# The VMO-Teichmüller space and the variant of Beurling–Ahlfors extension by
heat kernel

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 give a real-analytic section for the Teichmüller projection onto the VMO-
Teichmüller space by using the variant of Beurling–Ahlfors extension by heat
kernel introduced by Fefferman, Kenig and Pipher in 1991. Based on this
result, we prove that the VMO-Teichmüller space can be endowed with a real
Banach manifold structure that is real-analytically equivalent to its complex
Banach manifold structure. We also obtain that the VMO-Teichmüller space
admits a real-analytic contraction mapping.

###### Key words and phrases:

VMO Teichmüller space, Beurling–Ahlfors extension, vanishing Carleson measure,
$A_{\infty}$-weight, BMO function

###### 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 universal Teichmüller space

A sense-preserving homeomorphism $h$ of the unit circle
$\mathbb{S}=\\{z\in\mathbb{C}\mid|z|=1\\}$ onto itself is said to be
quasisymmetric if there exists a least positive constant $C(h)$, called the
quasisymmetry constant of $h$, such that

$\frac{\mid h(I_{1})\mid}{\mid h(I_{2})\mid}\leqslant C(h)$

for all pairs of adjacent intervals $I_{1}$ and $I_{2}$ on $\mathbb{S}$ with
the same length $|I_{1}|=|I_{2}|$. Beurling and Ahlfors [2] in 1956 proved
that a sense-preserving homeomorphism $h$ of $\mathbb{S}$ is quasisymmetric if
and only if there exists some quasiconformal homeomorphism of the unit disk
$\mathbb{D}=\\{z\in\mathbb{C}\mid|z|<1\\}$ onto itself that has boundary value
$h$. Later, Douady and Earle [9] in 1986 gave a quasiconformal extension of a
quasisymmetric homeomorphism of $\mathbb{S}$ in a conformally natural way.

The universal Teichmüller space $T$ is a universal parameter space of marked
complex structures on all Riemann surfaces and can be defined as the group
$\mbox{QS}(\mathbb{S})$ of all quasisymmetric homeomorphisms of $\mathbb{S}$
modulo the left action of the group $\mbox{\rm M\"{o}b}(\mathbb{S})$ of all
Möbius transformations of $\mathbb{S}$, i.e., $T=\mbox{\rm
M\"{o}b}(\mathbb{S})\backslash\mbox{QS}(\mathbb{S})$. The quotient by
$\mbox{\rm M\"{o}b}(\mathbb{S})$ is alternatively achieved by giving a
normalization to the elements in $\mbox{QS}(\mathbb{S})$. We identify $T$ as
the set of all quasisymmetric homeomorphisms of $\mathbb{S}$ that fix three
points, $1$, $-1$ and $i$. This is the real model of the universal Teichmüller
space $T$.

Let us recall other descriptions of the universal Teichmüller space $T$. Let
$M(\mathbb{D})$ denote the set of all Beltrami coefficients on $\mathbb{D}$.
By the measurable Riemann mapping theorem, for any $\mu\in M(\mathbb{D})$,
there is a unique quasiconformal homeomorphism $f_{\mu}$ of $\mathbb{D}$ onto
itself whose complex dilatation is $\mu$ and that is normalized by fixing $1$,
$-1$ and $i$. There is also a unique quasiconformal homeomorphism $f^{\mu}$ of
the Riemann sphere $\widehat{\mathbb{C}}$ with complex dilatation $\mu$ in
$\mathbb{D}$ and $0$ in
$\mathbb{D}^{*}=\hat{\mathbb{C}}-\overline{\mathbb{D}}$ that is normalized,
for example, by the Laurent expansion $f^{\mu}(z)=z+\frac{b_{1}}{z}+\cdots$ at
$\infty$. The image curve $\Gamma$ of the unit circle $\mathbb{S}$ under
$f^{\mu}$ is called a quasicircle. Let $\Omega$ denote the inner domain of
$\Gamma$, and let $g$ be the conformal mapping of $\mathbb{D}$ onto $\Omega$
such that $f_{\mu}|_{\mathbb{S}}=(g|_{\mathbb{S}})^{-1}\circ
f^{\mu}|_{\mathbb{S}}$. This gives the conformal welding with respect to
$f^{\mu}$, and $f_{\mu}|_{\mathbb{S}}$ belongs to $\mbox{QS}(\mathbb{S})$. We
say that two elements $\mu$ and $\nu$ in $M(\mathbb{D})$ are equivalent,
denoted by $\mu\sim\nu$, if
$f^{\mu}|_{\mathbb{D}^{*}}=f^{\nu}|_{\mathbb{D}^{*}}$, or equivalently,
$f_{\mu}|_{\mathbb{S}}=f_{\nu}|_{\mathbb{S}}$. Therefore, the universal
Teichmüller space $T$ can also be defined as follows.

  * •

$T$ is the set $M(\mathbb{D})/\sim$ of all equivalence classes $[\mu]$. This
is the Beltrami coefficient model of $T$ $([\mu]\mapsto
f_{\mu}|_{\mathbb{S}})$.

  * •

$T$ is the set of all quasicircles (up to a Möbius transformation of
$\hat{\mathbb{C}}$). Here, a Jordan curve $\Gamma$ is said to be a quasicircle
if, by definition, there is a positive constant $C(\Gamma)$ such that

$\frac{\mbox{diameter}(\widetilde{z_{1}z_{2}})}{|z_{1}-z_{2}|}\leq C(\Gamma)$

for the smaller subarc $\widetilde{z_{1}z_{2}}$ of $\Gamma$ joining any two
finite points $z_{1}$ and $z_{2}$ on $\Gamma$. This is the Jordan curve model
of $T$ $([\mu]\mapsto f^{\mu}(\mathbb{S}))$.

  * •

$T$ is the set of all conformal mappings $f^{\mu}$ on $\mathbb{D}^{*}$ (up to
a Möbius transformation of $\hat{\mathbb{C}}$) which can be extended to a
quasiconformal mapping in $\hat{\mathbb{C}}$. This is the conformal mapping
model of $T$ $([\mu]\mapsto f^{\mu}|_{\mathbb{D}^{*}})$.

The different models of $T$ are identified with each other by the one-to-one
maps given in the brackets. It is known that $T$ has a unique complex Banach
manifold structure such that the Teichmüller projection $\pi:M(\mathbb{D})\to
T$ sending $\mu$ to the equivalence class $[\mu]$ is a holomorphic split
submersion in the exact sense (see [18, Theorem V.5.3], [20, Sections 3.4,
3.5]).

###  1.2. Background on the VMO-Teichmüller space

The VMO-Teichmüller space $T_{v}$ is a subspace of the universal Teichmüller
space $T$ introduced from the viewpoint of harmonic analysis. Its real model
is the set of all normalized strongly symmetric homeomorphisms of
$\mathbb{S}$. Here, by a strongly symmetric homeomorphism $h$ of $\mathbb{S}$,
we mean that it is absolutely continuous with $\log
h^{\prime}\in\mbox{VMO}(\mathbb{S})$, the function space on $\mathbb{S}$ of
vanishing mean oscillation. By the results of Pommerenke [22] in 1978, Dyn′kin
[10] in 1997, and Shen and Wei [26] in 2013, the following conditions are
equivalent (see [26] for the comprehensive information):

  * $(V_{0})$

$f_{\mu}|_{\mathbb{S}}$ is a strongly symmetric homeomorphism of $\mathbb{S}$.

A positive measure $\lambda$ defined on $\mathbb{D}$ is called a Carleson
measure if

$\|\lambda\|_{c}=\sup\frac{\lambda(S_{h,\theta_{0}})}{h}$ (1)

is finite, where the supremum is taken over all sectors

$\displaystyle S_{h,\theta_{0}}$
$\displaystyle=\\{re^{i\theta}\in\mathbb{D}\mid 1-h\leqslant
r<1,\;|\theta-\theta_{0}|\leqslant\pi h\\}$ (2)

for $h\in(0,1]$ and $\theta_{0}\in[0,2\pi)$. A Carleson measure $\lambda$ is
called a vanishing Carleson measure if

$\lim_{h\to 0}\frac{\lambda(S_{h,\theta_{0}})}{h}=0$

uniformly for $\theta_{0}\in[0,2\pi)$. We denote by $\mbox{CM}(\mathbb{D})$
and $\mbox{CM}_{0}(\mathbb{D})$ the set of all Carleson measures and vanishing
Carleson measures on $\mathbb{D}$, respectively. Such measures on
$\mathbb{D}^{*}$ are defined similarly.

  * ($V_{1}$)

There exists some $f_{\mu}$ whose complex dilatation induces a vanishing
Carleson measure
$\lambda_{\mu}=|\mu(z)|^{2}/(1-|z|^{2})dxdy\in\mbox{CM}_{0}(\mathbb{D})$.

Let ${\mathcal{M}}_{0}(\mathbb{D})\subset M(\mathbb{D})$ be the subset of
Beltrami coefficients $\mu$ that induce vanishing Carleson measures
$\lambda_{\mu}$. The Beltrami coefficient model of the VMO-Teichmüller space
$T_{v}=T_{v}(\mathbb{D})$ is given by the image of the Teichmüller projection
$\pi:{\mathcal{M}}_{0}(\mathbb{D})\to T_{v}$.

  * ($V_{2}$)

$\Gamma=f^{\mu}(\mathbb{S})$ is a bounded asymptotically smooth curve in the
sense of Pommerenke [22]. Namely,

$\lim_{t\to 0}\sup_{|z_{1}-z_{2}|\leq t}\frac{\mbox{arc-
length}(\widetilde{z_{1}z_{2}})}{|z_{1}-z_{2}|}=1,\qquad\mbox{for any
}z_{1},z_{2}\in\Gamma.$

  * ($V_{3}$)

$\mathcal{L}_{f^{\mu}}=\log(f^{\mu})^{\prime}$ belongs to
$\mbox{VMOA}(\mathbb{D}^{*})$, the space of analytic functions in
$\mathbb{D}^{*}$ of vanishing mean oscillation. This can be also characterized
by the condition that
$\mathcal{S}_{f^{\mu}}=\mathcal{L}_{f^{\mu}}^{\prime\prime}-\frac{1}{2}(\mathcal{L}_{f^{\mu}}^{\prime})^{2}$
belongs to $\mathcal{B}_{0}(\mathbb{D}^{*})$.

Here, an analytic function $\phi$ on $\mathbb{D}^{*}$ belongs to
$\mbox{VMOA}(\mathbb{D}^{*})$ if and only if

$\lambda_{\phi}=|\phi^{\prime}(z)|^{2}(|z|^{2}-1)dxdy\in\mbox{CM}_{0}(\mathbb{D}^{*}).$
(3)

Moreover, let $\mathcal{B}_{0}(\mathbb{D}^{*})$ denote the Banach space of
holomorphic functions $\varphi$ in $\mathbb{D}^{*}$ each of which induces a
vanishing Carleson measure

$\lambda_{\varphi}=|\varphi(z)|^{2}(|z|^{2}-1)^{3}dxdy\in\mbox{CM}_{0}(\mathbb{D}^{*}).$
(4)

Via the Bers embedding given by the Schwarzian derivative
$\mathcal{S}_{f^{\mu}|_{\mathbb{D}^{*}}}$, $T_{v}$ is embedded into the space
$\mathcal{B}_{0}(\mathbb{D}^{*})$, which endows $T_{v}$ with a natural complex
Banach manifold structure, so that the Teichmüller projection $\pi$ from
$\mathcal{M}_{0}(\mathbb{D})$ to $T_{v}$ is holomorphic with a local
holomorphic right inverse (see [26, Theorem 5.1]).

