B_{p}}<\delta/2$ and if
$\|v\|_{B_{p}}<\delta/2$, then the denominator is uniformly bounded away from
$0$. This in particular shows that there is some $C>0$ such that if
$\|\tilde{u}-u_{0}\|_{B_{p}}<\delta$ then

$|\mu_{\tilde{u}}(x,y)|\leq C|\alpha_{y}\ast
e^{\tilde{u}-\tilde{u}_{I(x,y)}}(x)|.$ (24)

The right hand $|\alpha_{y}\ast e^{\tilde{u}-\tilde{u}_{I(x,y)}}(x)|$
coincides with $|\alpha_{y}\ast(e^{\tilde{u}-\tilde{u}_{I(x,y)}}-1)(x)|$ by
$\int_{\mathbb{R}}\alpha(x)dx=0$. Then, similarly to the above estimate, we
have

$\begin{split}&\quad|\alpha_{y}\ast(e^{\tilde{u}-\tilde{u}_{I(x,y)}}-1)(x)|\\\
&\leq\int_{\mathbb{R}}|\alpha_{y}(x-t)||\tilde{u}(t)-\tilde{u}_{I(x,y)}|e^{|\tilde{u}(t)-\tilde{u}_{I(x,y)}|}dt\\\
&\leq\left(\int_{\mathbb{R}}|\alpha_{y}(x-t)||\tilde{u}(t)-\tilde{u}_{I(x,y)}|^{2}dt\right)^{1/2}\left(\int_{\mathbb{R}}|\alpha_{y}(x-t)|e^{2|\tilde{u}(t)-\tilde{u}_{I(x,y)}|}dt\right)^{1/2}.\end{split}$
(25)

Again by Proposition 2.4 and Lemma 2.6, this is bounded if
$\|\tilde{u}-u_{0}\|_{B_{p}}\leq C_{JN}/8$. By (24), we see that
$\|\mu_{\tilde{u}}\|_{\infty}$ is bounded if
$\|\tilde{u}-u_{0}\|_{B_{p}}<\delta$. By (24) again, the boundedness of
$\|\mu_{\tilde{u}}\|_{p}$ follows from Lemma 4.2 below. Thus, the proof is
completed. ∎

######  Lemma 4.2.

Suppose that the complex dilatation $\mu$ on $\mathbb{U}$ is given so that

$|\mu(x,y)|\lesssim|\alpha_{y}\ast e^{u-u_{I(x,y)}}(x)|$

for $\alpha\in C^{\infty}(\mathbb{R})$ with $\int_{\mathbb{R}}\alpha(x)dx=0$
and $|\alpha(x)|\leq Ce^{-|x|}$ and for $u\in{\rm BMO}(\mathbb{R})$. If $u\in
B_{p}(\mathbb{R})$ is within norm $C_{JN}/(4q)$ from
$B^{\mathbb{R}}_{p}(\mathbb{R})$ for $1/p+1/q=1$, then

$\|\mu\|_{p}^{p}=\iint_{\mathbb{U}}\frac{|\mu(x,y)|^{p}}{y^{2}}dxdy\leq
C_{p}(u)\|u\|_{B_{p}}^{p},$

where the constant $C_{p}(u)>0$ is locally bounded in the neighborhood of
$B^{\mathbb{R}}_{p}(\mathbb{R})$.

###### Proof.

By $\int_{\mathbb{R}}\alpha(x)dx=0$ and the inequality
$|e^{z}-1|\leq|z|e^{|z|}$, we have

$\begin{split}|\mu(x,y)|^{p}&\lesssim|\alpha_{y}\ast(e^{u-u_{I(x,y)}}-1)(x)|^{p}\\\
&\leq\left(\int_{\mathbb{R}}|\alpha_{y}(x-t)||u(t)-u_{I(x,y)}|e^{|u(t)-u_{I(x,y)}|}dt\right)^{p}\\\
&\leq\left(\int_{\mathbb{R}}|\alpha_{y}(x-t)||u(t)-u_{I(x,y)}|^{p}dt\right)\left(\int_{\mathbb{R}}|\alpha_{y}(x-t)|e^{q|u(t)-u_{I(x,y)}|}dt\right)^{p/q}.\\\
\end{split}$ (26)

Since $u\in B_{p}(\mathbb{R})$ is in the small neighborhood of
$B_{p}^{\mathbb{R}}(\mathbb{R})$, Lemma 2.6 implies that the second factor of
the last line above is bounded by a locally bounded constant $C(u)>0$. Thus,
we have only to consider the first factor.

As in the proof of Proposition 2.4, we decompose the first factor into

$\begin{split}&\quad\int_{\mathbb{R}}|\alpha_{y}(x-t)||u(t)-u_{I(x,y)}|^{p}dt\\\
&\leq\frac{2C}{|I_{0}|}\int_{I_{0}}|u(t)-u_{I_{0}}|^{p}dt+\sum_{n=0}^{\infty}\frac{2^{n+2}C}{e^{2^{n}}|I_{n+1}|}\int_{I_{n+1}}|u(t)-u_{I_{0}}|^{p}dt\\\
&\leq
C\sum_{n=0}^{\infty}\frac{2^{n+2}}{e^{2^{n-1}}|I_{n}|}\int_{I_{n}}|u(t)-u_{I_{0}}|^{p}dt=C\sum_{n=0}^{\infty}\frac{2^{n+2}}{e^{2^{n-1}}}A_{n},\end{split}$
(27)

where we set

$A_{n}=\frac{1}{|I_{n}|}\int_{I_{n}}|u(t)-u_{I_{0}}|^{p}dt$

for $I_{n}=(x+2^{n}y,x-2^{n}y)$ for every integer $n\geq 0$. Moreover, by
using the Hölder inequality, we compute

