# BMO-Teichmüller spaces revisited

Huaying Wei Department of Mathematics and Statistics, Jiangsu Normal
University, Xuzhou 221116, PR China. Email<EMAIL_ADDRESS>Research
supported by the National Natural Science Foundation of China (Grant No.
11501259) and the Natural Science Foundation of Colleges of Jiangsu Province
(Grant No. 15KJB110006).  Michel Zinsmeister Corresponding author. MAPMO,
Université d’ Orléans, Orléans Cedex 2, France. Email<EMAIL_ADDRESS>
Abstract. In [6] the equivalence among three definitions of BMO-Teichmüller
spaces associated with a Fuchsian group was proven using the Douady-Earle
extension operator. In this paper, we show that these equivalences are
actually biholomorphisms. In [6] it was further shown that the Douady-Earle
extension operator is continuous at the origin. We improve this result by
showing Gâteaux-differentiability at this point.

Key words and phrases strongly quasisymmetric homeomorphisms, Carleson
measures, Douady-Earle extension.

2010 Mathematics Subject Classification 30C62, 30F60, 30H35.

##  1 Introduction

Let $h$ be a quasisymmetric homeomorphism of the unit circle $\mathbb{S}$.
Ahlfors and Beurling [5] have been the first to prove that $h$ may be extended
to a quasiconformal homeomorphism of the unit disk $\mathbb{D}$. Later Douady
and Earle [7] found a conformally natural way to extend $h$ to a
quasiconformal homeomorphism of $\mathbb{D}$. More precisely, their extension,
called Douady-Earle extension (or barycentric extension), denoted by $E(h)$,
satisfies

$E(\tau\circ h\circ\alpha)=\tau\circ E(h)\circ\alpha$

for any couple $\tau$, $\alpha$ of automorphisms of $\mathbb{D}$. The Douady-
Earle extension plays an important role applied to quasisymmetric
homeomorphisms of $\mathbb{S}$ in the complex analytic theory of Teichmüller
spaces. In this paper, our study on BMO-Teichmüller theory is based on good
properties of the Douady-Earle extension.

The universal Teichmüller space $T$ can be defined as the space
$QS_{*}(\mathbb{S})$ of all normalized quasisymmetric homeomorphisms of
$\mathbb{S}$. In this setting, the Teichmüller projection $\varPhi$ is
regarded as the boundary extension map on the space $QC_{*}(\mathbb{D})$ of
all normalized quasiconformal homeomorphisms of $\mathbb{D}$. By the
measurable Riemann mapping theorem, we can identity the latter space with the
space of Beltrami coefficients $M(\mathbb{D})=L^{\infty}(\mathbb{D})_{1}$,
which is the open unit ball of measurable functions on $\mathbb{D}$ with the
supremum norm. Then $\varPhi:M(\mathbb{D})\to T$ is continuous with respect to
the topology on $QS_{*}(\mathbb{S})$ induced by the quasisymmetry constant.
The Douady-Earle extension yields a continuous section $e:T\to M(\mathbb{D})$
for $\varPhi$. This section is called the Douady-Earle extension operator
which maps the quasisymmetric homeomorphism $h$ to the complex dilatation
$\mu$ of the Douady-Earle extension $E(h)$ of $h$. The continuity of this
section $e$ combined with conformally natural property of the Douady-Earle
extension allowed Douady and Earle [7] to give a much simpler proof of the
theorem of Tukia [19, 20] stating that the Teichmüller space of any Fuchsian
group is contractible.

The universal BMO-Teichmüller space $T_{b}$ is similarly defined as a subspace
of $T$. It is defined as the subspace $SQS_{*}(\mathbb{S})\subset
QS_{*}(\mathbb{S})$ of all normalized strongly quasisymmetric homeomorphisms.
The topology on $SQS_{*}(\mathbb{S})$ is induced by the BMO norm. On the other
hand, the corresponding subspace of Beltrami coefficients is
$\mathcal{M}(\mathbb{D})\subset M(\mathbb{D})$, which consists of all $\mu\in
M(\mathbb{D})$ such that

$\frac{|\mu|^{2}(z)}{1-|z|^{2}}dxdy$

is a Carleson measure in $\mathbb{D}$. Let’s consider a Fuchsian group $G$:
define $\mathcal{M}(G)=M(G)\cap\mathcal{M}(\mathbb{D})$, $SQS_{*}(G)=QS(G)\cap
SQS_{*}(\mathbb{S})$. The same equivalence relation as in the classical case
may be defined on $\mathcal{M}(G)$ and we denote by $\mathcal{T}_{S}$ the
quotient space ($S$ is the Riemann surface $\mathbb{D}/G$). Let $T(G)$ be the
space of Schwarzian derivatives of injective holomorphic functions in
$\hat{\mathbb{C}}\setminus\overline{\mathbb{D}}$ having a quasiconformal
extension to $\mathbb{C}$. Define $\mathcal{T}(G)=\\{\varphi\in
T(G);\,|\varphi|^{2}(z)(|z|^{2}-1)^{3}dxdy$ is a Carleson measure on
$\hat{\mathbb{C}}\setminus\overline{\mathbb{D}}\\}$.

Cui and Zinsmeister have proved in [6] that for any $h\in SQS_{*}(\mathbb{S})$
the complex dilatation $\mu$ of its Douady-Earle extension is in
$\mathcal{M}(\mathbb{D})$. Based on this well-defined property of the Douady-
Earle extension operator restricted in $SQS_{*}(\mathbb{S})$, Cui and
Zinsmeister have shown that the map $\varPsi:\mathcal{T}_{S}\to SQS_{*}(G)$ is
a bijection while the Bers embedding $\beta:\mathcal{T}_{S}\to\mathcal{T}(G)$
is bijective. In section 3, we will show that complex Banach manifold
structures can be provided for $\mathcal{T}_{S}$ and $SQS_{*}(G)$ through the
Bers embedding $\beta$ and the map $\beta\circ\varPsi^{-1}$. Then both the map
$\varPsi:\mathcal{T}_{S}\to SQS_{*}(G)$ and the Bers embedding
$\beta:\mathcal{T}_{S}\to\mathcal{T}(G)$ become biholomorphic.

By Cui and Zinsmeister $e(h)\in\mathcal{M}(\mathbb{D})$ if $h\in
SQS_{*}(\mathbb{S})$, and moreover $e$ is continuous at the origin. The global
continuity is not known. This property would imply contractibility of all BMO-
Teichmüller spaces; So far only the case $G=\\{I\\}$ is known [8]. In section
4, it is proven that at least the operator $e$ is also Gâteaux-differentiable
at the origin, and we identify its differential which happens to be a very
simple operator.

In section 2, we will explain the above mentioned concepts and results in more
detail.