###  1.3. Main results and plan of the paper

In 1974, Earle [11] proved that the Beurling–Ahlfors extension gives the
Teichmüller projection $\pi$ a global real-analytic section (right inverse)
for the universal Teichmüller space $T$. In 1986, Douady and Earle [9] proved
that this is also true for the conformally barycentric extension. In 2016,
Tang, Wei and Shen [27] showed that the conformally barycentric extension
yields a continuous section for the VMO-Teichmüller space $T_{v}$. However,
the existence of the real-analytic section for $T_{v}$ is not known yet in the
literature. In this paper, we provide a real-analytic section for $T_{v}$. Our
real-analytic section is given by the variant of Beurling–Ahlfors extension by
heat kernel introduced by Fefferman, Kenig and Pipher [13] in 1991 (see
Section 4).

We regard $T_{v}$ as its real model. For any $h\in T_{v}$ with
$u=\log|h^{\prime}|$, we denote $\mu_{u}$ the complex dilatation of the
variant of Beurling–Ahlfors extension by heat kernel of $h$. Precisely
speaking, this quasiconformal extension is done from the real line to the
upper half-plane, but under the lift and the projection by the universal cover
$z\mapsto e^{2\pi iz}$, we can also consider this extension from the unit
circle to the unit disk. Let ${\rm VMO}_{\mathbb{R}}(\mathbb{S})$ be the real
Banach space of all real-valued functions on $\mathbb{S}$ of vanishing mean
oscillation. Then, our main results can be summarized as follows. They are
given in the corresponding claims specified after the statements.

######  Theorem 1.1.

The following two statements hold:

  1. (i)

The map $\widetilde{L}:T_{v}\to{\rm VMO}_{\mathbb{R}}(\mathbb{S})$ sending $h$
to $u=\log|h^{\prime}|$ is real-analytic (Corollary 3.5).

  2. (ii)

The map $\widetilde{\Lambda}:{\rm
VMO}_{\mathbb{R}}(\mathbb{S})\to\mathcal{M}_{0}(\mathbb{D})$ sending $u$ to
$\mu_{u}$ is real-analytic (Theorem 6.4).

Consequently, the composition
$\widetilde{\Lambda}\circ\widetilde{L}:T_{v}\to\mathcal{M}_{0}(\mathbb{D})$ is
real-analytic such that $\pi\circ\widetilde{\Lambda}\circ\widetilde{L}$ is the
identity on $T_{v}$. This in particular gives a global real-analytic section
for the Teichmüller projection $\pi:\mathcal{M}_{0}(\mathbb{D})\to T_{v}$.

Based on the observation that
$\widetilde{L}^{-1}=\pi\circ\widetilde{\Lambda}$, we conclude that
$\widetilde{L}^{-1}:{\rm VMO}_{\mathbb{R}}(\mathbb{S})\to T_{v}$ is also real-
analytic.

######  Corollary 1.2.

$\widetilde{L}:T_{v}\to{\rm VMO}_{\mathbb{R}}(\mathbb{S})$ is a real-analytic
homeomorphism from $T_{v}$ onto ${\rm VMO}_{\mathbb{R}}(\mathbb{S})$ whose
inverse $\widetilde{L}^{-1}$ is also real-analytic.

This implies that the real model of $T_{v}$ can be endowed with a real Banach
manifold structure from ${\rm VMO}_{\mathbb{R}}(\mathbb{S})$ by
$\widetilde{L}$, which is real-analytically equivalent to the natural complex
Banach manifold structure on $T_{v}$ defined by ($V_{3}$).

The second immediate consequence of Theorem 1.1 is the contractibility of
$T_{v}$ in a stronger sense:

######  Corollary 1.3.

The VMO-Teichmüller space $T_{v}$ admits a real-analytic contraction.
Precisely, the contraction $\Phi:T_{v}\times[0,1]\to T_{v}$ is defined by
$\Phi(h,t)=\pi\left((1-t)\widetilde{\Lambda}\circ\widetilde{L}(h)\right)$, and
$\Phi$ is a real-analytic mapping. Alternatively, another real-analytic
contraction $\Phi^{\prime}$ can be given by
$\Phi^{\prime}(h,t)=\widetilde{L}^{-1}((1-t)\widetilde{L}(h))$.

Fan and Hu [12] proved that the image domain of $T_{v}$ under the Bers
embedding is holomorphically contractible in the sense that there is a
contraction $\Phi:T_{v}\times[0,1]\to T_{v}$ such that $\Phi$ is continuous
and $\Phi(\cdot,t)$ is holomorphic for each fixed $t$.

We remark that the corresponding claim to Theorem 1.1 for the BMO Teichmüller
space $T_{b}=\pi(\mathcal{M}(\mathbb{D}))$ is open, where
$\mathcal{M}(\mathbb{D})$ is the set of Beltrami coefficients $\mu$ on
$\mathbb{D}$ such that $\lambda_{\mu}\in{\rm CM}(\mathbb{D})$. The crucial
difference stems from the fact that $L_{\mathbb{R}}^{\infty}(\mathbb{R})$ is
dense in ${\rm VMO}_{\mathbb{R}}(\mathbb{S})$ in the BMO-norm (see Section 6).
Because of this property, the estimates performed in Section 5 will be
available.

We end this introduction section with noticing the organization of this paper.
In Section 2, we recall definitions and properties of the BMO and VMO spaces,
Muckenhoupt $A_{\infty}$-weights, and strongly quasisymmetric homeomorphisms,
which will be frequently used in the rest of the paper. In Section 3, we prove
the first statement of Theorem 1.1 (i.e. Corollary 3.5). In Section 4, we
review the variant of Beurling–Ahlfors extension by heat kernel, and in
Section 5, we show the local boundedness of the complex dilatations made by
such extensions of strongly symmetric homeomorphisms. Section 6 is devoted to
the proof of the second statement of Theorem 1.1 (i.e. Theorem 6.4).

##  2\. Preliminaries on BMO, VMO, and $A_{\infty}$-weights

We treat the cases of $\mathbb{R}$ and $\mathbb{S}$ simultaneously; $X$
denotes either of $\mathbb{R}$ or $\mathbb{S}$. Let $I$ be an interval in $X$.
A locally integrable complex-valued function $u$ on $I$ is of BMO if

$\|u\|_{{\rm BMO}(I)}=\sup_{J\subset
I}\frac{1}{|J|}\int_{J}|u(x)-u_{J}||dx|<\infty,$

where the supremum is taken over all bounded intervals $J$ on $I$ and $u_{J}$
denotes the integral mean of $u$ over $J$. The set of all BMO functions on $I$
is denoted by ${\rm BMO}(I)$. This is regarded as a Banach space with norm
$\|\cdot\|_{{\rm BMO}(I)}$ modulo constants. Hereafter, $\|\cdot\|_{{\rm
BMO}(X)}$ will be simplified to be $\|\cdot\|_{*}$. It is said that $u\in{\rm
BMO}(I)$ is of VMO if

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

and the set of all such functions is denoted by ${\rm VMO}(I)$. This is a
closed subspace of ${\rm BMO}(I)$. The real subspaces of ${\rm BMO}(I)$ and
${\rm VMO}(I)$ consisting of all real-valued BMO functions and VMO functions
on $I$ are denoted by ${\rm BMO}_{\mathbb{R}}(I)$ and ${\rm
VMO}_{\mathbb{R}}(I)$, respectively. The John–Nirenberg inequality for BMO
functions (see [14, VI.2], [28, IV.1.3]) asserts that there exists two
universal positive constants $C_{0}$ and $C_{JN}$ such that for any complex-
valued BMO function $u$, any bounded interval $J$ of $I$, and any $\lambda>0$,
it holds that

$\frac{1}{|J|}|\\{t\in J:|u(t)-u_{J}|\geq\lambda\\}|\leq
C_{0}\exp\left(\frac{-C_{JN}\lambda}{\|u\|_{{\rm BMO}(I)}}\right).$ (5)

A locally integrable non-negative measurable function $\omega\geq 0$ on $X$ is
called a weight. We say that $\omega$ is a Muckenhoupt $A_{\infty}$-weight (or
$A_{\infty}$-weight for simplicity) if there are positive constants
$\alpha(\omega)$, $K(\omega)>0$ such that

$\frac{\int_{E}\omega(x)|dx|}{\int_{I}\omega(x)|dx|}\leq
K(\omega)\left(\frac{|E|}{|I|}\right)^{\alpha(\omega)}$ (6)

for any bounded interval $I\subset X$ and for any measurable subset $E\subset
I$ (see [6, Theorem V] and [14, Lemma VI.6.11]). From this condition, we see
that an $A_{\infty}$-weight $\omega$ satisfies the doubling property: there
exists a constant $\rho(\omega)>1$ such that

$\int_{2I}\omega(x)|dx|\leq\rho(\omega)\int_{I}\omega(x)|dx|$ (7)

holds for any bounded intervals $I$ and $2I$ with the same center and
$|2I|=2|I|$. The doubling constant $\rho(\omega)$ can be given in terms of
$\alpha(\omega)$ and $K(\omega)$.

The Jensen inequality implies that

$\exp\left(\frac{1}{|I|}\int_{I}\log\omega(x)|dx|\right)\leq\frac{1}{|I|}\int_{I}\omega(x)|dx|.$
(8)

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

$\frac{1}{|I|}\int_{I}\omega(x)|dx|\leq
C_{\infty}(\omega)\exp\left(\frac{1}{|I|}\int_{I}\log\omega(x)|dx|\right)$ (9)

for every bounded interval $I\subset X$ (see [16]).

We call the least possible value of such $C_{\infty}(\omega)$ the
$A_{\infty}$-constant for $\omega$. If $\omega$ is an $A_{\infty}$-weight, the
constants $\alpha(\omega)$ and $K(\omega)$ in (6) are estimated by
$C_{\infty}(\omega)$ as is shown in [16, Theorem 1], and conversely,
$C_{\infty}(\omega)$ is estimated by $\alpha(\omega)$ and $K(\omega)$ (see [6,
Section 3]). One can also refer to [28, p.218] for these implications.

By (9) and the Jensen inequality, we see that if $\omega$ is an
$A_{\infty}$-weight on $X$, then $\log\omega$ belongs to ${\rm
BMO}_{\mathbb{R}}(X)$, and conversely, if $\|\log\omega\|_{*}$ is sufficiently
small, then $\omega$ is an $A_{\infty}$-weight (see [14, Lemma VI.6.5], [15,
p.409]).

A sense-preserving homeomorphism $h$ of $X$ is called strongly quasisymmetric
if $h$ is locally absolutely continuous such that $|h^{\prime}|$ is an
$A_{\infty}$-weight. In particular, $\log h^{\prime}\in{\rm BMO}(X)$. A
strongly quasisymmetric homeomorphism is quasisymmetric. We see from (6) that
the set of all strongly quasisymmetric homeomorphisms of $X$ forms a group.
Moreover, Jones [17] in 1983 characterized a strongly quasisymmetric
homeomorphism by the pull-back operator defined on the BMO space (see [1,
Lemma] for the case of the VMO space):

######  Proposition 2.1.

A sense-preserving homeomorphism $h$ of $X$ is strongly quasisymmetric if and
only if the pull-back operator $P_{h}:u\mapsto u\circ h$ gives an isomorphism
of ${\rm BMO}(X)$ onto itself, that is, $P_{h}$ and $(P_{h})^{-1}$ are bounded
linear operators. Moreover, in the case of $X=\mathbb{S}$, $P_{h}$ maps ${\rm
VMO}(\mathbb{S})$ onto ${\rm VMO}(\mathbb{S})$ if $h$ is strongly
quasisymmetric defined on $\mathbb{S}$.

It was noticed in our recent paper [31] that the pull-back operator $P_{h}$
does not necessarily maps ${\rm VMO}(\mathbb{R})$ into itself if $h$ is
strongly quasisymmetric defined on $\mathbb{R}$. However, under an extra
assumption that $h$ is uniformly continuous, the operator $P_{h}$ preserves
${\rm VMO}(\mathbb{R})$.

##  3\. Real-analytic mapping to the space of VMO (Theorem 1.1 (i))

In this section, we will prove the first part of Theorem 1.1 whose precise
statement will be given in Corollary 3.5 to Theorem 3.4.