$\begin{split}A_{n}&=\frac{1}{|I_{n}|}\int_{I_{n}}\left|u(t)-\frac{1}{|I_{0}|}\int_{I_{0}}u(s)ds\right|^{p}dt=\frac{1}{|I_{n}|}\int_{I_{n}}\left|\frac{1}{|I_{0}|}\int_{I_{0}}(u(t)-u(s))ds\right|^{p}dt\\\
&\leq\frac{1}{|I_{n}||I_{0}|}\int_{I_{n}}\int_{I_{0}}|u(t)-u(s)|^{p}dsdt.\\\
\end{split}$

By the translation and by $(a+b)^{p}\leq 2^{p-1}(a^{p}+b^{p})$ for $a,b\geq
0$, the last line above is further estimated by

$\begin{split}&\quad\frac{1}{|I_{n}||I_{0}|}\int_{I_{n}}\int_{I_{0}}|u(t)-u(s)|^{p}dsdt\\\
&\leq\frac{2^{p-1}}{2^{n+2}y^{2}}\int_{-2^{n}y}^{2^{n}y}\int_{-y}^{y}(|u(x+t)-u(x)|^{p}+|u(x+s)-u(x)|^{p})dsdt\\\
&=2^{p-1}\left(\frac{1}{2^{n+1}y}\int_{-2^{n}y}^{2^{n}y}|u(x+t)-u(x)|^{p}dt+\frac{1}{2y}\int_{-y}^{y}|u(x+s)-u(x)|^{p}ds\right)\\\
&=2^{p-1}(L_{n}+L_{0}),\end{split}$

where we set

$L_{n}=\frac{1}{2^{n+1}y}\int_{-2^{n}y}^{2^{n}y}|u(x+t)-u(x)|^{p}dt$

for every integer $n\geq 0$.

By the substitution of all the above computation into (26), we obtain that

$\begin{split}|\mu(x,y)|^{p}\lesssim
C(u)C\sum_{n=0}^{\infty}\frac{2^{n+2}}{e^{2^{n-1}}}A_{n}\leq
C_{p}(u)\sum_{n=0}^{\infty}\frac{1}{e^{2^{n}}}(L_{n}+L_{0})\end{split}$ (28)

for some constant $C_{p}(u)>0$, which is locally bounded in the neighborhood
of $B^{\mathbb{R}}_{p}(\mathbb{R})$. Moreover, the integral of each term
$L_{n}$ over $\mathbb{U}$ is explicitly given as follows:

$\begin{split}\iint_{\mathbb{U}}\frac{L_{n}}{y^{2}}dxdy&=\frac{1}{2^{n+1}}\iint_{\mathbb{U}}dxdy\int_{-2^{n}y}^{2^{n}y}\frac{|u(x+t)-u(x)|^{p}}{y^{3}}dt\\\
&=\frac{1}{2^{n+1}}\int_{-\infty}^{+\infty}dx\int_{0}^{+\infty}\frac{dy}{y^{3}}\int_{0}^{2^{n}y}(|u(x+t)-u(x)|^{p}+|u(x-t)-u(x)|^{p})dt\\\
&=\frac{1}{2^{n+1}}\int_{-\infty}^{+\infty}dx\int_{0}^{+\infty}(|u(x+t)-u(x)|^{p}+|u(x-t)-u(x)|^{p})dt\int_{2^{-n}t}^{+\infty}\frac{dy}{y^{3}}\\\
&=2^{n-2}\int_{-\infty}^{+\infty}dx\int_{0}^{+\infty}\frac{|u(x+t)-u(x)|^{p}+|u(x-t)-u(x)|^{p}}{t^{2}}dt\\\
&=2^{n-2}\int_{-\infty}^{+\infty}dx\int_{-\infty}^{+\infty}\frac{|u(x+t)-u(x)|^{p}}{t^{2}}dt=2^{n-2}\|u\|_{B_{p}}^{p}.\end{split}$
(29)

Therefore,

$\iint_{\mathbb{U}}\frac{|\mu(z)|^{p}}{y^{2}}dxdy\leq
C_{p}(u)\sum_{n=0}^{\infty}\frac{1}{e^{2^{n}}}\left(2^{n-2}+2^{-2}\right)\|u\|_{B_{p}}^{p}\leq
C_{p}(u)\|u\|_{B_{p}}^{p},$

which proves the statement of the lemma. ∎

######  Proposition 4.3.

Under the same circumstances as in Lemma 4.2,

$\lim_{t\to 0}\sup_{0<y<t}|\mu(x,y)|=0,$

that is, $\mu\in L_{0}(\mathbb{U})$.