##  2 Preliminaries

In this section, we summarize several results on the background of our
arguments. This includes definitions and properties of Teichmüller spaces,
preliminaries on BMO-Teichmüller spaces and fundamental results on groups of
divergence type and groups of convergence type.

2.1 Teichmüller theory. Let G be a Fuchsian group, i.e. a properly
discontinuous fixed point free group of Möbius transformations which keeps
$\mathbb{D}$ invariant. For such a group we define $M(G)$ as

$M(G)=\\{\mu\in L^{\infty}(\mathbb{D}):\|\mu\|_{\infty}<1\,\,and\,\,\forall
g\in G,\,\mu=\mu\circ g\frac{\overline{g^{\prime}}}{g^{\prime}}\\}.$

For any $\mu\in M(G)$, there exists a unique quasiconformal self-mapping
$f^{\mu}$ of $\mathbb{D}$ keeping 1, $i$ and -1 fixed and satisfying

$\frac{\partial f^{\mu}}{\partial\bar{z}}=\mu\frac{\partial f^{\mu}}{\partial
z}$

in $\mathbb{D}$. Similarly, there exists a unique quasiconformal homeomorphism
of $\hat{\mathbb{C}}$ which is holomorphic in $\mathbb{D}^{*}$ with the
normalization

$f_{\mu}(z)=z+\frac{b_{1}}{z}+\cdots$

at $\infty$ and such that

$\frac{\partial f_{\mu}}{\partial\bar{z}}=\mu\frac{\partial f_{\mu}}{\partial
z}$

in $\mathbb{D}$. If $g$ is a choice of a Riemann mapping from $\mathbb{D}$
onto $\Omega$, $\Omega=f_{\mu}(\mathbb{D})$, then $f^{\mu}=g^{-1}\circ
f_{\mu}$ is the conformal welding with respect to the boundary of the domain
$\Omega$. The mappings $f^{\mu}$ and $f_{\mu}$ respectively induce an
isomorphism of the group G onto the Fuchsian group

$G^{\mu}=\\{f^{\mu}\circ g\circ(f^{\mu})^{-1}\mid g\in G\\}$

and the quasi-Fuchsian group

$G_{\mu}=\\{f_{\mu}\circ g\circ(f_{\mu})^{-1}\mid g\in G\\},$

i.e., a Möbius transformation group acting properly discontinuous on the
quasidisk $f_{\mu}(\mathbb{D})$.

The mapping $f^{\mu}$ has a geometric interpretation: If we denote by $S$ the
Riemann surface $\mathbb{D}/G$, then $f^{\mu}$ is the lift (to the universal
covering) of a quasiconformal mapping from the Riemann surface $S$ onto
$S^{{}^{\prime}}=\mathbb{D}/G^{\mu}$. Conversely, if F is a quasiconformal
homeomorphism from S to a Riemann surface $S^{{}^{\prime}}$, it has a lift to
a quasiconformal homeomorphism $f$ of $\mathbb{D}$ and, replacing if necessary
$F$ by $\theta\circ F$, where $\theta:S^{{}^{\prime}}\rightarrow
S^{{}^{\prime\prime}}$ is a conformal isomorphism, we may assume that
$f=f^{\mu}$ for some $\mu\in M(G)$.

If $\mu\in M(G)$, then $f^{\mu}$ has a well-defined boundary value which is a
quasisymmetric homeomorphism of $\mathbb{S}$. We define an equivalence
relation on $M(G)$ by $\mu\sim\nu$ if
$f^{\mu}|_{\mathbb{S}}=f^{\nu}|_{\mathbb{S}}$. Again this equivalence relation
has a geometric interpretation: If F, G represent the quasiconformal mappings
on S whose lifts are precisely $f^{\mu}$, $f^{\nu}$, then $\mu\sim\nu$ is
equivalent to saying that $G\circ F^{-1}$ is homotopic to a conformal
isomorphism between $F(S)$ and $G(S)$, the homotopy being constant on the
(possibly empty) boundary of $F(S)$.

The Teichmüller space $T_{S}$ is the quotient space $M(G)/\sim$ . We refer to
[14] for details about this construction.

If $\mu\in M(G)$, then the Teichmüller space $T_{S}$ can be characterized as
the set of quasisymmetric homeomorphisms $f^{\mu}|_{\mathbb{S}}$. Since
$\mu\in M(G)$, the mappings $f^{\mu}$ and $f^{\mu}\circ g$ have the same
complex dilatation. It follows that $f^{\mu}\circ g\circ(f^{\mu})^{-1}$ is a
Möbius transformation. It is well known that $\Psi:[\mu]\mapsto
f^{\mu}|_{\mathbb{S}}$ is a bijection from $T_{S}$ onto $QS_{*}(G)$, the set
of quasisymmetric homeomorphisms $h$ of $\mathbb{S}$ keeping 1, $i$ and $-1$
fixed and such that $h\circ g\circ h^{-1}$ is a Möbius transformation.

There is a similar description of the Teichmüller space in terms of $f_{\mu}$.
Let the Banach space B(G) be the space consisting of all functions $\varphi$
holomorphic in $\mathbb{D}^{*}$ which are quadratic differentials for G and
have a finite hyperbolic supremum norm:

$\|\varphi\|_{B}=\sup_{z\in\mathbb{D}^{*}}|\varphi(z)|\rho_{\mathbb{D}^{*}}^{-2}(z)<\infty.$

If $\mu\in M(G)$, then the quasiconformal mapping $f_{\mu}\circ g\circ
f_{\mu}^{-1}$, $g\in G$ of the plane is a Möbius transformation. It follows
that

$\mathcal{S}_{f_{\mu}|_{\mathbb{D}^{*}}}=\mathcal{S}_{(f_{\mu}\circ g\circ
f_{\mu}^{-1})\circ f_{\mu}|_{\Delta^{*}}}=\mathcal{S}_{f_{\mu}\circ
g|_{\mathbb{D}^{*}}}=(\mathcal{S}_{f_{\mu}|_{\Delta^{*}}}\circ
g)(g^{{}^{\prime}})^{2}.$

So the Schwarzian derivative $\mathcal{S}_{f_{\mu}|_{\mathbb{D}^{*}}}$ is a
quadratic differential for G. It is also well known that the Bers embedding
$\beta:[\mu]\mapsto\mathcal{S}_{f_{\mu}|_{\mathbb{D}^{*}}}$ is a bijection
from $T_{S}$ onto $T(G)$, the space of Schwarzian derivatives of injective
holomorphic functions in $\mathbb{D}^{*}$ having a quasiconformal extension to
the complex plane which are quadratic differentials for G. It is known that
the set $T(G)$ is an open subset in the complex Banach space $B(G)$, and the
ball

$B(0,2)=\\{\phi\in B(G):\|\phi\|_{B}<2\\}$

lies in $T(G)$. With the aid of the Bers embedding, $T_{S}$ carries a natural
complex structure. The Teichmüller space $T_{S}$ thus becomes a complex
analytic Banach manifold. We refer to [14] for details about the Teichmüller
theory.