Let $M(\mathbb{D})$ denote the open unit ball of the Banach space
$L^{\infty}(\mathbb{D})$ of all essentially bounded measurable functions on
$\mathbb{D}$. An element in $M(\mathbb{D})$ is called a Beltrami coefficient.
For $\mu\in L^{\infty}(\mathbb{D})$, we set
$\lambda_{\mu}=|\mu(z)|^{2}/(1-|z|^{2})dxdy$ and define
$\|\mu\|_{c}=\|\lambda_{\mu}\|_{c}^{1/2}$. Here, $\|\lambda_{\mu}\|_{c}$ is
the Carleson norm of $\lambda_{\mu}$ defined in (1). Then, we introduce a new
norm $\|\mu\|_{\infty}+\|\mu\|_{c}$ for $\mu$. Let $\mathcal{L}(\mathbb{D})$
denote the Banach space consisting of all elements $\mu\in
L^{\infty}(\mathbb{D})$ with norm

$\|\mu\|_{\infty}+\|\mu\|_{c}<\infty.$

Let $\mathcal{L}_{0}(\mathbb{D})$ denote the subspace of
$\mathcal{L}(\mathbb{D})$ consisting of all $\mu$ such that $\lambda_{\mu}$ is
a vanishing Carleson measure. This is a Banach subspace of
$\mathcal{L}(\mathbb{D})$. Moreover, we define the corresponding spaces of
Beltrami coefficients as
$\mathcal{M}(\mathbb{D})=M(\mathbb{D})\cap\mathcal{L}(\mathbb{D})$ and
$\mathcal{M}_{0}(\mathbb{D})=M(\mathbb{D})\cap\mathcal{L}_{0}(\mathbb{D})$.
The relevant spaces on $\mathbb{D}^{*}$ and $\mathbb{U}$ can be defined by
replacing the hyperbolic density $1/(1-|z|^{2})$ on $\mathbb{D}$ with that on
$\mathbb{D}^{*}$ and $\mathbb{U}$.

It is well known that a quasiconformal homeomorphism of $\mathbb{D}$ onto
itself induces a biholomorphic automorphism of the universal Teichmüller space
$T$. Precisely, let $f_{\mu}:\mathbb{D}\to\mathbb{D}$ be a quasiconformal
homeomorphism with complex dilatation $\mu\in M(\mathbb{D})$. Then, $f_{\mu}$
induces a biholomorphic automorphism $r_{\mu}:M(\mathbb{D})\to M(\mathbb{D})$
defined as

$r_{\mu}(\nu)=\left(\frac{\nu-\mu}{1-\bar{\mu}\nu}\frac{\partial
f_{\mu}}{\overline{\partial f_{\mu}}}\right)\circ f_{\mu}^{-1},$

which is the complex dilatation of $f_{\nu}\circ f_{\mu}^{-1}$. Moreover,
$r_{\mu}$ descends down by the Teichmüller projection $\pi$ to a biholomorphic
automorphism $R_{[\mu]}:T\to T$ satisfying $R_{[\mu]}\circ\pi=\pi\circ
r_{\mu}$ (see [18, Section V.5.4], [20, Section 3.6.2]).

The corresponding result for
$T_{v}(\mathbb{D})=\pi(\mathcal{M}_{0}(\mathbb{D}))$ (and for
$T_{b}(\mathbb{D})=\pi(\mathcal{M}(\mathbb{D}))$) was asserted in [26, Remark
5.1], which will be used in the proof of Proposition 3.2 and Theorem 3.4.

######  Proposition 3.1.

Suppose $f_{\mu}$ is bi-Lipschitz under the hyperbolic metric with complex
dilatation $\mu\in\mathcal{M}_{0}(\mathbb{D})$. Then,
$r_{\mu}:\mathcal{M}_{0}(\mathbb{D})\to\mathcal{M}_{0}(\mathbb{D})$ is a
biholomorphic automorphism of $\mathcal{M}_{0}(\mathbb{D})$, and moreover
$R_{[\mu]}:T_{v}\to T_{v}$ is a biholomorphic automorphism of
$T_{v}=T_{v}(\mathbb{D})$.

######  Remark.

We have that an inequality $\|r_{\mu}(\nu)\|_{c}\leq
C(\|\mu\|_{\infty})\|\nu-\mu\|_{c}$ holds, which was observed in [26, Remark
5.1] by examining the proof of [8, Lemma 10] (see also [31]). This implies
that the map
$r_{\mu}:\mathcal{M}_{0}(\mathbb{D})\to\mathcal{M}_{0}(\mathbb{D})$ is locally
bounded. Following Remark after Theorem 6.2, the holomorphy of $r_{\mu}$ can
be verified. The inverse $r_{\mu}^{-1}$ is treated similarly. These arguments
show that the map
$r_{\mu}:\mathcal{M}_{0}(\mathbb{D})\to\mathcal{M}_{0}(\mathbb{D})$ is
biholomorphic. The local inverse of the Bers projection, which is equivalent
to the Teichmüller projection through the Bers embedding, constructed in [26,
Theorem 5.1] is holomorphic for the same reason.

A sense-preserving homeomorphism $h$ of $\mathbb{S}$ onto itself is called
strongly symmetric if $h$ is absolutely continuous with $\log
h^{\prime}\in{\rm VMO}(\mathbb{S})$. Then, $h$ must be strongly quasisymmetric
since the condition $\log h^{\prime}\in{\rm VMO}(\mathbb{S})$ implies
$|h^{\prime}|\in A_{\infty}$ by the John–Nirenberg inequality (5) (see [15,
p.474]). Let ${\rm SS}(\mathbb{S})$ be the set of all strongly symmetric
homeomorphisms $h$ of $\mathbb{S}$ onto itself which satisfy normalized
conditions

$h(1)=1\quad{\rm and}\quad\int_{-\pi}^{\pi}h(e^{it})dt=0.$ (10)

Here, we use these normalized conditions different from the traditional case
of the universal Teichmüller space for later consideration. By
$(V_{0})\Rightarrow(V_{1})$ in Section 1, any $h\in{\rm SS}(\mathbb{S})$ can
be extended to a quasiconformal mapping of $\mathbb{D}$ onto itself such that
its complex dilatation belongs to $\mathcal{M}_{0}(\mathbb{D})$. The
Douady–Earle extension $E(h)$ is such an extension, and moreover, it is a bi-
Lipschitz diffeomorphism under the hyperbolic metric (see [8, 27]). The second
condition in (10) implies $E(h)(0)=0$, and $E(h)(\infty)=\infty$ by reflection
across $\mathbb{S}$.

We say that a quasiconformal homeomorphism $G$ of the whole plane $\mathbb{C}$
onto itself is a VMO-quasiconformal mapping relative to $\mathbb{S}$ if
$\mu_{G|_{\mathbb{D}}}\in\mathcal{M}_{0}(\mathbb{D})$ and
$\mu_{G|_{\mathbb{D}^{*}}}\in\mathcal{M}_{0}(\mathbb{D}^{*})$. Let
$\gamma=G|_{\mathbb{S}}$ and $\Gamma=\gamma(\mathbb{S})$. Then, the
homeomorphism $\gamma:\mathbb{S}\to\Gamma$ is a parametrization of the curve
$\Gamma$, and we call $\gamma$ a strongly symmetric embedding. For
$\mu_{1}\in\mathcal{M}_{0}(\mathbb{D})$ and
$\mu_{2}\in\mathcal{M}_{0}(\mathbb{D}^{*})$, we denote by
$G=G(\mu_{1},\mu_{2})$ the normalized VMO-quasiconformal mapping $G$ relative
to $\mathbb{S}$ $(G(0)=0,G(1)=1,G(\infty)=\infty)$ with
$\mu_{G|_{\mathbb{D}}}=\mu_{1}$, $\mu_{G|_{\mathbb{D}^{*}}}=\mu_{2}$. Let
${\rm SS}_{\mathbb{C}}(\mathbb{S})$ be the set of all normalized strongly
symmetric embeddings of $\mathbb{S}$.

######  Proposition 3.2.

A normalized strongly symmetric embedding $\gamma:\mathbb{S}\to\mathbb{C}$ is
absolutely continuous and $\log\gamma^{\prime}$ belongs to ${\rm
VMO}(\mathbb{S})$. Moreover, the image $\Gamma=\gamma(\mathbb{S})$ is a
bounded asymptotically smooth curve.

###### Proof.

We take a normalized VMO-quasiconformal mapping $G(\mu_{1},\mu_{2})$ relative
to $\mathbb{S}$ such that $G|_{\mathbb{S}}=\gamma$. We extend $\mu_{2}$ to
$\mathbb{D}$ by the reflection

$\mu_{2}(z)=\overline{\mu_{2}\left(\frac{1}{\bar{z}}\right)}\frac{z^{2}}{\bar{z}^{2}},\qquad
z\in\mathbb{D},$ (11)

and take the unique normalized quasiconformal homeomorphism
$F:\mathbb{C}\to\mathbb{C}$ with complex dilatation $\mu_{2}$
$(F(0)=0,F(1)=1,F(\infty)=\infty)$. By $F(\mathbb{S})=\mathbb{S}$ and
$(V_{1})\Rightarrow(V_{0})$ in Section 1, we have $f=F|_{\mathbb{S}}$ is
strongly symmetric, and in particular, $\log f^{\prime}$ belongs to ${\rm
VMO}(\mathbb{S})$. Next, we take the unique normalized quasiconformal
homeomorphism $H:\mathbb{C}\to\mathbb{C}$ $(H(0)=0,H(1)=1,H(\infty)=\infty)$
that is conformal on $\mathbb{D}^{*}$ and whose complex dilatation on
$\mathbb{D}$ is the push-forward $F_{*}\mu_{1}$ of $\mu_{1}$ by $F$. Namely,
the complex dilatation of $H\circ F|_{\mathbb{D}}$ is $\mu_{1}$. Then, we have
$H\circ F=G$.

The complex dilatation $F_{*}\mu_{1}$ is Teichmüller equivalent to some
$\nu\in\mathcal{M}_{0}(\mathbb{D})$. This can be seen from Proposition 3.1 if
we replace $F|_{\mathbb{D}}$ with the Douady–Earle extension of $f$ which is a
bi-Lipschitz diffeomorphism under the hyperbolic metric and whose complex
dilatation belongs to $\mathcal{M}_{0}(\mathbb{D})$. Then, the curve $\Gamma$
is bounded asymptotically smooth by $(V_{1})\Rightarrow(V_{2})$ in Section 1.
Since $\log(H|_{\mathbb{D}^{*}})^{\prime}\in{\rm VMOA}(\mathbb{D}^{*})$ by
$(V_{1})\Rightarrow(V_{3})$, it has the finite non-tangential limit almost
everywhere on $\mathbb{S}$, and this boundary function belongs to ${\rm
VMO}(\mathbb{S})$ (see [32, Theorem 9.23]). This also implies that
$H|_{\mathbb{D}^{*}}$ has a finite angular derivative almost everywhere on
$\mathbb{S}$ by [23, Proposition 4.7]. Moreover, since $H(\mathbb{D}^{*})$ is
a quasidisk, [23, Theorem 5.5] asserts that the angular derivative at
$z\in\mathbb{S}$ coincides with

$h^{\prime}(z)=\lim_{\mathbb{S}\ni\xi\to z}\frac{h(\xi)-h(z)}{\xi-z}$

for $h=H|_{\mathbb{S}}$. This shows that the non-tangential limit of
$(H|_{\mathbb{D}^{*}})^{\prime}$ coincides with the ordinary derivative
$h^{\prime}$ almost everywhere on $\mathbb{S}$. By taking the logarithm, we
have that $\log h^{\prime}$ belongs to ${\rm VMO}(\mathbb{S})$. Hence,
$|h^{\prime}|$ is an $A_{\infty}$-weight, and in particular, $h$ is absolutely
continuous on $\mathbb{S}$.