###### Proof.

By (26) and (27) for $p=2$, we have

$|\mu(x,y)|^{2}\lesssim
C(u)C\sum_{n=0}^{\infty}\frac{2^{n+2}}{e^{2^{n-1}}|I_{n}|}\int_{I_{n}}|u(t)-u_{I_{0}}|^{2}dt,$
(30)

and by (9) and (10), we have

$\begin{split}&\quad\frac{1}{|I_{n}|}\int_{I_{n}}|u(t)-u_{I_{0}}|^{2}dt\\\
&\leq\frac{2}{|I_{n}|}\int_{I_{n}}|u(t)-u_{I_{n}}|^{2}dt+\frac{2}{|I_{n}|}\int_{I_{n}}|u_{I_{n}}-u_{I_{0}}|^{2}dt\leq
2\left(\frac{2C_{0}}{C_{JN}^{2}}+4n^{2}\right)\|u\|_{*}^{2}.\end{split}$ (31)

This implies that the infinite series in (30) converges. Hence, for an
arbitrarily given $\varepsilon>0$, we can choose $N\in\mathbb{N}$ such that

$C(u)C\sum_{n=N+1}^{\infty}\frac{2^{n+2}}{e^{2^{n-1}}|I_{n}|}\int_{I_{n}}|u(t)-u_{I_{0}}|^{2}dt<\varepsilon^{2}.$

Since $u\in{\rm VMO}(\mathbb{R})$ by Proposition 2.2, there is $\delta>0$ such
that for any interval $J\subset\mathbb{R}$ with $|J|\leq\delta$, we have

$\frac{1}{|J|}\int_{J}|u(t)-u_{J}|dt\leq\varepsilon.$ (32)

If $2^{N+1}y\leq\delta$, then $|I_{n}|\leq\delta$ for $0\leq n\leq N$, and
hence inequality (32) is valid for $J=I_{n}$ $(0\leq n\leq N)$. Then, we apply
the John–Nirenberg inequality (1) restricted to these small intervals. By
using (31), we can estimate

$C(u)C\sum_{n=0}^{N}\frac{2^{n+2}}{e^{2^{n-1}}|I_{n}|}\int_{I_{n}}|u(t)-u_{I_{0}}|^{2}dt$

from above by a multiple of $\|u\|_{*}^{2}$, but when $y\leq\delta/2^{N+1}$ we
see from (32) that $\|u\|_{*}$ can be replaced with $\varepsilon$. ∎

For any $u\in B_{p}(\mathbb{R})$, by setting $\Lambda(u)=\mu_{u}$, we define a
map $\Lambda$ on $B_{p}(\mathbb{R})$. More explicitly,

$\Lambda(u)(x,y)=\frac{\alpha_{y}\ast e^{u}(x)}{\beta_{y}\ast e^{u}(x)}$ (33)

for the functions $\alpha,\beta$ in (14). Now we are ready for showing the
main part of our result, the holomorphy of $\Lambda$.

######  Theorem 4.4.

There exists a neighborhood $U(B_{p}^{\mathbb{R}}(\mathbb{R}))$ of the
subspace $B_{p}^{\mathbb{R}}(\mathbb{R})$ of the real-valued functions in
$B_{p}(\mathbb{R})$ such that
$\Lambda:U(B_{p}^{\mathbb{R}}(\mathbb{R}))\to\mathcal{L}_{p}(\mathbb{U})$ is
holomorphic and the image of $U(B_{p}^{\mathbb{R}}(\mathbb{R}))$ under
$\Lambda$ is contained in $\mathcal{M}_{p}(\mathbb{U})\cap M_{0}(\mathbb{U})$.

It should be pointed out that the statement of Theorem 4.4 and its proof are
inspired by Shen and Tang [23, Lemma 6.1]. This is originally in Takhtajan and
Teo [27, p.30]. It is worthwhile to compare our arguments with theirs. They
proved that $\Lambda:U_{\delta}(0)\subset
B_{2}(\mathbb{R})\to\mathcal{M}_{2}(\mathbb{U})$ is holomorphic for some small
$\delta$-neighborhood $U_{\delta}(0)$ of the origin under the premise that
$\Lambda(u)$ is the complex dilatation of the modified Beurling–Ahlfors
extension due to Semmes [21], while we show that the variant of the
Beurling–Ahlfors extension by the heat kernel has the better property in the
sense that $\Lambda$ can be defined in some neighborhood of the entire
$B^{\mathbb{R}}_{p}(\mathbb{R})$. We also generalize their result to the case
of the $p$-integrable class $\mathcal{M}_{p}(\mathbb{U})$ and the $p$-Besov
space $B_{p}(\mathbb{R})$. More importantly, our strategy is that we first
prove the holomorphy of $\Lambda$ in a certain larger domain by the local
boundedness of $\Lambda$ and the holomorphy of $\Lambda$ in a weak sense, and
then we see that $\Lambda$ is continuous from this stronger property. By the
continuity of $\Lambda$, we obtain a smaller domain whose image under
$\Lambda$ is contained in the appropriate space.

###### Proof of Theorem 4.4.

We first show that for each $u_{0}\in B_{p}^{\mathbb{R}}(\mathbb{R})$,
$\Lambda$ is a Gâteaux holomorphic function from the $\delta$-neighborhood
$U_{\delta}(u_{0})$ of $u_{0}$ to $\mathcal{L}_{p}(\mathbb{U})$, where
$\delta>0$ is the constant chosen in Lemma 4.1 depending on $u_{0}$. Namely,
we prove that for every $\tilde{u}\in U_{\delta}(u_{0})$ and every non-trivial
$\tilde{v}\in B_{p}(\mathbb{R})$, the function
$\lambda(t)=\Lambda(\tilde{u}+t\tilde{v})$ of $t\in\mathbb{C}$ is holomorphic
in some neighborhood of $0\in\mathbb{C}$ with the image in
$\mathcal{L}_{p}(\mathbb{U})$. We choose $\epsilon>0$ with
$2\epsilon<(\delta-\|\tilde{u}\|_{B_{p}})/\|\tilde{v}\|_{B_{p}}$ so that
$\tilde{u}+t\tilde{v}\in U_{\delta}(u_{0})$ when $|t|\leq 2\epsilon$. Then, by
(33), the complex-valued function $\lambda(t)(z)$ for each fixed
$z\in\mathbb{U}$ is holomorphic on $|t|\leq 2\epsilon$. By using the Cauchy
integral formula, we have