2.2 BMO-Teichmüller theory. Recall that a positive measure $\lambda$ defined
in a simply connected domain $\Omega$ is called a Carleson measure (see [11])
if

$\|\lambda\|_{c}=\sup\\{\frac{\lambda(\Omega\cap
D(z,r))}{r}:z\in\partial\Omega,0<r<diameter(\Omega)\\}<\infty,$ (1)

where $D(z,r)$ is the disk with center $z$ and radius $r$. A Carleson measure
$\lambda$ is called a vanishing Carleson measure if $\lim_{r\rightarrow
0}\lambda(\Omega\cap D(z,r))/r=0$ uniformly for $z\in\partial\Omega$. We
denote by $CM(\Omega)$ and $CM_{0}(\Omega)$ the set of all Carleson measures
and vanishing Carleson measures on $\Omega$, respectively.

We denote by $\mathcal{L}(\mathbb{D})$ the Banach space of essentially bounded
measurable functions $\mu$ on $\mathbb{D}$ such that the measure

$\lambda_{\mu}=\frac{|\mu|^{2}(z)}{1-|z|^{2}}dxdy$

in $CM(\mathbb{D})$. The norm on $\mathcal{L}(\Delta)$ is defined as

$\|\mu\|_{c}=\|\mu\|_{\infty}+\|\lambda_{\mu}\|_{c}^{1/2},$

where $\|\lambda_{\mu}\|_{c}$ is the Carleson norm of $\lambda_{\mu}$ defined
in (1). Set
$\mathcal{M}(\mathbb{D})=\\{\mu\in\mathcal{L}(\mathbb{D}):\|\mu\|_{\infty}<1\\}$.
Define $\mathcal{M}(G)=M(G)\cap\mathcal{M}(\mathbb{D})$. The same equivalence
relation as in the classical case may be defined on $\mathcal{M}(G)$ and we
denote by $\mathcal{T}_{S}$ the quotient space which can be called BMO-
Teichmüller spaces.

An homeomorphism $h$ of $\mathbb{S}$ is called strongly quasisymmetric (see
[13]) if there exist two positive constants $C_{1}(h)$, $C_{2}(h)$, called the
strongly quasisymmetric constants of $h$ such that

$\frac{|h(E)|}{|h(I)|}\leqslant C_{1}(h)(\frac{|E|}{|I|})^{C_{2}(h)}$

whenever $I\subset\mathbb{S}$ is an interval and $E\subset I$ a measurable
subset. In other words, $h$ is strongly quasisymmetric if and only if $h$ is
absolutely continuous so that $|h^{{}^{\prime}}|$ belongs to the class of
weights $A^{\infty}$ introduced by Muckenhoupt, in particular, $\log
h^{{}^{\prime}}$ belongs to $BMO(\mathbb{S})$, the space of integrable
functions on $\mathbb{S}$ of bounded mean oscillation. Let $SQS(\mathbb{S})$
denote the set of all strongly quasisymmetric homeomorphisms on $\mathbb{S}$.
We define $SQS_{*}(G)=QS_{*}(G)\cap SQS(\mathbb{S})$.

We denote by $\mathcal{B}(\mathbb{D}^{*})$ the Banach space of function
$\varphi$ holomorphic in $\mathbb{D}^{*}$ such that the measure

$\lambda_{\varphi}=|\varphi(z)|^{2}(|z|^{2}-1)^{3}dxdy$

in $CM(\mathbb{D}^{*})$. The norm on $\mathcal{B}(\mathbb{D}^{*})$ is

$\|\varphi\|_{\mathcal{B}}=\|\lambda_{\varphi}\|_{c}.$

Define $\mathcal{T}(G)=\\{\varphi\in T(G):\lambda_{\varphi}\in
CM(\mathbb{D}^{*})\\}$. Let $\mathcal{B}(G)$ be the space consisting of all
functions $\varphi$ in $\mathcal{B}(\mathbb{D}^{*})$ which are quadratic
differentials for G. We can see that $\mathcal{T}(G)$ is a subset of
$\mathcal{B}(G)$.

In 2004, Cui and Zinsmeister [6] proved the following theorem based on the
well-defined property of the Douady-Earle extension operator restricted in
$SQS_{*}(\mathbb{S})$.

Theorem A. The mapping $\Psi:[\mu]\mapsto f^{\mu}$ is a bijection from
$\mathcal{T}_{S}$ onto $SQS_{*}(G)$, while
$\beta:[\mu]\mapsto\mathcal{S}_{f_{\mu}}$ is bijective from $\mathcal{T}_{S}$
onto $\mathcal{T}(G)$.

One of the goals of this paper is to make this theorem precise using methods
in [18] by showing both two maps above are actually homeomorphisms (even
biholomorphic automorphisms). Before proceeding we end this section by a
discussion on Fuchsian groups leading to non-trivial BMO-Teichmüller spaces

2.3 Groups of convergence type. In contrast to the classical Teichmüller
spaces, $\mathcal{T}_{S}$ can be trivial, as shown by Astala and Zinsmeister
[4]. For completeness, let us recall some related facts.

Let $G$ be a discrete group of Möbius transformation on $\hat{\mathbb{C}}$. We
say that $G$ has the Mostow rigidity property if for each homeomorphism
$h:\mathbb{S}\rightarrow\mathbb{S}$ with $h\circ G\circ h^{-1}$ a Möbius
group, it holds that either $h$ is completely singular or else is a Möbius
transformation.

We say that the group $G$ acting on $\hat{\mathbb{C}}$ is of divergence type
if

$\sum_{\gamma\in G}(1-|\gamma(0)|)=\infty,$

and of convergence type, if the series converge. It was shown by Agard [1] and
Sullivan [17] that groups of divergence type have the Mostow rigidity
property. In 1990, Astala and Zinsmeister [4] proved the converse:

Theorem B.  A Fuchsian group $G$ has Mostow rigidity property if and only if
it is of divergence type.

Combining with Theorem A, we see that $\mathcal{T}_{S}$ is trivial if $G$ is
of divergence type, while $\mathcal{T}_{S}$ is never trivial if $G$ is of
convergence type. In what follows, we henceforth shall not deal with Fuchsian
groups of divergence type. We shall focus on Fuchsian groups of convergence
type.

##  3 Complex structure on $\mathcal{T}_{S}$

In this section, we adopt methods from [18] to prove that $\mathcal{T}_{S}$
has a natural complex Banach manifold structure. As a byproduct, we shall
strengthen the above conclusion (Theorem A) by Cui and Zinsmeister.