By $H\circ F=G$, we see that $\gamma=G|_{\mathbb{S}}$ is also absolutely
continuous, and taking the derivative on $\mathbb{S}$, we have

$\log h^{\prime}\circ f+\log f^{\prime}=\log\gamma^{\prime}.$

We have seen that $\log f^{\prime}\in{\rm VMO}(\mathbb{S})$. Since $f$ is
strongly symmetric and $\log h^{\prime}\in{\rm VMO}(\mathbb{S})$, Proposition
2.1 shows that $\log h^{\prime}\circ f\in{\rm VMO}(\mathbb{S})$. Thus, we
obtain that $\log\gamma^{\prime}\in{\rm VMO}(\mathbb{S})$. This completes the
proof. ∎

The VMO-Teichmüller space $T_{v}(\mathbb{D})$ on $\mathbb{D}$ is defined to be
the set of all Teichmüller equivalence classes $[\mu]$ for
$\mu\in\mathcal{M}_{0}(\mathbb{D})$. The quotient map
$\pi:\mathcal{M}_{0}(\mathbb{D})\to T_{v}(\mathbb{D})$ is called the
Teichmüller projection. The VMO-Teichmüller space $T_{v}(\mathbb{D}^{*})$ on
$\mathbb{D}^{*}$ and related concepts are defined similarly.

We define a map

$\widetilde{\iota}:\mathcal{M}_{0}(\mathbb{D})\times\mathcal{M}_{0}(\mathbb{D}^{*})\to{\rm
SS}_{\mathbb{C}}(\mathbb{S})$

by $\widetilde{\iota}(\mu_{1},\mu_{2})=G(\mu_{1},\mu_{2})|_{\mathbb{S}}$.
Then, by the argument of simultaneous uniformization due to Bers, we see the
following fact.

######  Proposition 3.3.

The space ${\rm SS}_{\mathbb{C}}(\mathbb{S})$ is identified with
$T_{v}(\mathbb{D})\times T_{v}(\mathbb{D}^{*})$. More precisely,
$\widetilde{\iota}$ splits into a well-defined bijection

$\iota:T_{v}(\mathbb{D})\times T_{v}(\mathbb{D}^{*})\to{\rm
SS}_{\mathbb{C}}(\mathbb{S})$

by the product of the Teichmüller projections

$\widetilde{\pi}:\mathcal{M}_{0}(\mathbb{D})\times\mathcal{M}_{0}(\mathbb{D}^{*})\to
T_{v}(\mathbb{D})\times T_{v}(\mathbb{D}^{*}),$

such that $\widetilde{\iota}=\iota\circ\widetilde{\pi}$.

Moreover, via the Bers embedding $\beta$ given by the Schwarzian derivative
$\mathcal{S}_{G(\mu,0)|_{\mathbb{D}^{*}}}$, $T_{v}(\mathbb{D})$ is embedded
into the complex Banach space $\mathcal{B}_{0}(\mathbb{D}^{*})$ with norm
defined by $\|\varphi\|_{\mathcal{B}}=\|\lambda_{\varphi}\|_{c}$ for
$\lambda_{\varphi}$ as in (4), and in the same way $T_{v}(\mathbb{D}^{*})$ is
embedded into the complex Banach space $\mathcal{B}_{0}(\mathbb{D})$, which is
defined similarly to $\mathcal{B}_{0}(\mathbb{D}^{*})$:

$\displaystyle T_{v}(\mathbb{D})$
$\displaystyle\cong\beta(T_{v}(\mathbb{D}))=\\{{\mathcal{S}}_{G(\mu,0)}\in\mathcal{B}_{0}(\mathbb{D}^{*})\mid\mu\in\mathcal{M}_{0}(\mathbb{D})\\};$
$\displaystyle T_{v}(\mathbb{D}^{*})$
$\displaystyle\cong\beta(T_{v}(\mathbb{D}^{*}))=\\{{\mathcal{S}}_{G(0,\mu)}\in\mathcal{B}_{0}(\mathbb{D})\mid\mu\in\mathcal{M}_{0}(\mathbb{D}^{*})\\}.$

Thus, $T_{v}(\mathbb{D})$ and $T_{v}(\mathbb{D}^{*})$ are endowed with the
complex Banach manifold structures, respectively (see [26, Theorem 5.2]), and
thus, $T_{v}(\mathbb{D})\times T_{v}(\mathbb{D}^{*})$ is endowed with the
product complex Banach manifold structure. Hence, by the identification ${\rm
SS}_{\mathbb{C}}(\mathbb{S})\cong T_{v}(\mathbb{D})\times
T_{v}(\mathbb{D}^{*})$ in Proposition 3.3, we may regard ${\rm
SS}_{\mathbb{C}}(\mathbb{S})$ as a domain of
$\mathcal{B}_{0}(\mathbb{D}^{*})\times\mathcal{B}_{0}(\mathbb{D})$.

Let us recall some results on $\rm VMOA$ space (see [14, 23, 32]). It is known
that $\phi\in{\rm VMOA}(\mathbb{D}^{*})$ if $\phi\in H^{2}(\mathbb{D}^{*})$
and if the boundary values $b(\phi)$ of $\phi$ on $\mathbb{S}$ belongs to
${\rm VMO}(\mathbb{S})$. Moreover, ${\rm VMOA}(\mathbb{D}^{*})$ is a Banach
space modulo constants, and the norm defined by $\|\phi\|=\|b(\phi)\|_{*}$ is
equivalent to the norm defined by the Carleson measure $\lambda_{\phi}$ given
by (3). Then, the boundary extension $b:{\rm VMOA}(\mathbb{D}^{*})\to{\rm
VMO}(\mathbb{S})$ is regarded as a linear isometric operator. Let

$\mathcal{T}_{v}(\mathbb{D}^{*})=\\{\mathcal{L}_{G(\mu,0)}\in{\rm
VMOA}(\mathbb{D}^{*})\mid\mu\in\mathcal{M}_{0}(\mathbb{D})\\}.$

By the normalization of the VMO-quasiconformal mapping $G(\mu,0)$ relative to
$\mathbb{S}$, the map

$\alpha:\,\mathcal{T}_{v}(\mathbb{D}^{*})\to\beta(T_{v}(\mathbb{D}))$

is bijective by sending $\mathcal{L}_{G(\mu,0)}$ to $\mathcal{S}_{G(\mu,0)}$.
Moreover, from the arguments in [26, Section 6], we see that $\alpha$ is a
biholomorphic homeomorphism.

By Proposition 3.2, we can consider a map $L:{\rm
SS}_{\mathbb{C}}(\mathbb{S})\to{\rm VMO}(\mathbb{S})$ defined by
$L(\gamma)=\log\gamma^{\prime}$. Then, with respect to the complex structure
of ${\rm SS}_{\mathbb{C}}(\mathbb{S})$ given as above, we see the following:

######  Theorem 3.4.

The map $L:{\rm SS}_{\mathbb{C}}(\mathbb{S})\to{\rm VMO}(\mathbb{S})$ is
holomorphic.

###### Proof.

We will prove that $L$ is holomorphic at any point
$\gamma=G(\mu_{1},\mu_{2})|_{\mathbb{S}}$ in ${\rm
SS}_{\mathbb{C}}(\mathbb{S})$. Since ${\rm SS}_{\mathbb{C}}(\mathbb{S})$ can
be regarded as a domain of the product
$\mathcal{B}_{0}(\mathbb{D}^{*})\times\mathcal{B}_{0}(\mathbb{D})$ of the
Banach spaces, the Hartogs theorem for Banach spaces (see [5, Theorem 14.27]
and [19, Theorem 36.8]) implies that we have only to prove that $L$ is
separately holomorphic. Thus, by fixing $[\mu_{2}]\in T_{v}(\mathbb{D}^{*})$,
we will show that $\log(G(\mu,\mu_{2})|_{\mathbb{S}})^{\prime}\in{\rm
VMO}(\mathbb{S})$ depends holomorphically on $[\mu]\in T_{v}(\mathbb{D})$. The
other case is similarly treated.

By the proof of Proposition 3.2, we have

$\log(G(\mu,\mu_{2})|_{\mathbb{S}})^{\prime}=\log h^{\prime}\circ f+\log
f^{\prime},$

where $f:\mathbb{S}\to\mathbb{S}$ is the boundary extension of the fixed bi-
Lipschitz diffeomorphism $F:\mathbb{D}\to\mathbb{D}$ defined by the
Douady–Earle extension for the Teichmüller class $[\mu_{2}]$, and
$h:\mathbb{S}\to\mathbb{C}$ is the restriction of the quasiconformal
homeomorphism $H$ of $\mathbb{C}$ that is conformal on $\mathbb{D}^{*}$ and
has the complex dilatation $F_{*}\mu$ on $\mathbb{D}$. Since $F$ is a bi-
Lipschitz diffeomorphism, we conclude by Proposition 3.1 that $F_{*}$ acts on
${\mathcal{M}}_{0}(\mathbb{D})$ as a biholomorphic automorphism, and its
action projects down to $T_{v}(\mathbb{D})$, which is biholomorphically
equivalent to $\mathcal{T}_{v}(\mathbb{D}^{*})$ through $\alpha$ and $\beta$,
also as a biholomorphic automorphism. Hence,
$\log(H|_{\mathbb{D}^{*}})^{\prime}\in\mathcal{T}_{v}(\mathbb{D}^{*})\subset{\rm
VMOA}(\mathbb{D}^{*})$ depends on $[\mu]\in T_{v}(\mathbb{D})$
holomorphically.

We see that the boundary extension $b:{\rm VMOA}(\mathbb{D}^{*})\to{\rm
VMO}(\mathbb{S})$ is a bounded linear operator. Moreover, by Proposition 2.1,
the pull-back operator $P_{f}:{\rm VMO}(\mathbb{S})\to{\rm VMO}(\mathbb{S})$
induced by $f\in{\rm SS}(\mathbb{S})$ is also a bounded linear operator.
Therefore,

$\log h^{\prime}\circ f=P_{f}\circ
b({\mathcal{L}}_{H|_{\mathbb{D}^{*}}})\in{\rm VMO}(\mathbb{S})$

in particular depends on $[\mu]\in T_{v}(\mathbb{D})$ holomorphically, and so
does $\log(G(\mu,\mu_{2})|_{\mathbb{S}})^{\prime}$. ∎

By Proposition 3.3, ${\rm SS}_{\mathbb{C}}(\mathbb{S})$ is identified with
$T_{v}(\mathbb{D})\times T_{v}(\mathbb{D}^{*})$. Then, we can represent any
element of ${\rm SS}_{\mathbb{C}}(\mathbb{S})$ by a pair
$([\mu_{1}],[\mu_{2}])$. For $\mu_{1}\in\mathcal{M}_{0}(\mathbb{D})$, the same
notation $\mu_{1}$ denotes the Beltrami coefficient on $\mathbb{D}^{*}$ given
by the reflection (11), and similarly for
$\mu_{2}\in\mathcal{M}_{0}(\mathbb{D}^{*})$,
$\mu_{2}\in\mathcal{M}_{0}(\mathbb{D})$ denotes its reflection. Let $j$ be the
anti-holomorphic involution of ${\rm SS}_{\mathbb{C}}(\mathbb{S})$ defined by
$([\mu_{1}],[\mu_{2}])\mapsto([\mu_{2}],[\mu_{1}])$. Then, the fixed point
locus of $j$ is

${\rm SS}(\mathbb{S})\cong\\{([\mu],[\mu])\in T_{v}(\mathbb{D})\times
T_{v}(\mathbb{D}^{*})\\},$

which is thus the real-analytic submanifold of ${\rm
SS}_{\mathbb{C}}(\mathbb{S})$, and the map $T_{v}(\mathbb{D})\to{\rm
SS}(\mathbb{S})$ given by $[\mu]\mapsto([\mu],[\mu])$ is a real-analytic
homeomorphism onto ${\rm SS}(\mathbb{S})$.