$\displaystyle\quad\
\lambda(t)(z)-\lambda(t_{0})(z)-(t-t_{0})\left.\frac{d}{dt}\right|_{t=t_{0}}\lambda(t)(z)$
$\displaystyle=\frac{1}{2\pi
i}\oint_{|\tau|=2\epsilon}\lambda(\tau)(z)\left(\frac{1}{\tau-t}-\frac{1}{\tau-
t_{0}}-\frac{t-t_{0}}{(\tau-t_{0})^{2}}\right)d\tau$
$\displaystyle=\frac{(t-t_{0})^{2}}{2\pi
i}\oint_{|\tau|=2\epsilon}\frac{\lambda(\tau)(z)}{(\tau-
t_{0})^{2}(\tau-t)}d\tau.$

Since $\tilde{u}+t\tilde{v}\in U_{\delta}(u_{0})$, we see from Lemma 4.1 that
$\|\lambda(t)(z)\|_{\infty}\leq M$ for all $|t|=2\epsilon$, and then we have

$\displaystyle\quad\left\|\frac{\lambda(t)(z)-\lambda(t_{0})(z)}{t-t_{0}}-\left.\frac{d}{dt}\right|_{t=t_{0}}\lambda(t)(z)\right\|_{\infty}$
$\displaystyle\leq\frac{|t-t_{0}|}{2\pi\epsilon^{3}}\oint_{|\tau|=2\epsilon}\|\lambda(\tau)(z)\|_{\infty}|d\tau|\leq\frac{2M}{\epsilon^{2}}|t-t_{0}|.$

Moreover, by Lemma 4.1 again, we have

$\displaystyle\quad\iint_{\mathbb{U}}\frac{1}{y^{2}}\left|\frac{\lambda(t)(z)-\lambda(t_{0})(z)}{t-t_{0}}-\left.\frac{d}{dt}\right|_{t=t_{0}}\lambda(t)(z)\right|^{p}dxdy$
$\displaystyle\leq\frac{|t-t_{0}|^{p}}{(2\pi)^{p}\epsilon^{3}}\iint_{\mathbb{U}}\frac{1}{y^{2}}\left(\oint_{|\tau|=2\epsilon}|\lambda(\tau)(z)||d\tau|\right)^{p}dxdy$
$\displaystyle\leq\frac{2^{p-1}|t-t_{0}|^{p}}{2\pi\epsilon^{4-p}}\oint_{|\tau|=2\epsilon}\left(\iint_{\mathbb{U}}\frac{|\lambda(\tau)(z)|^{p}}{y^{2}}dxdy\right)|d\tau|\leq\frac{(2M)^{p}}{\epsilon^{3-p}}|t-t_{0}|^{p}.$

Consequently, the limit

$\lim_{t\to
t_{0}}\frac{\lambda(t)-\lambda(t_{0})}{t-t_{0}}=\left.\frac{d}{dt}\right|_{t=t_{0}}\lambda(t)$

exists in $\mathcal{L}_{p}(\mathbb{U})$ and thus $\Lambda$ is Gâteaux
holomorphic in $U_{\delta}(u_{0})$.

It is known that a locally bounded Gâteaux holomorphic function is holomorphic
in the sense that it is Fréchet differentiable (see [4, Theorem 14.9], [7,
Proposition 3.7] and [19, Theorem 36.5]). Thus, we see that $\Lambda$ is
holomorphic on some neighborhood $U(B_{p}^{\mathbb{R}}(\mathbb{R}))$ of
$B_{p}^{\mathbb{R}}(\mathbb{R})$. In particular, $\Lambda$ is continuous
there. By Theorem 3.1 and Proposition 4.1, we have
$\Lambda(B_{p}^{\mathbb{R}}(\mathbb{R}))\subset\mathcal{M}_{p}(\mathbb{U})$.
Since $\mathcal{M}_{p}(\mathbb{U})$ is open in $\mathcal{L}_{p}(\mathbb{U})$,
the image of $U(B_{p}^{\mathbb{R}}(\mathbb{R}))$ under $\Lambda$ is contained
in $\mathcal{M}_{p}(\mathbb{U})$ by making the neighborhood
$U(B_{p}^{\mathbb{R}}(\mathbb{R}))$ smaller if necessary. Since we have
$|\mu_{\tilde{u}}(x,y)|\leq C|\alpha_{y}\ast
e^{\tilde{u}-\tilde{u}_{I(x,y)}}(x)|$ by (24), Proposition 4.3 implies that
$\mu_{\tilde{u}}\in M_{0}(\mathbb{U})$. That is all what we have to prove. ∎

Finally, we verify that the mapping $F_{u}$ defined on $\mathbb{R}$ by (12)
and on $\mathbb{U}$ and $\mathbb{L}$ by formulae (13) and (17) respectively
for $u\in U(B_{p}^{\mathbb{R}}(\mathbb{R}))$ is a quasiconformal homeomorphism
of $\mathbb{C}$. Here, the neighborhood $U(B_{p}^{\mathbb{R}}(\mathbb{R}))$
should be taken smaller if necessary so that the same result on $\mathbb{L}$
as Theorem 4.4 on $\mathbb{U}$ is also satisfied. The spaces $M(\mathbb{L})$,
$M_{0}(\mathbb{L})$, and $\mathcal{M}_{p}(\mathbb{L})$ of Beltrami
coefficients on $\mathbb{L}$ are defined in the same way.