We begin with some basic definitions and notations. Let $C$ denote the
universal constant that might change from one line to another. While
$C(\cdot)$, $C_{1}(\cdot)$, $C_{2}(\cdot)$, $\cdots$ will denote constants
that depend only on the elements put in the brackets. Denote by $\rho_{D}(z)$
the hyperbolic metric in the Jordan domain $D$. The notation $A\approx B$
means that there exists a universal constant $C$ such that
$\frac{B}{C}\leqslant A\leqslant CB$.

Let $\Omega=f_{\mu}(\mathbb{D})$ and $\Omega^{*}=f_{\mu}(\mathbb{D}^{*})$,
then $\Omega$ and $\Omega^{*}$ are complementary Jordan regions bounded by a
quasicircle. Let

$z^{*}=\gamma(\Omega)(z)=f_{\mu}\circ j\circ f_{\mu}^{-1}(z)$ (2)

be a quasiconformal reflection that fixes $\partial\Omega$ pointwise and
interchanges $\Omega$ and $\Omega^{*}$, where $j(z)=\frac{1}{\bar{z}}$. If $g$
is a Möbius transformation of the extended complex plane, then, according to
(2),

$\gamma(g(\Omega))=(g\circ f_{\mu})\circ j\circ(g\circ
f_{\mu})^{-1}=g\circ\gamma(\Omega)\circ g^{-1}.$

Similar to the Ahlfors map (see [10]), the mapping

$\varphi\mapsto\mu_{\Omega}(\varphi)$

is defined on the Banach space

$B(\Omega^{*})=\\{\varphi\,\text{ is holomorphic in
}\Omega^{*}:\sup_{z\in\Omega^{*}}|\varphi(z)|\rho_{\Omega^{*}}^{-2}(z)<\infty\\}$

by putting

$\mu_{\Omega}(\varphi)(z)=\frac{\varphi(z^{*})(z^{*}-z)^{2}\gamma(\Omega)_{\bar{z}}(z)}{2+\varphi(z^{*})(z^{*}-z)^{2}\gamma(\Omega)_{z}(z)},\,\,z\in\Omega.$
(3)

######  Lemma 1.

$\mu_{g(\Omega)}(\varphi)(g(z))\frac{\overline{g^{{}^{\prime}}(z)}}{g^{{}^{\prime}}(z)}=\mu_{\Omega}(\varphi\circ
g(g^{{}^{\prime}})^{2})(z)$.

###### Proof.

We borrow the method from Earle-Nag [10]. According to (3),

$\mu_{g(\Omega)}(\varphi)(g(z))\frac{\overline{g^{{}^{\prime}}(z)}}{g^{{}^{\prime}}(z)}=\frac{\varphi(g(z)^{*})(g(z)^{*}-g(z))^{2}\gamma(g(\Omega))_{\bar{z}}(g(z))}{2+\varphi(g(z)^{*})(g(z)^{*}-g(z))^{2}\gamma(g(\Omega))_{z}(g(z))}\frac{\overline{g^{{}^{\prime}}(z)}}{g^{{}^{\prime}}(z)}.$

Here $g(z)^{*}=\gamma(g(\Omega))(g(z))=g\circ\gamma(\Omega)\circ
g^{-1}(g(z))=g\circ\gamma(\Omega)(z)=g(z^{*})$, if $z^{*}=\gamma(\Omega)(z)$.
Differentiating both sides of

$\gamma(g(\Omega))(g(z))=g\circ\gamma(\Omega)(z)$

with respect to $\bar{z}$ and $z$ respectively, we find that

$\gamma(g(\Omega))_{\bar{z}}(g(z))\overline{g^{{}^{\prime}}(z)}=g^{{}^{\prime}}(z^{*})\gamma(\Omega)_{\bar{z}}(z)$

and

$\gamma(g(\Omega))_{z}(g(z))g^{{}^{\prime}}(z)=g^{{}^{\prime}}(z^{*})\gamma(\Omega)_{z}(z).$

Therefore,

$\begin{split}\mu_{g(\Omega)}(\varphi)(g(z))\frac{\overline{g^{{}^{\prime}}(z)}}{g^{{}^{\prime}}(z)}&=\frac{\varphi(g(z^{*}))(z^{*}-z)^{2}g^{{}^{\prime}}(z^{*})^{2}\gamma(\Omega)_{\bar{z}}(z)}{2+\varphi(g(z^{*}))(z^{*}-z)^{2}g^{{}^{\prime}}(z^{*})^{2}\gamma(\Omega)_{z}(z)}\\\
&=\mu_{\Omega}(\varphi\circ g(g^{{}^{\prime}})^{2})(z).\\\ \end{split}$

∎

######  Lemma 2 (see [18]).

Let $\alpha>0$, $\beta>0$. For a positive function $\lambda$ in $\mathbb{D}$,
set

$\tilde{\lambda}(z)=\iint_{\mathbb{D}}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}}{|1-\bar{z}w|^{\alpha+\beta+2}}\lambda(w)dudv.$
(4)

Then, using the same notation for a function $\tau$ and the associated measure
$\tau dxdy$, we have that $\tilde{\lambda}\in\text{CM}(\mathbb{D})$ if
$\lambda\in\text{CM}(\mathbb{D})$, and $\|\tilde{\lambda}\|_{c}\leqslant
C\|\lambda\|_{c}$, while $\tilde{\lambda}\in\text{CM}_{0}(\mathbb{D})$ if
$\lambda\in\text{CM}_{0}(\mathbb{D})$.

######  Lemma 3 (see [21]).

Let $f$ be conformal in $\mathbb{D}$. Then $\log
f^{\prime}\in\text{BMOA}(\mathbb{D})$ if and only if for each Carleson measure
$\lambda\in\text{CM}(\Omega),\,\Omega=f(\mathbb{D})$, the inverse image
$\lambda\circ f|f^{\prime}|\in\text{CM}(\mathbb{D})$. Besides, the norm of
$\lambda\circ f|f^{\prime}|$ is dominated by the norm of $\lambda$.

Let us denote by $T(1)$, $\mathcal{T}(1)$, the spaces $T(G)$, $\mathcal{T}(G)$
for $G=\\{I\\}$, and also denote by $B(\mathbb{D}^{*})$,
$\mathcal{B}(\mathbb{D}^{*})$, the spaces $B(G)$, $\mathcal{B}(G)$ for
$G=\\{I\\}$. We will also need the spaces $B(\mathbb{D})$ and
$\mathcal{B}(\mathbb{D})$. $B(\mathbb{D})$ and $\mathcal{B}(\mathbb{D})$ can
be defined respectively just $\rho_{\mathbb{D}^{*}}(z)$ replaced by
$\rho_{\mathbb{D}}(z)$ in the definitions of $B(\mathbb{D}^{*})$ and
$\mathcal{B}(\mathbb{D}^{*})$. The following Lemma shows that
$\mathcal{B}(\mathbb{D})\subset B(\mathbb{D})$.