We consider the restriction of $L:{\rm SS}_{\mathbb{C}}(\mathbb{S})\to{\rm
VMO}(\mathbb{S})$ to ${\rm SS}(\mathbb{S})$. We also look at this by composing
the real analytic equivalence $T_{v}(\mathbb{D})\cong{\rm SS}(\mathbb{S})$ and
taking the real part of ${\rm VMO}(\mathbb{S})$. The resulting map is

$\widetilde{L}:T_{v}(\mathbb{D})\to{\rm VMO}_{\mathbb{R}}(\mathbb{S}),$

which sends a strongly symmetric homeomorphism $h$ of $\mathbb{S}$ to
$\log|h^{\prime}|$.

######  Corollary 3.5.

Under the identification $T_{v}(\mathbb{D})\cong{\rm SS}(\mathbb{S})$, the map
$\widetilde{L}:T_{v}(\mathbb{D})\to{\rm VMO}_{\mathbb{R}}(\mathbb{S})$ sending
$h\in{\rm SS}(\mathbb{S})$ to $\log|h^{\prime}|\in{\rm
VMO}_{\mathbb{R}}(\mathbb{S})$ is real-analytic.

This follows immediately from Theorem 3.4. As stated in Corollary 1.2,
$\widetilde{L}$ is in fact a real-analytic homeomorphism whose inverse
$\widetilde{L}^{-1}$ is also real-analytic.

##  4\. The variant of Beurling–Ahlfors extension by heat kernel

In order to prove the second statement of Theorem 1.1, we need the variant of
Beurling–Ahlfors extension by heat kernel introduced in [13]. In this section,
we recall the definitions and results on this extension.

We begin with the classical Beurling–Ahlfors extension given in [2]. 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 $f$ of the real line $\mathbb{R}$, the Beurling–Ahlfors
extension $F(x,y)=(U(x,y),V(x,y))$ for $(x,y)\in\mathbb{U}$ defined by the
convolutions $U(x,y)=(f\ast\phi_{y})(x)$ and $V(x,y)=(f\ast\psi_{y})(x)$ is a
quasiconformal homeomorphism of the upper half-plane $\mathbb{U}$ onto itself.
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 $u\in\rm
BMO(\mathbb{R})$ in general, we consider a mapping
$\gamma_{u}=\gamma:\mathbb{R}\to\mathbb{C}$ given by

$\gamma(x)=\gamma(0)+\int_{0}^{x}e^{u(t)}dt.$ (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
$\mathbb{U}$ as a differentiable map $F_{u}=F:\mathbb{U}\to\mathbb{C}$ defined
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

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

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 $F=F_{u}$ defined by
$(\ref{F})$ is not necessarily a quasiconformal homeomorphism.

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 Beurling–Ahlfors extension by heat kernel. The following
results are obtained in [13, Theorem 4.2] and [29, 30].

######  Theorem 4.1.

For a real-valued $u\in{\rm BMO}_{\mathbb{R}}(\mathbb{R})$ such that $e^{u}$
is an $\rm A_{\infty}$-weight, the map $F_{u}$ given by the variant of
Beurling–Ahlfors extension by heat kernel is a quasiconformal diffeomorphism
of $\mathbb{U}$ onto itself whose complex dilatation $\mu_{u}$ belongs to
$\mathcal{M}(\mathbb{U})$. Moreover, it is bi-Lipschitz with respect to the
hyperbolic metric. If $u\in{\rm VMO}_{\mathbb{R}}(\mathbb{R})$ in addition,
then $\mu_{u}\in\mathcal{M}_{0}(\mathbb{U})$.

We note that a Carleson measure $\lambda$ on $\mathbb{U}$ can be defined by
the condition

$\|\lambda\|_{c}=\sup\frac{\lambda(I\times(0,|I|))}{|I|}<\infty,$

where the supremum is taken over all bounded intervals $I$ on $\mathbb{R}$. A
vanishing Carleson measure and the spaces of the Beltrami coefficients
$\mathcal{M}(\mathbb{U})$ and $\mathcal{M}_{0}(\mathbb{U})$ are defined in the
same way as in the case of $\mathbb{D}$.

##  5\. Local boundedness of the complex dilatations

In this section, we prepare the essential step for the proof of the second
part of Theorem 1.1. This is based on the variant of Beurling-Ahlfors
extension by heat kernel reviewed in Section 4. We will show the local
boundedness of the complex dilatations of these extensions of certain strongly
symmetric embeddings.

First, we state the following two claims on ${\rm BMO}(\mathbb{R})$ given in
[30, Proposition 2.4, Lemma 2.6] which play key roles in the next two lemmas.
Let $I(x,y)\subset\mathbb{R}$ be the interval $(x-y,x+y)$ for any
$x\in\mathbb{R}$ and $y>0$.

######  Proposition 5.1.

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 $k\in\mathbb{N}$ be a positive integer. Then,

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

for a positive constant $C(k)$ depending on $k$.

######  Proposition 5.2.

Let $\varphi$ be a complex-valued function on $\mathbb{R}$ such that
$|\varphi(x)|\leq Ce^{-|x|}$ for some constant $C>0$. For each real-valued
essentially bounded function $u_{0}\in L^{\infty}_{\mathbb{R}}(\mathbb{R})$,
there exists a positive constant $C(u_{0})$ depending on $\|u_{0}\|_{\infty}$
such that every complex-valued function $u\in{\rm BMO}(\mathbb{R})$ with
$\|u-u_{0}\|_{*}\leq C_{JN}/2$ satisfies

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

In the following two lemmas, we prove the local boundedness of the complex
dilatation $\mu_{w}$ of the variant of Beurling–Ahlfors extension by heat
kernel given by $w\in{\rm BMO}(\mathbb{R})$ when $w$ is sufficiently close to
$L^{\infty}_{\mathbb{R}}(\mathbb{R})$. Hereafter, we use the comparability
symbols $\asymp$ and $\lesssim$ instead of mentioning the existence of some
constants that are uniform with respect to the obvious quantities in the
situation.

######  Lemma 5.3.

For any real-valued bounded function $u_{0}\in
L^{\infty}_{\mathbb{R}}(\mathbb{R})$, there are $\delta_{1}>0$ and $M_{1}>0$
such that if $w\in{\rm BMO}(\mathbb{R})$ satisfies
$\|w-u_{0}\|_{*}<\delta_{1}$, then $\|\mu_{w}\|_{\infty}\leq M_{1}$.

###### Proof.

For any $w\in{\rm BMO}(\mathbb{R})$, we set $w=u+iv$ 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

$|\mu_{w}(x,y)|=\frac{|\alpha_{y}\ast e^{w}(x)|}{|\beta_{y}\ast
e^{w}(x)|}=\frac{|\alpha_{y}\ast e^{w-w_{I(x,y)}}(x)|}{|\beta_{y}\ast
e^{w-w_{I(x,y)}}(x)|}.$ (15)

The denominator is estimated from below as

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

Here, since $u$ is real-valued, we see that

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

for $\phi(x)=\frac{1}{\sqrt{\pi}}e^{-x^{2}}$. On the contrary, the
Cauchy–Schwarz inequality and $|e^{ix}-1|\leq|x|$ yield that

$\begin{split}&\quad|\beta_{y}\ast(e^{u-u_{I(x,y)}}(e^{i(v-v_{I(x,y)})}-1))(x)|\\\
&\leq\int_{\mathbb{R}}|\beta_{y}(x-t)|e^{u(t)-u_{I(x,y)}}|e^{i(v(t)-v_{I(x,y)})}-1|dt\\\
&\leq\left(\int_{\mathbb{R}}|\beta_{y}(x-t)|e^{2(u(t)-u_{I(x,y)})}dt\right)^{1/2}\left(\int_{\mathbb{R}}|\beta_{y}(x-t)||v(t)-v_{I(x,y)}|^{2}dt\right)^{1/2}.\end{split}$
(16)

By Proposition 5.2, the first factor of the last line of (16) is bounded if
$\|u-u_{0}\|_{*}$ is sufficiently small. By Proposition 5.1, the second factor
is bounded by a multiple of $\|v\|_{*}$. Thus, the denominator in the fraction
of (15) is bounded away from $0$ if $\|v\|_{*}$ is sufficiently small. In
particular, there is some $0<\delta_{1}<C_{JN}/4$ such that if
$\|u-u_{0}\|_{*}<\delta_{1}/2$ and if $\|v\|_{*}<\delta_{1}/2$, then the
denominator is uniformly bounded away from $0$. Hence, there is some constant
$A>0$ such that if $\|w-u_{0}\|_{*}<\delta_{1}$ then

$|\mu_{w}(x,y)|\leq A|\alpha_{y}\ast e^{w-w_{I(x,y)}}(x)|.$ (17)

In (17), $|\alpha_{y}\ast e^{w-w_{I(x,y)}}(x)|$ coincides with
$|\alpha_{y}\ast(e^{w-w_{I(x,y)}}-1)(x)|$ by $\int_{\mathbb{R}}\alpha(x)dx=0$.
Then, similarly to the above estimate, the Cauchy–Schwarz inequality and
$|e^{z}-1|\leq|z|e^{|z|}$ yield that

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

Again by Propositions 5.1 and 5.2, this is bounded if $\|w-u_{0}\|_{*}\leq
C_{JN}/4$. Thus, we see that $\|\mu_{w}\|_{\infty}\leq M_{1}$ for some
constant $M_{1}>0$ if $\|w-u_{0}\|_{*}<\delta_{1}$. ∎

In the next lemma, the assumption on $\|w-u_{0}\|_{*}$ is necessary for using
lemma 5.3 at the beginning. In the rest of the proof, only the smallness of
$\|u-u_{0}\|_{*}$ is required.

######  Lemma 5.4.

For any real-valued bounded function $u_{0}\in
L^{\infty}_{\mathbb{R}}(\mathbb{R})$, there are $\delta_{2}>0$ and $M_{2}>0$
such that if $w\in{\rm BMO}(\mathbb{R})$ satisfies
$\|w-u_{0}\|_{*}<\delta_{2}$, then $\|\mu_{w}\|_{c}\leq M_{2}$.

###### Proof.

By inequality (17) in the proof of Lemma 5.3, we have

$|\mu_{w}(x,y)|\lesssim|\alpha_{y}\ast
e^{w-w_{I(x,y)}}(x)|=\frac{|\alpha_{y}\ast e^{w}(x)|}{e^{u_{I(x,y)}}}$ (19)

for $w=u+iv$ if $\|w-u_{0}\|_{*}$ is sufficiently small. We also note that if
$\|u-u_{0}\|_{*}$ is sufficiently small, then $e^{ru}$ is an
$A_{\infty}$-weight for $r\in[-4,4]$ (see [15, Corollary IV.5.15]), and
moreover, the $A_{\infty}$-constant $C_{\infty}(e^{ru})$ for $e^{ru}$ depends
only on $u_{0}$ (see also the proof of [30, Lemma 2.6]). By the property of
$A_{\infty}$-weights in (9) and the Cauchy–Schwarz inequality, we have

$\frac{|\alpha_{y}\ast e^{w}(x)|}{e^{u_{I(x,y)}}}\\\
\leq\frac{C_{\infty}(e^{u})|\alpha_{y}\ast e^{w}(x)|}{(e^{u})_{I(x,y)}}\leq
C_{\infty}(e^{u})|\alpha_{y}\ast e^{w}(x)|(e^{-u})_{I(x,y)}$ (20)

for $I(x,y)=(x-y,x+y)$.