######  Theorem 4.5.

For any $u\in U(B_{p}^{\mathbb{R}}(\mathbb{R}))$, the mapping $F_{u}$ defined
on $\mathbb{C}$ is a quasiconformal homeomorphism onto $\mathbb{C}$ such that
its complex dilatations on $\mathbb{U}$ and on $\mathbb{L}$ are in
${\mathcal{M}}_{p}(\mathbb{U})\cap M_{0}(\mathbb{U})$ and in
${\mathcal{M}}_{p}(\mathbb{L})\cap M_{0}(\mathbb{L})$, respectively, and they
both depend holomorphically on $u$.

###### Proof.

We can choose a sequence $\\{u_{j}\\}$ in $U(B_{p}^{\mathbb{R}}(\mathbb{R}))$
such that each $u_{j}$ is continuous and compactly supported and $u_{j}$
converges to $u$ in $B_{p}(\mathbb{R})$. Indeed, for each $j\in\mathbb{N}$, we
set $u_{j}=\eta_{1/j}\ast(u1_{[-j,j]})$, where $\eta_{1/j}$ is the mollifier
defined in (20). We see that $u1_{[-j,j]}$ converges to $u$ in
$B_{p}(\mathbb{R})$ as $j\to\infty$. Moreover, it is known (see [15,
Proposition 14.5]) that for $\tilde{u}\in B_{p}(\mathbb{R})$ in general,
$\eta_{1/j}\ast\tilde{u}$ converges to $\tilde{u}$ in $B_{p}(\mathbb{R})$.
This shows that $u_{j}\to u$ in $B_{p}(\mathbb{R})$. See also [2, Theorem 5.3]
for the claim that smooth and compactly supported functions are dense in the
$p$-Besov space.

By the same argument as in the proof of Theorem 3.7, we see that $F_{u_{j}}$
is a quasiconformal homeomorphism of $\mathbb{C}$. Moreover, by Theorem 4.4
also applied on $\mathbb{L}$, the complex dilatation $\mu_{u_{j}}$ of
$F_{u_{j}}$ converges to $\mu_{u}$ of $F_{u}$ in the $L^{\infty}$ norm. Let
$\widetilde{F}$ be the quasiconformal homeomorphism of $\mathbb{C}$ whose
complex dilatation is $\mu_{u}$. If we normalize $F_{u_{j}}$, $F_{u}$ and
$\widetilde{F}$ suitably, then $F_{u_{j}}$ converges locally uniformly to
$\widetilde{F}$. Since $u_{j}$ converges to $u$ in $B_{p}(\mathbb{R})$, we see
that $e^{u_{j}}$ converges to $e^{u}$ locally in $L^{1}$. Then, by the
definition of $F_{u_{j}}$ and $F_{u}$ in (13) and (17), $F_{u_{j}}$ converges
to $F_{u}$. Therefore, $F_{u}$ coincides with $\widetilde{F}$, which proves
that $F_{u}$ is a quasiconformal homeomorphism of $\mathbb{C}$.

The rest of the statements has been shown in Theorem 4.4 if we extend it also
to $\mathbb{L}$. ∎

######  Remark.

There are remaining problems of showing that for every $\tilde{u}\in
U(B_{p}^{\mathbb{R}}(\mathbb{R}))$ in Theorems 4.4 and 4.5, the quasiconformal
homeomorphism $F_{\tilde{u}}$ of $\mathbb{U}$ (and $\mathbb{L}$) onto its
image is bi-Lipschitz with respect to the hyperbolic metrics as well as
$\frac{|\mu_{\tilde{u}}(x,y)|^{2}}{y}dxdy$ is a vanishing Carleson measure. On
the contrary, we can obtain a little stronger consequence about the
quasiconformality; $F_{\tilde{u}}$ is asymptotically conformal on $\mathbb{U}$
(and $\mathbb{L}$). This means that its complex dilatation $\mu_{\tilde{u}}$
satisfies

$\inf\,\\{\|\mu_{\tilde{u}}|_{\mathbb{U}\backslash
K}\|_{\infty}:K\subset\mathbb{U}\;{\rm compact}\\}=0.$

This can be proved also on the unit disk $\mathbb{D}$ if we consider the same
problem there.

By Theorem 4.5, we can extend $\Lambda$ in Theorem 4.4 to a holomorphic map

$\widetilde{\Lambda}:U(B^{\mathbb{R}}_{p}(\mathbb{R}))\to\mathcal{M}_{p}(\mathbb{U})\times\mathcal{M}_{p}(\mathbb{L})$

such that the Beltrami coefficients in the image correspond to the
quasiconformal homeomorphism $F_{u}$ extending $\gamma_{u}$. Then, this
satisfies the properties of Theorem 1.2 and Corollaries 1.3 and 1.4 in Section
1.