######  Lemma 4.

For any $\phi\in\mathcal{B}(\mathbb{D})$,
$\|\phi\|_{B}\leqslant\frac{32\sqrt{2}}{3}\|\phi\|_{\mathcal{B}}$.

###### Proof.

Examine the proof of Lemma 4.1 in [18] carefully, we can find that this
conclusion is valid. ∎

######  Lemma 5.

The space $\mathcal{B}(G)$ is a Banach space, and the set $\mathcal{T}(G)$ is
open in $\mathcal{B}(G)$. Furthermore, there exists a constant $k$, so that
the ball $B(0,k)=\\{\varphi\in\mathcal{B}(G):\,\|\varphi\|_{\mathcal{B}}<k\\}$
lies in $\mathcal{T}(G)$.

###### Proof.

We consider functions $\varphi_{n}\in\mathcal{B}(G)$ which converge to
$\varphi$ in $\mathcal{B}(\mathbb{D}^{*})$. Given a $g\in G$, we then have
$\varphi_{n}(z)\rightarrow\varphi(z)$,
$\varphi_{n}(g(z))\rightarrow\varphi(g(z))$, uniformly on every compact subset
of $\mathbb{D}^{*}$. It follows that

$\varphi(g(z))g^{{}^{\prime}}(z)^{2}=\lim_{n\rightarrow\infty}\varphi_{n}(g(z))g^{{}^{\prime}}(z)^{2}=\lim_{n\rightarrow\infty}\varphi_{n}(z)=\varphi(z).$

Consequently, $\varphi\in\mathcal{B}(G)$. Therefore, $\mathcal{B}(G)$ is a
closed subspace of $\mathcal{B}(\mathbb{D}^{*})$ and hence a Banach space.

The definition of $\mathcal{T}(G)$ and Lemma 4 imply that

$\mathcal{T}(1)=T(1)\cap\mathcal{B}(\mathbb{D}^{*})\subset
B(\mathbb{D}^{*})\cap\mathcal{B}(\mathbb{D}^{*})=\mathcal{B}(\mathbb{D}^{*}),$

so $\mathcal{T}(1)$ is an open subset of $\mathcal{B}(\mathbb{D}^{*})$ by the
openness of $T(1)$ in $B(\mathbb{D}^{*})$. The definition of $\mathcal{T}(G)$
also implies that

$\mathcal{T}(G)=\mathcal{T}(1)\cap\mathcal{B}(G)\subset\mathcal{B}(\mathbb{D}^{*})\cap\mathcal{B}(G)=\mathcal{B}(G),$

so $\mathcal{T}(G)$ is an open subset of Banach space $\mathcal{B}(G)$ by the
openness of $\mathcal{T}(1)$ in $\mathcal{B}(\mathbb{D}^{*})$.

Now for $\varphi\in\mathcal{B}({\mathbb{D}^{*}})$, define $\phi=\varphi\circ
g(g^{{}^{\prime}})^{2}$, where $z=g(w)=\frac{1}{w}$, $w\in\mathbb{D}$. Then
$\|\phi\|_{B}=\|\varphi\|_{B}$. Let
$\lambda_{\phi}(w)=|\phi(w)|^{2}(1-|w|^{2})^{3}$ and
$\lambda_{\varphi}(z)=|\varphi(z)|^{2}(|z|^{2}-1)^{3}$ as above. then
$\lambda_{\phi}(\frac{1}{z})|\frac{1}{z}|^{2}=\lambda_{\varphi}(z)$,
$z\in\mathbb{D}^{*}$. So
$\|\phi\|_{\mathcal{B}}\approx\|\varphi\|_{\mathcal{B}}$ by Lemma 3. According
to Lemma 4, there is a constant $k$, such that
$\|\varphi\|_{B}\leqslant\frac{2}{k}\|\varphi\|_{\mathcal{B}}$. Suppose
$\|\varphi\|_{B}\leqslant\frac{2}{k}\|\varphi\|_{\mathcal{B}}<2$. Set
$\mathcal{S}_{f}=\varphi$. According to Ahlfors-Weill Theorem, $f$ is
univalent and can be extended to a quasiconformal mapping of the complex plane
with complex dilatation

$\mu(z)=-\frac{1}{z}\varphi(\frac{1}{\bar{z}})(1-|z|^{2})^{2}\frac{1}{(\bar{z})^{4}},\,\,z\in\mathbb{D}.$

Lemma 3 implies that

$\begin{split}\lambda_{\varphi}(\frac{1}{z})|\frac{1}{z}|^{2}&=|\varphi(\frac{1}{z})|^{2}(\frac{1}{|z|^{2}}-1)^{3}\frac{1}{|z|^{2}}\\\
&=|\varphi(\frac{1}{z})|^{2}(1-|z|^{2})^{3}\frac{1}{|z|^{8}}\\\ \end{split}$

is a Carleson measure in $\mathbb{D}$. Hence

$\frac{|\mu(z)|^{2}}{1-|z|^{2}}=\frac{1}{4}|\varphi(\frac{1}{\bar{z}})|^{2}(1-|z|^{2})^{3}\frac{1}{|z|^{8}}$

is a Carleson measure in $\mathbb{D}$. Consequently, the set
$\\{\varphi\in\mathcal{B}(\mathbb{D}^{*}):\,\|\varphi\|_{\mathcal{B}}<k\\}$
lies in $\mathcal{T}(1)$.

For any $\varphi\in B(0,k)$, since $\mu(z)$ above can be written as

$\mu(z)=\frac{1}{z}\varphi(z^{*})(z-z^{*})^{2}\gamma(\mathbb{D})_{\bar{z}}(z),$

where $z^{*}=\gamma(\mathbb{D})(z)=\frac{1}{\bar{z}}$. Then $\varphi\circ
g(g^{{}^{\prime}})^{2}=\varphi$ implies $\mu\circ
g\frac{\bar{g^{{}^{\prime}}}}{g^{{}^{\prime}}}=\mu$ by Lemma 1. So $\mu$ is a
Beltrami differential for $G$. Now we can come to the conclusion that for any
$\varphi\in B(0,k)$, there exists a holomorphic function $f$ in
$\mathbb{D}^{*}$ which can be extended to a quasiconformal mapping in the
complex plane whose complex dilatation $\mu\in\mathcal{M}(G)$ and
$\mathcal{S}_{f}=\varphi$. ∎

######  Theorem 1.

The function

$S:\mu\mapsto\mathcal{S}_{f_{\mu}|_{\Delta^{*}}}$

which maps $\mathcal{M}(G)$ into $\mathcal{B}(G)$ is holomorphic, and it has
local holomorphic sections everywhere in $\mathcal{T}(G)$.