Using inequalities (19) and (20), for any bounded interval $I=I(x_{0},t/2)$
for $x_{0}\in\mathbb{R}$ and $t>0$, we have

$\begin{split}&\quad\frac{1}{|I|}\int_{0}^{|I|}\\!\int_{I}|\mu_{w}(x,y)|^{2}\frac{dxdy}{y}\\\
&\lesssim\frac{C_{\infty}(e^{u})^{2}}{|I|}\int_{0}^{|I|}\\!\int_{I}|\alpha_{y}\ast
e^{w}(x)|^{2}((e^{-u})_{I(x,y)})^{2}\frac{dxdy}{y}\\\
&=\frac{C_{\infty}(e^{u})^{2}}{t}\int_{I(x_{0},t/2)}\\!\int_{0}^{t}\left(\frac{1}{|I(x,y)|}\int_{I(x,y)}e^{-u(s)}ds\right)^{2}|\alpha_{y}\ast
e^{w}(x)|^{2}\frac{dy}{y}dx\\\
&=\frac{C_{\infty}(e^{u})^{2}}{t}\int_{I(x_{0},t/2)}\\!\int_{0}^{t}\left(\frac{1}{2y}\int_{|x-s|<y}e^{-u(s)}1_{I(x_{0},N_{0}t)}(s)ds\right)^{2}|\alpha_{y}\ast
e^{w}(x)|^{2}\frac{dy}{y}dx.\end{split}$

Here, we have replaced $e^{-u(s)}$ with the product
$e^{-u(s)}1_{I(x_{0},N_{0}t)}(s)$ of the characteristic function by a fact
that if $x\in I(x_{0},t/2)$ and $|x-s|<y\leq t$ then $s\in I(x_{0},N_{0}t)$
for any $N_{0}\geq 2$. We will choose $N_{0}$ sufficiently large and fix it
later. The last line above is estimated further as follows.

$\begin{split}&\quad\frac{C_{\infty}(e^{u})^{2}}{t}\int_{I(x_{0},t/2)}\\!\int_{0}^{t}\left(\frac{1}{2y}\int_{|x-s|<y}e^{-u(s)}1_{I(x_{0},N_{0}t)}(s)ds\right)^{2}|\alpha_{y}\ast
e^{w}(x)|^{2}\frac{dy}{y}dx\\\
&\leq\frac{C_{\infty}(e^{u})^{2}}{t}\int_{I(x_{0},t/2)}\sup_{y>0}\left(\frac{1}{2y}\int_{|x-s|<y}e^{-u(s)}1_{I(x_{0},N_{0}t)}(s)ds\right)^{2}\left(\int_{0}^{t}|\alpha_{y}\ast
e^{w}(x)|^{2}\frac{dy}{y}\right)dx\\\ &\leq
C_{\infty}(e^{u})^{2}\left[\frac{1}{t}\int_{I(x_{0},t/2)}\left(\sup_{y>0}\left\\{\frac{1}{2y}\int_{|x-s|<y}e^{-u(s)}1_{I(x_{0},N_{0}t)}(s)ds\right\\}\right)^{4}dx\right]^{1/2}\\\
&\qquad\qquad\qquad\qquad\times\left[\frac{1}{t}\int_{I(x_{0},t/2)}\left(\int_{0}^{t}|\alpha_{y}\ast
e^{w}(x)|^{2}\frac{dy}{y}\right)^{2}dx\right]^{1/2}.\end{split}$ (21)

First, we consider the first integral factor of the last line of (21). Here,
we use the $L^{4}$-boundedness of the Hardy–Littlewood maximal function

$M(e^{-u}1_{I(x_{0},N_{0}t)})(x)=\sup_{y>0}\left(\frac{1}{2y}\int_{|x-s|<y}e^{-u(s)}1_{I(x_{0},N_{0}t)}(s)ds\right).$

Then, there is a constant $C_{1}>0$ such that

$\begin{split}&\quad\frac{1}{t}\int_{I(x_{0},t/2)}\left(\sup_{y>0}\left\\{\frac{1}{2y}\int_{|x-s|<y}e^{-u(s)}1_{I(x_{0},N_{0}t)}(s)ds\right\\}\right)^{4}dx\\\
&\leq\frac{1}{t}\int_{\mathbb{R}}\left(\sup_{y>0}\left\\{\frac{1}{2y}\int_{|x-s|<y}e^{-u(s)}1_{I(x_{0},N_{0}t)}(s)ds\right\\}\right)^{4}dx\\\
&=\frac{1}{t}\int_{\mathbb{R}}\left(M(e^{-u}1_{I(x_{0},N_{0}t)})(x)\right)^{4}dx\leq\frac{C_{1}}{t}\int_{I(x_{0},N_{0}t)}e^{-4u(s)}ds.\end{split}$
(22)

For the estimate of the last term, we recall that, if $\|u-u_{0}\|_{*}$ is
sufficiently small, then $e^{-4u}$ is an $A_{\infty}$-weight. In addition,
throughout the estimate of the complex dilatation $\mu_{w}(x,y)$, we may
assume that $u_{I(x_{0},N_{0}t)}=0$ for the given and fixed constants $x_{0}$,
$t$ and $N_{0}$ because adding a constant to $u$ corresponds to a dilation of
$F$, which does not change the complex dilatation $\mu_{w}$. (When $x_{0}$,
$t$, or $N_{0}$ change, we regard that the assumption is renewed accordingly.)
These facts yield that

$\begin{split}&\quad\frac{1}{|I(x_{0},N_{0}t)|}\int_{I(x_{0},N_{0}t)}e^{-4u(s)}ds\\\
&=\frac{1}{|I(x_{0},N_{0}t)|}\int_{I(x_{0},N_{0}t)}e^{-4(u(s)-u_{I(x_{0},N_{0}t)})}ds\\\
&\leq
C_{\infty}(e^{-4u})\exp\left(\frac{4}{|I(x_{0},N_{0}t)|}\int_{I(x_{0},N_{0}t)}|u(s)-u_{I(x_{0},N_{0}t)}|ds\right)\leq
C_{\infty}(e^{-4u})e^{4\|u\|_{*}}.\end{split}$ (23)

Therefore, by (22) and (23), we have

$\begin{split}\frac{1}{t}\int_{I(x_{0},t/2)}\left(\sup_{y>0}\left\\{\frac{1}{2y}\int_{|x-s|<y}e^{-u(s)}1_{I(x_{0},N_{0}t)}(s)ds\right\\}\right)^{4}dx\leq
2C_{1}N_{0}C_{\infty}(e^{-4u})e^{4\|u\|_{*}},\end{split}$

which shows the local boundedness of the first integral factor of the last
line of (21).

Next, we consider the second integral factor of the last line of (21). In the
following, we decompose it further by dividing $\mathbb{R}$ into the interval
$I(x_{0},Nt)$ and its complement $I(x_{0},Nt)^{c}$ for a sufficiently large
$N>0$:

$\begin{split}&\quad\frac{1}{t}\int_{I(x_{0},t/2)}\left(\int_{0}^{t}|\alpha_{y}\ast
e^{w}(x)|^{2}\frac{dy}{y}\right)^{2}dx\\\
&\leq\frac{8}{t}\int_{I(x_{0},t/2)}\left(\int_{0}^{t}|\alpha_{y}\ast(e^{w}1_{I(x_{0},Nt)})(x)|^{2}\frac{dy}{y}\right)^{2}dx\\\
&\qquad\qquad+\frac{8}{t}\int_{I(x_{0},t/2)}\left(\int_{0}^{t}|\alpha_{y}\ast(e^{w}1_{I(x_{0},Nt)^{c}})(x)|^{2}\frac{dy}{y}\right)^{2}dx.\end{split}$
(24)

We consider the first term of the right side of inequality (24). We utilize
the Littlewood–Paley operator defined by the rapidly decreasing function
$\alpha$ with $\int_{\mathbb{R}}\alpha(x)dx=0$:

$S_{\alpha}(e^{w}1_{I(x_{0},Nt)})(x)=\left(\int_{0}^{\infty}|\alpha_{y}\ast(e^{w}1_{I(x_{0},Nt)})(x)|^{2}\frac{dy}{y}\right)^{1/2}.$

By the $L^{4}$-boundedness of the Littlewood–Paley operator $S_{\alpha}$ (see
[3, p.363]), we have the estimate of the first term: there is some constant
$C_{2}>0$ such that

$\begin{split}&\quad\frac{8}{t}\int_{I(x_{0},t/2)}\left(\int_{0}^{t}|\alpha_{y}\ast(e^{w}1_{I(x_{0},Nt)})(x)|^{2}\frac{dy}{y}\right)^{2}dx\\\
&\leq\frac{8}{t}\int_{\mathbb{R}}\left(\int_{0}^{\infty}|\alpha_{y}\ast(e^{w}1_{I(x_{0},Nt)})(x)|^{2}\frac{dy}{y}\right)^{2}dx=\frac{8}{t}\int_{\mathbb{R}}(S_{\alpha}(e^{w}1_{I(x_{0},Nt)})(x))^{4}dx\\\
&\leq\frac{8C_{2}}{t}\int_{\mathbb{R}}|e^{w(x)}|^{4}1_{I(x_{0},Nt)}(x)dx=\frac{8C_{2}}{t}\int_{I(x_{0},Nt)}e^{4u(x)}dx.\end{split}$
(25)

We consider the second term of the right side of inequality (24). We have an
inequality

$\begin{split}|\alpha_{y}\ast(e^{w}1_{I(x_{0},Nt)^{c}})(x)|&\leq\int_{I(x_{0},Nt)^{c}}|\alpha_{y}(x-s)|e^{u(s)}ds\end{split}$

by assuming $x\in I(x_{0},t/2)$. For this estimate, we use a fact that the
$A_{\infty}$-weight $e^{u}$ has the doubling property with some constant
$\rho=\rho(e^{u})>1$ as in (7). This constant can be estimated in terms of the
$A_{\infty}$-constant $C_{\infty}(e^{u})$, and hence this depends only on
$u_{0}$ when $\|u-u_{0}\|_{*}$ is sufficiently small. We also note that
$|\alpha|$ is an even function.

Let $n_{0}=n_{0}(y,t,N)\in\mathbb{N}$ satisfy $2^{n_{0}-1}=(N-1)t/y$ (we may
adjust $N$ so that $n_{0}$ becomes an integer). Then, for $x\in I(x_{0},t/2)$,
we see that

$\displaystyle\int_{I(x_{0},Nt)^{c}}|\alpha_{y}(x-s)|e^{u(s)}ds$
$\displaystyle\leq\int_{|s-x_{0}|\geq(N-1)t}e^{u(s)}|\alpha_{y}(s)|ds$
$\displaystyle=\sum_{n=n_{0}}^{\infty}\int_{2^{n-1}y\leq|s-x_{0}|<2^{n}y}e^{u(s)}|\alpha_{y}(s)|ds$
$\displaystyle\leq\sum_{n=n_{0}}^{\infty}\rho^{n}|\alpha_{y}(2^{n-1}y)|\int_{|s-x_{0}|<y}e^{u(s)}ds.$

Here, by $|\alpha(x)|\leq Ce^{-|x|}$, we have
$\rho^{n}|\alpha_{y}(2^{n-1}y)|\leq C\rho^{n}e^{-2^{n-1}}/y$. For $n\geq
n_{0}(y,t,N)$, we may assume that $\rho^{n}e^{-2^{n-2}}\leq 1$. In fact, by
$2^{n_{0}-1}\geq N-1$, this holds when $N$ is sufficiently large. Moreover, if
$\|u-u_{0}\|_{*}<C_{JN}/2$, then the John–Nirenberg inequality (5) implies
that

$\begin{split}\int_{|s-x_{0}|<y}e^{u(s)}ds&\leq\int_{I(x_{0},Nt)}e^{u(s)}ds=\int_{I(x_{0},Nt)}e^{u(s)-u_{I(x_{0},Nt)}}ds\\\
&\leq\frac{2Nte^{2\|u_{0}\|_{\infty}}}{|I(x_{0},Nt)|}\int_{I(x_{0},Nt)}e^{|u(s)-u_{0}(s)-u_{I(x_{0},Nt)}+(u_{0})_{I(x_{0},Nt)}|}dt\\\
&\leq 2Nte^{2\|u_{0}\|_{\infty}}(2C_{0}+1)=DNt\end{split}$

for some constant $D>0$ depending on $\|u_{0}\|_{\infty}$. Therefore, we
obtain that if $x\in I(x_{0},t/2)$ then

$\displaystyle|\alpha_{y}\ast(e^{w}1_{I(x_{0},Nt)^{c}})(x)|$
$\displaystyle\leq\frac{CDNt}{y}\sum_{n=n_{0}}^{\infty}\frac{\rho^{n}}{e^{2^{n-2}}}\frac{1}{e^{2^{n-2}}}\leq\frac{CDNt}{y}\exp\left(-\frac{Nt}{4y}\right).$

Hence, for $x\in I(x_{0},t/2)$,

$\begin{split}\int_{0}^{t}|\alpha_{y}\ast(e^{w}1_{I(x_{0},Nt)^{c}})(x)|^{2}\frac{dy}{y}\leq(CDNt)^{2}\int_{0}^{t}\exp\left(-\frac{Nt}{2y}\right)\frac{dy}{y^{3}}\leq(CD)^{2}N^{2}e^{-N/2}.\end{split}$

From this, we obtain that

$\frac{8}{t}\int_{I(x_{0},t/2)}\left(\int_{0}^{t}|\alpha_{y}\ast(e^{w}1_{I(x_{0},Nt)^{c}})(x)|^{2}\frac{dy}{y}\right)^{2}dx\to
0$

as $N\to\infty$ uniformly for $x_{0}\in\mathbb{R}$ and $t>0$.