##  5\. Appendix: The $p$-Weil–Petersson class on the unit circle

Let $W_{p}(\mathbb{S})$ denote the set of all quasisymmetric homeomorphisms
$g$ of the unit circle $\mathbb{S}$ onto itself that has a quasiconformal
extension $G$ to the unit disk $\mathbb{D}$ whose complex dilatation $\nu$ is
$p$-integrable in the hyperbolic metric, namely,

$\iint_{\mathbb{D}}\frac{|\nu(w)|^{p}}{(1-|w|^{2})^{2}}dudv<\infty.$

We call $W_{p}(\mathbb{S})$ the $p$-Weil–Petersson class on $\mathbb{S}$. The
class $W_{2}(\mathbb{S})$ for $p=2$ was first introduced and studied by Cui
[6] and then investigated by Takhtajan and Teo [27]. For $p\geq 2$,
$W_{p}(\mathbb{S})$ appeared in Guo [13] (see aslo [33]).

Shen [22] characterized intrinsically the elements in the Weil–Petersson class
$W_{2}(\mathbb{S})$ without using quasiconformal extensions, which solved the
problem proposed in [27]. Later on, Tang and Shen [26] generalized this result
to any $p\geq 2$. Let $B_{p}(\mathbb{S})$ be the $p$-Besov space of all
locally integrable functions $v$ on $\mathbb{S}$ with $\|v\|_{B_{p}}<\infty$,
where

$\|v\|_{B_{p}}^{p}=\int_{\mathbb{S}}\\!\int_{\mathbb{S}}\frac{|v(z)-v(w)|^{p}}{|z-w|^{2}}\frac{|dz|}{2\pi}\frac{|dw|}{2\pi}=\int_{0}^{1}\\!\int_{0}^{1}\frac{|v(e^{2\pi
ix})-v(e^{2\pi iy})|^{p}}{|e^{2\pi ix}-e^{2\pi iy}|^{2}}dxdy.$

Then, the results of [22] and [26] can be stated as follows.

######  Theorem 5.1.

Let $g$ be a sense-preserving homeomorphism of $\mathbb{S}$ onto itself. Then,
$g$ is absolutely continuous and $\log g^{\prime}$ belongs to the $p$-Besov
space $B_{p}(\mathbb{S})$ $(p\geq 2)$ if and only if $g$ belongs to the
$p$-Weil–Petersson class $W_{p}(\mathbb{S})$.

Recently, Wu, Hu and Shen [32] gave an alternative proof for the case of $p=2$
by exporting the result on $\mathbb{R}$ obtained by the modified
Beurling–Ahlfors extension due to Semmes [21], which is much different from
the method given previously in [22].

The purpose of this appendix is to show that the variant of the
Beurling–Ahlfors extension by the heat kernel, which is translated to the
setting of the unit disk, also yields the desired quasiconformal extension.
This in particular gives an alternative proof of the only-if part of Theorem
5.1 for a general $p>1$. As the extension used in [32] is valid only for such
$g$ with small norm, the decomposition of $g$ and the composition of such
extensions are required, but our method gives a straight extension and certain
properties of the complex dilatation are thus inherited.

For a sense-preserving homeomorphism $g:\mathbb{S}\to\mathbb{S}$ with $\log
g^{\prime}\in B_{p}(\mathbb{S})$, our quasiconformal extension
$G:\mathbb{D}\to\mathbb{D}$ is precisely defined as follows. We first note
that $\log g^{\prime}\in B_{p}(\mathbb{S})$ implies that $v=\log|g^{\prime}|$
belongs to the subspace $B^{\mathbb{R}}_{p}(\mathbb{S})$ of real-valued
functions. In fact, $\|v\|_{B_{p}}\leq\|\log g^{\prime}\|_{B_{p}}$. In
addition, by the same proof of Proposition 2.2 applied to the unit circle
case, we have $v\in{\rm VMO}(\mathbb{S})$ and $\|v\|_{*}\leq\|v\|_{B_{p}}$.

We take a continuous lift $f:\mathbb{R}\to\mathbb{R}$ of $g$ satisfying that
$g(e^{2\pi ix})=e^{2\pi if(x)}$ for $x\in\mathbb{R}$. Let $u(x)=\log
f^{\prime}(x)=\log|g^{\prime}(e^{2\pi ix})|$. This satisfies $u(x+1)=u(x)$ for
any $x\in\mathbb{R}$. From this periodicity, we also see that $u\in{\rm
VMO}(\mathbb{R})$, but $u$ does not necessarily belong to $B_{p}(\mathbb{R})$.
Moreover, we can verify that$\|v\|_{*}\leq\|u\|_{*}\leq 3\|v\|_{*}$ (see [20,
Lemma 2.2]).

Let $F:\mathbb{U}\to\mathbb{U}$ be the variant of the Beurling–Ahlfors
extension by the heat kernel such that $F|_{\mathbb{R}}=f$, and let
$\mu(z)=F_{\bar{z}}/F_{z}$ be the complex dilatation of $F$. Since
$f(x+1)=f(x)+1$, we see from definition (13) that the quasiconformal extension
$F$ of $f$ satisfies $F(z+1)=F(z)+1$. Thus, $F$ can be projected to a
quasiconformal homeomorphism $G$ of the punctured disk
$\mathbb{D}\backslash\\{0\\}$ onto itself such that $G(e^{2\pi iz})=e^{2\pi
iF(z)}$ for $z\in\mathbb{U}$ and $G|_{\mathbb{S}}=g$. Clearly, $G$ can be
extended quasiconformally to $0$, and the resulting mapping from $\mathbb{D}$
onto itself is still denoted by $G$.

Concerning this quasiconformal extension $G$ of $g$, we prove the following
properties. The properties that the complex dilatation $\nu$ on $\mathbb{D}$
vanishes at the boundary and induces a vanishing Carleson measure are defined
similarly to the case of $\mathbb{U}$.