###### Proof.

We first show that $S:\mathcal{M}(G)\rightarrow\mathcal{B}(G)$ is continuous.
We borrow some discussion from Astala and Zinsmeister [3]. By an integral
representation of the Schwarzian derivative by means of the representation
theorem of quasiconformal mappings, Astala and Zinsmeister [3] proved that for
any two elements $\mu$ and $\nu$ in $M(\mathbb{D})$, there exists some
constant $C_{1}(\|\mu\|_{\infty})$ such that

$|S(\nu)(z)-S(\mu)(z)|^{2}\leq\frac{C_{1}(\|\mu\|_{\infty})}{(|z|^{2}-1)^{2}}\iint_{\mathbb{D}}\frac{|\nu(\zeta)-\mu(\zeta)|^{2}+\|\nu-\mu\|_{\infty}^{2}|\mu(\zeta)|^{2}}{|\zeta-z|^{4}}d\xi
d\eta.$

Then,

$\begin{split}|S(\nu)(z)-S(\mu)(z)|^{2}&(|z|^{2}-1)^{3}\\\ &\leq
C_{1}(\|\mu\|_{\infty})\Big{(}\iint_{\mathbb{D}}\frac{|\nu(\zeta)-\mu(\zeta)|^{2}}{1-|\zeta|^{2}}\frac{(1-|\zeta|^{2})(|z|^{2}-1)}{|\zeta-z|^{4}}d\xi
d\eta\\\
&+\|\nu-\mu\|_{\infty}^{2}\iint_{\mathbb{D}}\frac{|\mu(\zeta)|^{2}}{1-|\zeta|^{2}}\frac{(1-|\zeta|^{2})(|z|^{2}-1)}{|\zeta-z|^{4}}d\xi
d\eta\Big{)}.\\\ \end{split}$

According to Lemma 2, there exists some constant $C_{2}(\|\mu\|_{\infty})$
such that

$\begin{split}\|S(\nu)-S(\mu)\|_{\mathcal{B}}^{2}&\leq
C_{2}(\|\mu\|_{\infty})(\|\lambda_{\nu-\mu}\|^{2}_{c}+\|\nu-\mu\|_{\infty}^{2}\|\lambda_{\mu}\|^{2}_{c})\\\
&\leq C_{2}(\|\mu\|_{\infty})(1+\|\mu\|_{c}^{2})\|\nu-\mu\|_{c}^{2}.\\\
\end{split}$

Consequently, $S:\mathcal{M}(G)\rightarrow\mathcal{B}(G)$ is continuous.

To prove that $S:\mathcal{M}(G)\rightarrow\mathcal{B}(G)$ is a holomorphic
map, we use a general result about the infinite dimensional holomorphy (see
[14, 15]). It says that a continuous map $f$ from a domain $U$ in a complex
Banach space $X$ into another complex Banach space $Y$ is holomorphic if for
each pair $(u,x)$ in $U\times X$ and each element $y^{*}$ from a total subset
$Y_{0}^{*}$ of the dual space $Y^{*}$, $y^{*}(f(u+tx))$ is a holomorphic
function in a neighborhood of zero in the complex plane. Here a subset
$Y_{0}^{*}$ of $Y^{*}$ is total if $y^{*}(y)=0$ for all $y^{*}\in Y_{0}^{*}$
implies that $y=0$.

We have already seen from Lemma 5 that $\mathcal{B}(G)$ is a Banach space. The
ball $\mathcal{M}(G)$ is an open subset of the Banach space $\mathcal{L}(G)$.
Now for each $z\in\mathbb{D}^{*}$, define
$l_{z}(\phi)=\phi(z),\;\forall\phi\in\mathcal{B}(G).$ The proof of Lemma 5
says that $|\phi(z)|(|z|^{2}-1)^{2}\leq C\|\phi\|_{\mathcal{B}},$ which
implies that $\|l_{z}\|\leq C(|z|^{2}-1)^{-2}.$ Thus,
$l_{z}\in\mathcal{B}^{*}(G)$. Set $A=\\{l_{z}:z\in\mathbb{D}^{*}\\}.$ Clearly,
A is a total subset of $\mathcal{B}^{*}(G)$. Now for each
$z\in\mathbb{D}^{*}$, each pair
$(\mu,\nu)\in\mathcal{M}(G)\times\mathcal{L}(G)$ and small $t$ in the complex
plane, by the well known holomorphic dependence of quasiconformal mappings on
parameters (see [2, 14, 15]), we conclude that
$l_{z}(S(\mu+t\nu))=S(\mu+t\nu)(z)$ is a holomorphic function of $t$.
Consequently, $S:\mathcal{M}(G)\rightarrow\mathcal{B}(G)$ is holomorphic.

Finally, we prove $S:\mathcal{M}(G)\rightarrow\mathcal{B}(G)$ has local
holomorphic sections everywhere in $\mathcal{T}(G)$. Fix
$\phi\in\mathcal{T}(G)$. There exists a univalent function $f$ on
$\mathbb{D}^{*}$ such that $\mathcal{S}_{f}=\phi$. Let $F$ be a Riemann
mapping from $\mathbb{D}$ onto
$\mathbb{C}\setminus\overline{f(\mathbb{D}^{*})}$ and $h=F^{-1}\circ f$ the
conformal welding. Then we have $h\in SQS(\mathbb{S})$. Now the proof in [14,
P.199] gives that $h\in SQS(G)$. Let $f_{\mu}=F\circ E(h)$. Then $f_{\mu}$ is
a quasiconformal extension of $f$ whose dilatation $\mu$ is in
$\mathcal{M}(G)$. Thus $S(\mu)=\phi$. Let $\Omega=f_{\mu}(\mathbb{D})$ and
$\Omega^{*}=f_{\mu}(\mathbb{D}^{*})$. We have
$\rho_{\Omega^{*}}(f_{\mu}(z))|f_{\mu}^{{}^{\prime}}(z)|=(|z|^{2}-1)^{-1}$ for
$z\in\mathbb{D}^{*}$. Consider
$U_{\epsilon}(\phi)=\\{\psi\in\mathcal{B}(G):\|\psi-\phi\|_{\mathcal{B}}<\epsilon\\}$
for $\epsilon>0$. Then for each $\psi\in U_{\epsilon}(\phi)$ there exists a
unique locally univalent function $f_{\psi}$ in $\mathbb{D}^{*}$ with
$f_{\mu}(z)=z+\frac{b_{1}}{z}+\cdots$ as $z\rightarrow\infty$ such that
$\mathcal{S}_{f_{\psi}}=\psi$. Set $g_{\psi}=f_{\psi}\circ(f_{\mu})^{-1}$.
Then $\mathcal{S}_{g_{\psi}}=((\psi-\phi)\circ
f_{\mu}^{-1})((f_{\mu}^{-1})^{{}^{\prime}})^{2}$ and
$\sup_{z\in\Omega^{*}}\rho_{\Omega^{*}}^{-2}(z)|\mathcal{S}_{g_{\psi}}(z)|=\|\psi-\phi\|_{B}$.
Since $\psi,\phi\in\mathcal{B}(G)$, $\psi\circ gg^{{}^{\prime}2}=\psi$ and
$\phi\circ gg^{{}^{\prime}2}=\phi$ for any $g\in G$. Hence,