By the above argument, we can make the second term of the right side of
inequality (24) arbitrarily small if we choose $N$ sufficiently large. We fix
such an $N$ as $N_{0}$. Then, we return to the estimate of the first term of
the right side of inequality (24) as in (25). By the assumption
$u_{I(x_{0},N_{0}t)}=0$, we have

$\begin{split}&\quad\frac{8C_{2}}{t}\int_{I(x_{0},N_{0}t)}e^{4u(x)}dx\\\
&=\frac{16C_{2}N_{0}}{|I(x_{0},N_{0}t)|}\int_{I(x_{0},N_{0}t)}e^{4(u(x)-u_{I(x_{0},N_{0}t)})}dx\\\
&\leq
16C_{2}N_{0}C_{\infty}(e^{4u})\exp\left(\frac{4}{|I(x_{0},N_{0}t)|}\int_{I(x_{0},N_{0}t)}|u(x)-u_{I(x_{0},N_{0}t)}|dx\right)\\\
&\leq 16C_{2}N_{0}C_{\infty}(e^{4u})e^{4\|u\|_{*}}.\end{split}$ (26)

This shows the local boundedness of the second integral factor of the last
line of (21).

The required constants $\delta_{2}>0$ and $M_{2}>0$ can be found from the
above arguments. This completes the proof of Lemma 5.4. ∎

##  6\. Arguments for holomorphy and real-analycity (Theorem 1.1 (ii))

In this section, we will complete the proof of the second part of Theorem 1.1,
whose precise statement is given in Theorem 6.4. In the previous section, we
have seen the local boundedness of the complex dilatations of the variant of
Beurling–Ahlfors extension by heat kernel of strongly symmetric embeddings.
Our remaining task is to show the holomorphy of this dependence. This step has
already appeared in several work, but here we prove this claim from a general
viewpoint.

In addition, since the variant of Beurling–Ahlfors extension by heat kernel is
available on the real line but not on the unit circle, we first prove our main
result of this section in the real line (Theorem 6.2), and then transfer it to
the case of the unit circle by using lift and projection maps.

In the theory of infinite dimensional Teichmüller spaces, there are several
occasions where we have to show the holomorphy of mappings to function spaces
$F$. Useful criteria and applications can be found in [18, Lemma V.5.1]
(Remark: an assumption on the local boundedness of $f:U\to F$ is dropped in
its statement (i)). In the typical case where $F$ is a complex Banach space of
certain measurable functions on a domain $\Omega$ in $\mathbb{C}$, we can
formulate the lemma below, which is based on the general result cited above
claiming that a weak holomorphy implies the holomorphy under the local
boundedness (see also [4, p.28]). In fact, this kind of arguments appeared in
[25, Lemma 6.1], and we extract essential ideas from it and arrange them in a
widely applicable form. In the case where $F$ is of the supremum norm, we can
find a similar result in [11, Lemma 3.4].

Let $F(\Omega)$ be a complex Banach space of measurable functions on a domain
$\Omega\subset\mathbb{C}$. We say that the norm $\|\cdot\|$ of this space is
of integral type if any measurable function

$f:\Omega\times[0,1]\to{\mathbb{C}},\qquad(z,s)\mapsto f_{s}(z)$

such that $f_{s}(z)$ belongs to $F(\Omega)$ for all $s\in[0,1]$ satisfies

$\left\|\int_{0}^{1}f_{s}(z)dt\right\|\leq\sup_{0\leq s\leq 1}\|f_{s}(z)\|.$

######  Lemma 6.1.

Let $E$ be a complex Banach space and $F(\Omega)$ a complex Banach space of
measurable functions on a domain $\Omega\subset\mathbb{C}$ with integral type
norm $\|\cdot\|$. Let $\Lambda:U\to F(\Omega)$ be a bounded function on an
open subset $U$ of $E$ that satisfies the following property: For any
$w_{0}\in U$ and any $w_{1}\in E$, there is an $\epsilon>0$ such that

$\lambda(\zeta)(z)=\Lambda(w_{0}+\zeta w_{1})(z)\in\mathbb{C}$

is a holomorphic function on $\\{\zeta\in\mathbb{C}\mid|\zeta|\leq
2\epsilon\\}$ for almost every $z\in\Omega$. Then, $\Lambda$ is holomorphic on
$U$.

###### Proof.

We will show that $\Lambda$ is Gâteaux holomorphic. By assumption,
$\lambda(\zeta)(z)$ is holomorphic on $\\{\zeta\in\mathbb{C}\mid|\zeta|\leq
2\epsilon\\}$ for almost every $z\in\Omega$. Fix such $z$. By using the Cauchy
integral formula, we have

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

for any $\zeta_{0}$ and $\zeta$ with $|\zeta_{0}|<\epsilon$ and
$|\zeta|<\epsilon$. Let $M=\sup_{w\in U}\|\Lambda(w)\|<\infty$. Then, the
integral type property of the norm $\|\cdot\|$ implies that

$\displaystyle\left\|\frac{\lambda(\zeta)(z)-\lambda(\zeta_{0})(z)}{\zeta-\zeta_{0}}-\left.\frac{d}{d\zeta}\right|_{\zeta=\zeta_{0}}\lambda(\zeta)(z)\right\|\leq\frac{2M}{\epsilon^{2}}|\zeta-\zeta_{0}|.$

Consequently, the limit

$\lim_{\zeta\to\zeta_{0}}\frac{\lambda(\zeta)-\lambda(\zeta_{0})}{\zeta-\zeta_{0}}=\left.\frac{d}{d\zeta}\right|_{\zeta=\zeta_{0}}\lambda(\zeta)$

exists in $F(\Omega)$. This shows that $\Lambda$ is Gâteaux holomorphic on
$U$. It is known that a locally bounded Gâteaux holomorphic function is
holomorphic in the sense that it is Fréchet differentiable (see [5, Theorem
14.9] and [19, Theorem 36.5]). Thus, we see that $\Lambda$ is holomorphic on
$U$. ∎

We return to the setting in Section 5 concerning the investigation of the
variant of Beurling–Ahlfors extension by heat kernel. We apply the above lemma
to obtain the following theorem.

######  Theorem 6.2.

For each $u_{0}\in L_{\mathbb{R}}^{\infty}(\mathbb{R})$, there is a
neighborhood $U(u_{0})$ in ${\rm BMO}(\mathbb{R})$ such that the complex
dilatation $\mu_{w}$ of the variant of Beurling–Ahlfors extension by heat
kernel $F_{w}:\mathbb{U}\to\mathbb{C}$ belongs to $\mathcal{M}(\mathbb{U})$
for every $w\in U(u_{0})$. Moreover, the map
$\Lambda_{u_{0}}:U(u_{0})\to\mathcal{M}(\mathbb{U})$ defined by
$w\mapsto\mu_{w}$ is holomorphic. This in particular implies that the map
$\Lambda_{u_{0}}$ restricted on $U(u_{0})\cap{\rm
BMO}_{\mathbb{R}}(\mathbb{R})$ is real-analytic.

###### Proof.

We first prove that $\Lambda_{u_{0}}$ is a holomorphic map to
$\mathcal{L}(\mathbb{U})$ on a neighborhood

$\widetilde{U}(u_{0})=\\{w\in{\rm
BMO}(\mathbb{R})\mid\|w-u_{0}\|_{*}<\delta\\}$

of $u_{0}$, where $\delta=\min\\{\delta_{1},\delta_{2}\\}>0$ is defined from
Lemmas 5.3 and 5.4. Then, $\Lambda_{u_{0}}$ is bounded on
$\widetilde{U}(u_{0})$. Moreover, we choose $\epsilon$ with
$0<\epsilon<(\delta-\|w_{0}\|_{*})/(2\|w_{1}\|_{*})$ so that $w_{0}+\zeta
w_{1}\in\widetilde{U}(u_{0})$ when $|\zeta|\leq 2\epsilon$. Explicit
representation (14) shows that the complex-valued function
$\lambda(\zeta)(z)=\Lambda_{u_{0}}(w_{0}+\zeta w_{1})(z)$ is holomorphic on
$\\{\zeta\in\mathbb{C}\mid|\zeta|\leq 2\epsilon\\}$ with $z\in\mathbb{U}$
fixed. To prove the holomorphy of $\Lambda_{u_{0}}$, it suffices to see that
the norm $\|\cdot\|_{\infty}+\|\cdot\|_{c}$ on $\mathcal{L}(\mathbb{U})$ is of
integral type by Lemma 6.1.

For a measurable function

$\mu:\mathbb{U}\times[0,1]\to{\mathbb{C}},\qquad(z,s)\mapsto\mu_{s}(z)$

such that $\mu_{s}(z)$ belongs to $\mathcal{L}(\mathbb{U})$ for all
$s\in[0,1]$, we have

$\left\|\int_{0}^{1}\mu_{s}(z)ds\right\|_{\infty}\leq\sup_{0\leq s\leq
1}\|\mu_{s}(z)\|_{\infty};$
$\displaystyle\left\|\int_{0}^{1}\mu_{s}(z)ds\right\|_{c}^{2}$
$\displaystyle=\sup_{x_{0}\in{\mathbb{R}},\
t>0}\frac{1}{t}\int_{0}^{t}\int_{I(x_{0},t/2)}\left|\int_{0}^{1}\mu_{s}(z)ds\right|^{2}\frac{dxdy}{y}$
$\displaystyle\leq\sup_{x_{0}\in{\mathbb{R}},\
t>0}\frac{1}{t}\int_{0}^{t}\int_{I(x_{0},t/2)}\left(\int_{0}^{1}|\mu_{s}(z)|^{2}ds\right)\frac{dxdy}{y}$
$\displaystyle\leq\int_{0}^{1}\left(\sup_{x_{0}\in{\mathbb{R}},\
t>0}\frac{1}{t}\int_{0}^{t}\int_{I(x_{0},t/2)}|\mu_{s}(z)|^{2}\frac{dxdy}{y}\right)ds\leq\sup_{0\leq
s\leq 1}\|\mu_{s}(z)\|_{c}^{2}.$

These inequalities show that $\|\cdot\|_{\infty}+\|\cdot\|_{c}$ is of integral
type.