######  Theorem 5.2.

Let $g$ be a sense-preserving absolutely continuous homeomorphism of the unit
circle $\mathbb{S}$ onto itself with $v=\log|g^{\prime}|\in
B_{p}^{\mathbb{R}}(\mathbb{S})$. Then, the complex dilatation $\nu$ of the
quasiconformal homeomorphism $G$ of $\mathbb{D}$ onto itself defined above
satisfies

$\iint_{\mathbb{D}}\frac{|\nu(w)|^{p}}{(1-|w|^{2})^{2}}dudv\leq
C_{p}(v)\|v\|_{B_{p}}^{p}$

for a locally bounded constant $C_{p}(v)>0$ depending on $v\in
B_{p}^{\mathbb{R}}(\mathbb{S})$. In particular, $g\in W_{p}(\mathbb{S})$.
Moreover, $\nu$ vanishes at the boundary, and

$\frac{1}{1-|w|^{2}}|\nu(w)|^{2}dudv$

is a vanishing Carleson measure on $\mathbb{D}$.

###### Proof.

The complex dilatation $\nu(w)=G_{\bar{w}}/G_{w}$ $(w\in\mathbb{D})$ satisfies
$\nu(e^{2\pi iz})\overline{e^{2\pi iz}}/e^{2\pi iz}=-\mu(z)$ for the complex
dilatation $\mu(z)=F_{\bar{z}}/F_{z}$ $(z\in\mathbb{U})$, and in particular,
$\|\nu\|_{\infty}=\|\mu\|_{\infty}$. We fix some constant $r_{0}$ with
$e^{-\pi}<r_{0}<1$ so that $c=\frac{1}{2\pi}\log\frac{1}{r_{0}}<\frac{1}{2}$.
Noting that $e^{2\pi y}-1\geq 2\pi y$ for $y>0$, we have

$\begin{split}\iint_{r_{0}<|w|<1}\frac{|\nu(w)|^{p}}{(1-|w|^{2})^{2}}dudv&=4\pi^{2}\int_{0}^{1}dx\int_{0}^{c}\frac{|\nu(e^{2\pi
iz})|^{p}}{(1-|e^{2\pi iz}|^{2})^{2}}|e^{2\pi iz}|^{2}dy\\\
&\leq\int_{0}^{1}dx\int_{0}^{c}\frac{|\mu(z)|^{p}}{y^{2}}dy.\\\ \end{split}$
(34)

Here, we can estimate $|\mu(z)|^{p}$ by (28). Then, similarly to (29), we
obtain by using the periodicity $u(x+1)=u(x)$ and $c<1/2$ that

$\begin{split}\int_{0}^{1}dx\int_{0}^{c}\frac{L_{n}}{y^{2}}dy&=\frac{1}{2^{n+1}}\int_{0}^{1}dx\int_{0}^{c}\frac{1}{y^{3}}dy\int_{-2^{n}y}^{2^{n}y}|u(x+t)-u(x)|^{p}dt\\\
&=\frac{1}{2^{n+1}}\int_{0}^{1}dx\int_{-2^{n}c}^{2^{n}c}|u(x+t)-u(x)|^{p}dt\int_{2^{-n}t}^{c}\frac{1}{y^{3}}dy\\\
&\leq
2^{n-2}\int_{0}^{1}dx\int_{-2^{n}c}^{2^{n}c}\frac{|u(x+t)-u(x)|^{p}}{t^{2}}dt\\\
&\leq
2^{2n-2}\int_{0}^{1}dx\int_{-1/2}^{1/2}\frac{|u(x+t)-u(x)|^{p}}{t^{2}}dt=2^{2n-2}\|v\|_{B_{p}}.\\\
\end{split}$ (35)

Hence, by (28), (34), and (35), we conclude that

$\begin{split}\iint_{r_{0}<|w|<1}\frac{|\nu(w)|^{p}}{(1-|w|^{2})^{2}}dudv&\leq
C_{p}(v)\sum_{n=0}^{\infty}\frac{1}{e^{2^{n}}}\int_{0}^{1}dx\int_{0}^{c}\frac{L_{n}+L_{0}}{y^{2}}dy\\\
&\leq
C_{p}(v)\sum_{n=0}^{\infty}\frac{2^{2n-2}+2^{-2}}{e^{2^{n}}}\|v\|_{B_{p}}\end{split}$

for a locally bounded constant $C_{p}(v)>0$ depending on $v\in
B_{p}^{\mathbb{R}}(\mathbb{S})$.

On the other hand, we see from Proposition 3.5 that
$\|\mu\|_{\infty}\lesssim\|u\|_{*}$ in any case without restricting
$\|u\|_{*}$ to be small since $\|\mu\|_{\infty}<1$ holds in the present
situation. Therefore,

$\|\nu\|_{\infty}=\|\mu\|_{\infty}\lesssim\|u\|_{*}\lesssim\|v\|_{*}\leq\|v\|_{B_{p}},$

and hence,

$\iint_{|w|<r_{0}}\frac{|\nu(w)|^{p}}{(1-|w|^{2})^{2}}dudv\leq\frac{\pi}{(1-r_{0}^{2})^{2}}\|\nu\|_{\infty}^{p}\lesssim\|v\|_{B_{p}}^{p}.$