$\mathcal{S}_{g_{\psi}}\circ\widetilde{g}(\widetilde{g})^{{}^{\prime}2}=\mathcal{S}_{g_{\psi}},$
(5)

for any $\widetilde{g}\in G_{\mu}$. Since the inclusion map
$i:\mathcal{B}(G)\rightarrow B(G)$ is continuous,
$\|\psi-\phi\|_{B}<C\epsilon$ for $\psi\in U_{\epsilon}(\phi)$. The Earle-Nag
reflection [12, P.263] associated with the curve $\Gamma=\partial\Omega$ is
given by the formula

$\gamma(z)=\begin{cases}f\circ j\circ f_{\mu}^{-1}(z)=f\circ j\circ
E(h)^{-1}\circ F^{-1},&z\in\Omega\\\ z,&z\in\Gamma\\\
\gamma^{-1}(z),&z\in\Omega^{*}\end{cases}$

where $j(z)=1/\bar{z}$, and [12, P.265] says

$C_{3}^{-1}(\|\mu\|_{\infty})\leqslant|\gamma(z)-z|^{2}\rho_{\Omega^{*}}^{-2}(\gamma(z))|\bar{\partial}\gamma(z)|\leqslant
C_{3}(\|\mu\|_{\infty}).$ (6)

Under the condition that
$\sup_{z\in\Omega^{*}}\rho_{\Omega^{*}}^{-2}(z)|\mathcal{S}_{g_{\psi}}(z)|$ is
sufficiently small (when $\epsilon$ is sufficiently small), Ahlfors [2], Earle
and Nag [10] (see also [12, P.266]) proved that $g_{\psi}$ is univalent and
can be extended to a quasiconformal mapping in the whole plane whose complex
dilatation $\mu_{\psi}$ has the form

$\mu_{\psi}(z)=\frac{\mathcal{S}_{g_{\psi}}(\gamma(z))(\gamma(z)-z)^{2}\bar{\partial}\gamma(z)}{2+\mathcal{S}_{g_{\psi}}(\gamma(z))(\gamma(z)-z)^{2}\partial{\gamma}(z)},\,\,z\in\Omega.$
(7)

Then by means of (6) we have

$|\mu_{\psi}(z)|\leqslant
C_{4}(\|\mu\|_{\infty})|\mathcal{S}_{g_{\psi}}(\gamma(z))|\rho_{\Omega^{*}}^{-2}(\gamma(z)),\,\,z\in\Omega.$
(8)

Consequently, $f_{\psi}=g_{\psi}\circ f_{\mu}$ is univalent in
$\mathbb{D}^{*}$ and has a quasiconformal extension to the whole plane whose
complex dilatation $\nu_{\psi}$ is

$\nu_{\psi}=\frac{\mu+(\mu_{\psi}\circ
f_{\mu})\tau}{1+\bar{\mu}(\mu_{\psi}\circ
f_{\mu})\tau},\,\,\,\tau=\frac{\overline{\partial f_{\mu}}}{\partial
f_{\mu}}.$ (9)

According to Lemma 1, (5) implies that

$\mu_{\psi}\circ\widetilde{g}\frac{\overline{(\widetilde{g})^{{}^{\prime}}}}{(\widetilde{g})^{{}^{\prime}}}=\mu_{\psi}.$

Again we know that $\mu\circ
g\frac{\bar{g^{{}^{\prime}}}}{g^{{}^{\prime}}}=\mu$. By direct computation, we
can get

$\nu_{\psi}\circ g\frac{\bar{g^{{}^{\prime}}}}{g^{{}^{\prime}}}=\nu_{\psi}.$
(10)

Now, it follows from (8) that

$\begin{split}|\mu_{\psi}(f_{\mu}(z))|&\leqslant
C_{4}(\|\mu\|_{\infty})|\mathcal{S}_{g_{\psi}}(\gamma(f_{\mu}(z)))|\rho_{\Omega^{*}}^{-2}(\gamma(f_{\mu}(z)))\\\
&=C_{4}(\|\mu\|_{\infty})|\mathcal{S}_{g_{\psi}}(f_{\mu}(j(z)))|\rho_{\Omega^{*}}^{-2}(\gamma(f_{\mu}(z)))\\\
&=C_{4}(\|\mu\|_{\infty})|\psi(j(z))-\phi(j(z))|(1-|j(z)|^{2})^{2},\\\
\end{split}$ (11)

which implies that $\|\lambda_{\mu_{\psi}\circ f_{\mu}}\|_{c}\leqslant
C_{5}(\|\mu\|_{\infty})\|\psi-\phi\|_{\mathcal{B}}$. Thus, $\mu_{\psi}\circ
f_{\mu}\in\mathcal{M}(\mathbb{D})$, and we conclude by (9) that
$\nu_{\psi}\in\mathcal{M}(\mathbb{D})$. Combining with (10), we can get
$\nu_{\psi}\in\mathcal{M}(G)$. On the other hand, from (7) and (9) it is easy
to see that $\nu_{\psi}$ depends holomorphically on $\psi$. Since
$S(\nu_{\psi})=\psi$, we conclude that
$\nu:U_{\epsilon}(\phi)\rightarrow\mathcal{B}(G)$ is a local holomorphic
section to $S:\mathcal{M}(G)\rightarrow\mathcal{B}(G).$ This completes the
proof. ∎

######  Corollary 1.

Via the Bers embedding $\beta$, $\mathcal{T}_{S}$ carries a natural complex
structure. Under this complex structure, the Bers embedding

$\beta:\mathcal{T}_{S}\rightarrow\mathcal{T}(G)$

is biholomorphic.

###### Proof.

First of all, $\beta:\mathcal{T}_{S}\rightarrow\mathcal{T}(G)$ is one-to-one
by Theorem A.