Thus, we obtain that
$\Lambda_{u_{0}}:\widetilde{U}(u_{0})\to\mathcal{L}(\mathbb{U})$ is
holomorphic. In particular, $\Lambda_{u_{0}}$ is continuous there. Since
$\Lambda_{u_{0}}(u_{0})\in\mathcal{M}(\mathbb{U})$ by Theorem 4.1 and
$\mathcal{M}(\mathbb{U})$ is open in $\mathcal{L}(\mathbb{U})$, by taking a
smaller neighborhood $U(u_{0})\subset\widetilde{U}(u_{0})$, the image of
$U(u_{0})$ under $\Lambda_{u_{0}}$ is contained in $\mathcal{M}(\mathbb{U})$.
∎

######  Remark.

Lemma 6.1 can be widely applied for showing holomorphy in the infinite
dimensional Teichmüller theory because almost all complex Banach spaces used
in this theory are of integral type norms such as the spaces of Beltrami
differentials and the spaces of holomorphic differentials. The point-wise
holomorphy is also easily verified. Proposition 3.1 and Theorem 6.2 are two
cases of all.

Finally, we transfer the result of Theorem 6.2 applied to ${\rm
VMO}_{\mathbb{R}}(\mathbb{R})\subset{\rm BMO}_{\mathbb{R}}(\mathbb{R})$ to the
case of the unit circle $\mathbb{S}$. We consider the lift $\tilde{u}$ of an
element $u$ of ${\rm VMO}_{\mathbb{R}}(\mathbb{S})$ to $\mathbb{R}$ under the
universal covering $\mathbb{R}\to\mathbb{S}$ given by $x\mapsto e^{2\pi ix}$.
Namely, we define $\tilde{u}(x)=u(e^{2\pi ix})$ so that it satisfies
$\tilde{u}(x+1)=\tilde{u}(x)$. From this periodicity,
$\|u\|_{*}\leq\|\tilde{u}\|_{*}\leq 3\|u\|_{*}$ holds by [21, Lemma 2.2], and
$\tilde{u}$ belongs to ${\rm VMO}_{\mathbb{R}}(\mathbb{R})$ by $u\in{\rm
VMO}_{\mathbb{R}}(\mathbb{S})$. Since this correspondence ${\rm
VMO}_{\mathbb{R}}(\mathbb{S})\to{\rm VMO}_{\mathbb{R}}(\mathbb{R})$ is a
bounded linear map by this property, it is in particular real-analytic.

We denote the set of all such lifts by ${\rm
VMO}_{\mathbb{R}}^{\\#}(\mathbb{R})$. Then, ${\rm
VMO}_{\mathbb{R}}^{\\#}(\mathbb{R})$ is a real Banach subspace of ${\rm
VMO}_{\mathbb{R}}(\mathbb{R})$. Moreover, we see that
$L^{\infty}_{\mathbb{R}}(\mathbb{R})$ is dense in ${\rm
VMO}^{\\#}_{\mathbb{R}}(\mathbb{R})$ since
$L^{\infty}_{\mathbb{R}}(\mathbb{S})$ is dense in ${\rm
VMO}_{\mathbb{R}}(\mathbb{S})$ under the BMO topology induced by the norm
$\|\cdot\|_{*}$ (see [24], [14, Theorem VI.5.1]). Hence, the map
$\Lambda_{u_{0}}$, which is given locally in Theorem 6.2, can be extended
globally to some neighborhood of ${\rm VMO}^{\\#}_{\mathbb{R}}(\mathbb{R})$ by
the same correspondence $w\mapsto\mu_{w}$. We denote its restriction to ${\rm
VMO}^{\\#}_{\mathbb{R}}(\mathbb{R})$ by $\Lambda$. Moreover, we also consider
the closed subspace of $\mathcal{M}_{0}(\mathbb{U})$ consisting of all
elements $\tilde{\mu}$ satisfying the periodicity
$\tilde{\mu}(z+1)=\tilde{\mu}(z)$, which is denoted by
$\mathcal{M}^{\\#}_{0}(\mathbb{U})$.

######  Proposition 6.3.

$\Lambda$ is a real-analytic map from ${\rm
VMO}^{\\#}_{\mathbb{R}}(\mathbb{R})$ to $\mathcal{M}^{\\#}_{0}(\mathbb{U})$.

###### Proof.

By Theorem 6.2, we see that the map $\Lambda$ from ${\rm
VMO}_{\mathbb{R}}^{\\#}(\mathbb{R})$ into $\mathcal{M}(\mathbb{U})$ defined by
$u\mapsto\mu_{u}$ is real-analytic. Further, by Theorem 4.1 together with the
period preserving property of this extension, the image of ${\rm
VMO}_{\mathbb{R}}^{\\#}(\mathbb{R})$ under $\Lambda$ is contained in
$\mathcal{M}^{\\#}_{0}(\mathbb{U})$. Consequently, $\Lambda:{\rm
VMO}_{\mathbb{R}}^{\\#}(\mathbb{R})\to\mathcal{M}^{\\#}_{0}(\mathbb{U})$ is
real-analytic. ∎

By the periodicity of $\tilde{u}\in{\rm VMO}_{\mathbb{R}}^{\\#}(\mathbb{R})$
under $x\mapsto x+1$, we see that the variant of Beurling–Ahlfors extension by
heat kernel $F_{\tilde{u}}$ satisfies $F_{\tilde{u}}(z+1)=F_{\tilde{u}}(z)+1$.
Thus, $F_{\tilde{u}}$ can be projected to a quasiconformal homeomorphism
$G_{u}$ of the punctured disk $\mathbb{D}\backslash\\{0\\}$ onto itself such
that $G_{u}(e^{2\pi iz})=e^{2\pi iF_{\tilde{u}}(z)}$ for $z\in\mathbb{U}$.
Clearly, $G_{u}$ can be extended quasiconformally to $0$, and the resulting
mapping from $\mathbb{D}$ onto itself is still denoted by $G_{u}$. Its complex
dilatation is denoted by $\nu_{u}$.

Summing up, for any $u\in{\rm VMO}_{\mathbb{R}}(\mathbb{S})$, we have the
quasiconformal homeomorphism $G_{u}$ of $\mathbb{D}$ onto itself with complex
dilatation $\nu_{u}$. We regard this $G_{u}$ as the variant of
Beurling–Ahlfors extension by heat kernel transferred to the unit disk. The
map defined by the correspondence $u\mapsto\nu_{u}$ is denoted by
$\widetilde{\Lambda}$.

######  Theorem 6.4.

$\widetilde{\Lambda}$ is a real-analytic map from ${\rm
VMO}_{\mathbb{R}}(\mathbb{S})$ to $\mathcal{M}_{0}(\mathbb{D})$.

###### Proof.

The complex dilatations $\mu_{\tilde{u}}$ of $F_{\tilde{u}}$ and $\nu_{u}$ of
$G_{u}$ satisfy

$\nu_{u}(e^{2\pi iz})\overline{e^{2\pi iz}}/e^{2\pi iz}=-\mu_{\tilde{u}}(z)$

for $z\in\mathbb{U}$, and in particular,
$\|\nu_{u}\|_{\infty}=\|\mu_{\tilde{u}}\|_{\infty}$. We will further prove
that there exists a constant $C>0$ such that

$\|\nu_{u}\|_{\infty}+\|\nu_{u}\|_{c}\leq
C(\|\mu_{\tilde{u}}\|_{\infty}+\|\mu_{\tilde{u}}\|_{c}),$

and moreover $\nu_{u}\in\mathcal{M}_{0}(\mathbb{D})$ by
$\mu_{\tilde{u}}\in\mathcal{M}^{\\#}_{0}(\mathbb{U})$. These properties show
that the projection from $\mathcal{M}^{\\#}_{0}(\mathbb{U})$ to
$\mathcal{M}_{0}(\mathbb{D})$ defined by $\mu_{\tilde{u}}\mapsto\nu_{u}$ is a
bounded linear map, and thus it is real-analytic.

For any $h\in(0,1]$ and $\theta_{0}\in[0,2\pi)$, we take a sector
$S_{h,\theta_{0}}$ defined by (2). If $h\in(\frac{1}{2},1]$, then by $e^{2\pi
y}-1\geq 2\pi y$ for $y>0$, we have

$\begin{split}&\quad\frac{1}{h}\iint_{S_{h,\theta_{0}}}\frac{|\nu_{u}(\zeta)|^{2}}{1-|\zeta|^{2}}d\xi
d\eta\leq\frac{1}{h}\iint_{|\zeta|<\frac{1}{2}}\frac{|\nu_{u}(\zeta)|^{2}}{1-|\zeta|^{2}}d\xi
d\eta+\frac{1}{h}\iint_{\frac{1}{2}\leq|\zeta|<1}\frac{|\nu_{u}(\zeta)|^{2}}{1-|\zeta|^{2}}d\xi
d\eta\\\
&\leq\frac{2\pi}{h}\int_{0}^{1/2}\frac{rdr}{1-r^{2}}\,\|\nu_{u}\|_{\infty}^{2}+\frac{4\pi^{2}}{h}\int_{0}^{\log
2/(2\pi)}\int_{-1/2}^{1/2}\frac{|\nu_{u}(e^{2\pi iz})|^{2}}{1-|e^{2\pi
iz}|^{2}}|e^{2\pi iz}|^{2}dxdy\\\
&\leq\frac{\pi}{3h}\|\mu_{\tilde{u}}\|_{\infty}^{2}+\frac{\pi}{h}\int_{0}^{1}\int_{-1/2}^{1/2}\frac{|\mu_{\tilde{u}}(z)|^{2}}{y}dxdy\lesssim\|\mu_{\tilde{u}}\|_{\infty}^{2}+\|\mu_{\tilde{u}}\|_{c}^{2}.\end{split}$
(27)

Similarly, if $h\in(0,\frac{1}{2}]$, we have

$\begin{split}\quad\frac{1}{h}\iint_{S_{h,\theta_{0}}}\frac{|\nu_{u}(\zeta)|^{2}}{1-|\zeta|^{2}}d\xi
d\eta&\leq\frac{\pi}{h}\int_{0}^{\frac{1}{2\pi}\log\frac{1}{1-h}}\int_{\theta_{0}/(2\pi)-h/2}^{\theta_{0}/(2\pi)+h/2}\frac{|\mu_{\tilde{u}}(z)|^{2}}{y}dxdy\\\
&\leq\frac{\pi}{h}\int_{0}^{h}\int_{\theta_{0}/(2\pi)-h/2}^{\theta_{0}/(2\pi)+h/2}\frac{|\mu_{\tilde{u}}(z)|^{2}}{y}dxdy\lesssim\|\mu_{\tilde{u}}\|_{c}^{2}.\end{split}$
(28)

Therefore, we have $\|\nu_{u}\|_{\infty}+\|\nu_{u}\|_{c}\leq
C(\|\mu_{\tilde{u}}\|_{\infty}+\|\mu_{\tilde{u}}\|_{c})$ for some constant
$C>0$. On the other hand, we also see that the left side of the first line of
(28) tends to $0$ uniformly as $h\to 0$ since the right side of the first line
of (28) tends to $0$ uniformly as $h\to 0$ by
$\mu_{\tilde{u}}\in\mathcal{M}_{0}(\mathbb{U})$. Then, we have
$\nu_{u}\in\mathcal{M}_{0}(\mathbb{D})$.

We can conclude that $\widetilde{\Lambda}$ is the composition of the following
three maps:

$\displaystyle{\rm VMO}_{\mathbb{R}}(\mathbb{S})\ni u$
$\displaystyle\mapsto\tilde{u}\in{\rm VMO}_{\mathbb{R}}^{\\#}(\mathbb{R}),$
$\displaystyle{\rm VMO}_{\mathbb{R}}^{\\#}(\mathbb{R})\ni\tilde{u}$
$\displaystyle\mapsto\mu_{\tilde{u}}\in\mathcal{M}^{\\#}_{0}(\mathbb{U}),\
{\rm and}$ $\displaystyle\mathcal{M}_{0}^{\\#}(\mathbb{U})\ni\mu_{\tilde{u}}$
$\displaystyle\mapsto\nu_{u}\in\mathcal{M}_{0}(\mathbb{D}),$

each of which is real-analytic. Thus, the composition $\widetilde{\Lambda}$ is
real-analytic. This completes the proof of Theorem 6.4. ∎

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