Consequently,

$\displaystyle\iint_{\mathbb{D}}\frac{|\nu(w)|^{p}}{(1-|w|^{2})^{2}}dudv$
$\displaystyle=\iint_{r_{0}<|w|<1}\frac{|\nu(w)|^{p}}{(1-|w|^{2})^{2}}dudv+\iint_{|w|<r_{0}}\frac{|\nu(w)|^{p}}{(1-|w|^{2})^{2}}dudv$
$\displaystyle\leq C_{p}(v)\|v\|_{B_{p}}^{p}.$

To see that $\nu$ vanishes at the boundary, it suffices to see that $\mu$
vanishes at the boundary. This was already proved in Proposition 4.3. To see
that $\frac{1}{1-|w|^{2}}|\nu(w)|^{2}dudv$ is a vanishing Carleson measure on
$\mathbb{D}$, it suffices to see that $\frac{1}{y}|\mu(z)|^{2}dxdy$ is a
vanishing Carleson measure on $\mathbb{U}$. This was already proved in Theorem
3.1. This completes the proof of Theorem 5.2. ∎

## References

  * [1] A. Beurling and L.V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142.
  * [2] L. Brasco, D. Gómez-Castro and J.L. Vázquez, Characterisation of homogeneous fractional Sobolev spaces, Calc. Var. Partial Differential Equations 60, 60 (2021).
  * [3] C.J. Bishop, Weil–Petersson curves, conformal energies, $\beta$-numbers, and minimal surfaces, preprint.
  * [4] S.B. Chae, Holomorphy and Calculus in Normed Spaces, Pure and Applied Math. 92, Marcel Dekker, 1985.
  * [5] R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.
  * [6] G. Cui, Integrably asymptotic affine homeomorphisms of the circle and Teichmüller spaces, Sci. China Ser. A 43 (2000), 267–279.
  * [7] S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer, 1999.
  * [8] A. Douady and C. J. Earle, Conformal natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), 23–48.
  * [9] J. Fan and J. Hu, Holomorphic contractibility and other properties of the Weil–Petersson and VMOA Teichmüller spaces, Ann. Acad. Sci. Fenn. Math. 41 (2016), 587–600.
  * [10] R.A. Fefferman, C.E. Kenig and J. Pipher, The theory of weights and the Dirichlet problems for elliptic equations, Ann. of Math. 134 (1991), 65–124.
  * [11] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
  * [12] F. Gay-Balmaz and T.S. Ratiu, The geometry of the universal Teichmüller space and the Euler-Weil-Petersson equation, Adv. Math. 279 (2015), 717–778.
  * [13] H. Guo, Integrable Teichmüller spaces, Sci. China Ser. A 43 (2000), 47–58.
  * [14] S.V. Hruščev, A description of weights satisfying the $A_{\infty}$ condition of Muckenhoupt, Proc. Amer. Math. Soc. 90 (1984), 253–257.
  * [15] G. Leoni, A First Course in Sobolev Spaces, Graduate Studies in Math. 105, American Math. Soc., 2009.
  * [16] K. Matsuzaki, Rigidity of groups of circle diffeomorphisms and Teichmüller spaces, J. Anal. Math. 40 (2020), 511–548.
  * [17] K. Matsuzaki, Circle diffeomorphisms, rigidity of symmetric conjugation and affine foliation of the universal Teichmüller space, Geometry, dynamics, and foliations 2013, Advanced Studies in Pure Mathematics vol. 72, pp. 145–180, Mathematical Society of Japan, 2017.
  * [18] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.
  * [19] J. Mujica, Complex Analysis in Banach Spaces, Dover, 2010.
  * [20] D. Partyka, Eigenvalues of quasisymmetric automorphisms determined by VMO functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 52 (1998), 121–135.
  * [21] S. Semmes, Quasiconformal mappings and chord-arc curves, Trans. Amer. Math. Soc. 306 (1988), 233–263.
  * [22] Y. Shen, Weil–Petersson Teichmüller space, Amer. J. Math. 140 (2018), 1041–1074.
  * [23] Y. Shen and S. Tang, Weil–Petersson Teichmüller space II: smoothness of flow curves of $H^{\frac{3}{2}}$-vector fields, Adv. Math. 359 (2020).
  * [24] Y. Shen and L. Wu, Weil–Petersson Teichmüller space III: dependence of Riemann mappings for Weil–Petersson curves, Math. Ann. 381 (2021), 875–904.
  * [25] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993.
  * [26] S. Tang and Y. Shen, Integrable Teichmüller space, J. Math. Anal. Appl. 465 (2018), 658–672.
  * [27] L. Takhtajan and L.P. Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc. 183 (861) (2006).
  * [28] J.F. Toland, A few remarks about the Hilbert transform, J. Funct. Anal. 145 (1997), 151–174.
  * [29] Y. Wang, Equivalent descriptions of the Loewner energy. Invent. Math. 218 (2019), 573–621.
  * [30] H. Wei and K. Matsuzaki, Beurling–Ahlfors extension by heat kernel, ${\rm A}_{\infty}$-weights for VMO, and vanishing Carleson measures, Bull. London Math. Soc. 53 (2021), 723–739.
  * [31] H. Wei and K. Matsuzaki, Parametrization of the $p$-Weil–Petersson curves: holomorphic dependence, arXiv:2111.14011.
  * [32] L. Wu, Y. Hu and Y. Shen, Weil–Petersson Teichmüller space revisited, J. Math. Anal. Appl. 491 (2020), 124304.
  * [33] M. Yanagishita, Introduction of a complex structure on the $p$-integrable Teichmüller space, Ann. Acad. Sci. Fenn. Math. 39 (2014), 947–971.