Next, $S:\mathcal{M}(G)\rightarrow\mathcal{B}(G)$ is continuous by Theorem 1
and $\Phi:\mu\mapsto[\mu]$ which maps $\mathcal{M}(G)$ onto $\mathcal{T}_{S}$
is a projection. Thus, $\beta:\mathcal{T}_{S}\rightarrow\mathcal{B}(G)$ is
continuous by the topological property of the projection mapping. We fix a
point $\phi\in\mathcal{T}(G)$, consider
$U_{\epsilon}(\phi)=\\{\psi\in\mathcal{B}(G):\|\psi-\phi\|_{\mathcal{B}}<\epsilon\\}$
for $\epsilon>0$. According to the proof of Theorem 1, there exists a mapping
$\nu:\psi\rightarrow\nu_{\psi}$ which maps $U_{\epsilon}(\phi)$ into
$\mathcal{M}(G)$, and it is continuous. The projection
$\Phi:\mathcal{M}(G)\rightarrow\mathcal{T}_{S}$ is also continuous. So the
composition
$\beta^{-1}=\Phi\circ\nu:U_{\epsilon}(\phi)\rightarrow\mathcal{T}_{S}$ is
continuous. Thus, $\beta:\mathcal{T}_{S}\rightarrow\mathcal{T}(G)$ is a
homeomorphism, i.e., $\mathcal{T}_{S}$ is actually homeomorphic to an open
subset in the complex Banach space $\mathcal{B}(G)$.

Hence, by using this result, we define on $\mathcal{T}_{S}$ a complex analytic
Banach manifold structure. The Bers embedding
$\beta:\mathcal{T}_{S}\rightarrow\mathcal{B}(G)$ is biholomorphic under this
complex structure. ∎

######  Corollary 2.

The canonical projection

$\Phi:\mathcal{M}(G)\rightarrow\mathcal{T}_{S}$

is holomorphic, and it has local holomorphic sections everywhere in
$\mathcal{T}_{S}$.

###### Proof.

First of all, we have

$\Phi=\beta^{-1}\circ S.$

Since $\beta^{-1}$ and $S$ are holomorphic, it follows that $\Phi$ is
holomorphic.

Next, let us consider the mapping $\Phi\circ\nu\circ\beta$ which is the
identity on $\beta^{-1}(U_{\epsilon}(\phi)).$ According to the proof of
Theorem 1, $\nu\circ\beta$ is the desired section. ∎

######  Corollary 3.

$\Psi:\mathcal{T}_{S}\rightarrow SQS(G)$ is a homeomorphism. Consequently,
$SQS(G)$ possesses a complex structure so that
$\Psi:\mathcal{T}_{S}\rightarrow SQS(G)$ is a biholomorphic isomorphism.

###### Proof.

We know that $\Psi:\mathcal{T}_{S}\rightarrow SQS(G)$ is one-to-one by Theorem
A. Shen and Wei [18] have proved that $\Psi:\mathcal{T}_{S}\rightarrow SQS(G)$
is a homeomorphism when $G=\\{I\\}$. So, $\Psi:\mathcal{T}_{S}\rightarrow
SQS(G)$ is a homeomorphism for any Fuchsian group $G$. The homeomorphism
$\beta\circ\Psi^{-1}:SQS(G)\rightarrow\mathcal{T}(G)$ endows the spaces
$SQS(G)$ with the structure of complex Banach manifolds modeled on the Banach
space $\mathcal{B}(G)$. Naturally, under this complex structure,
$\Psi:\mathcal{T}_{S}\rightarrow SQS(G)$ is a biholomorphic isomorphism. ∎

##  4 Gâteaux-differentiability of $e$ at the origin

In this section, we will show that the Douady-Earle extension operator
$e:SQS(\mathbb{S})\to\mathcal{M}(\mathbb{D})$ is Gâteaux-differentiable at the
origin, and compute its differential.

Let $E(h)$ denote the Douady-Earle extension of a quasisymmetric homeomorphism
$h$ on $\mathbb{S}$. The definition of $E(h)$ is very simple: given
$z\in\mathbb{D}$, $E(h)(z)$ is the unique $w\in\mathbb{D}$ such that

$F_{h}(z,w)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{h(e^{iu})-w}{1-\bar{w}h(e^{iu})}\frac{1-|z|^{2}}{|z-e^{iu}|^{2}}du=0.$

Coming to BMO-Teichmüller theory, Cui and Zinsmeister [6] have shown that if
$h\in SQS(\mathbb{S})$ then, if $\mu$ denotes the complex dilatation of the
Douady-Earle extension $E(h)$, it holds that

$\frac{|\mu|^{2}(z)}{1-|z|}dxdy$

is a Carleson measure in $\mathbb{D}$, which implies that the Douady-Earle
extension operator $h\mapsto\mu$ is a bijection from $SQS(\mathbb{S})$ onto
its image in $\mathcal{M}(\mathbb{D})$. Lemma 2 combined with [16, Proposition
4.2] shows continuity at $h=Id$ of the operator, but global continuity is
still unknown.

Suppose $X$ and $Y$ are Banach spaces, $U\subset X$ is open, and $G:X\to Y$.
The differential $dG(u;b)$ of $G$ at $u\in U$ in the direction $b\in X$ is
defined as

$dG(u;b)=\lim_{t\to 0}\frac{G(u+tb)-G(u)}{t}=\frac{d}{dt}G(u+tb)|_{t=0}.$

If the limit exists for all $b\in X$, and if $dG(u;\cdot):X\to Y$ is linear
and continuous, then one says that $G$ is Gâteaux differentiable at the point
$u$.

In the following we show that the operator $e:\,h\mapsto\mu$ from
$SQS(\mathbb{S})$ onto its image in $\mathcal{M}(\mathbb{D})$ is Gâteaux
differentiable at the origin. In order to make this statement precise we first
write, for $h\in SQS(\mathbb{S})$, $h(e^{it})=e^{i\phi(t)}$, where $\phi$ is
an increasing homeomorphism of $[0,2\pi]$ such that $\phi^{{}^{\prime}}(t)$ is
a Muckenhoupt weight. The topology of $SQS(\mathbb{S})$ is the one inherited
from $BMO_{\mathbb{R}}(\mathbb{S})$ the space of real-valued $2\pi$-periodic
BMO functions:

$d(h_{1},h_{2})=\|\log\phi_{1}^{{}^{\prime}}-\log\phi_{2}^{{}^{\prime}}\|_{BMO}.$

Put $b(t)=\log\phi^{{}^{\prime}}(t)$. There is no loss of generality to assume
that $\int_{0}^{2\pi}b(t)dt=0$ (since any $\varphi\in BMO$ can be identified
with $\varphi+\alpha$, $\alpha$ constant).

Conversely, suppose now that $b\in BMO_{\mathbb{R}}(\mathbb{S})$ with
$\int_{0}^{2\pi}b(t)dt=0$. Let $t>0$ be small. We define $c(t)$ as being the
unique real number such that

$\int_{0}^{2\pi}e^{tb(u)-tc(t)}du=2\pi.$

That is

$c(t)=\log[(\frac{1}{2\pi}\int_{0}^{2\pi}e^{tb(u)}du)^{\frac{1}{t}}].$

We have $\lim_{t\to 0}c(t)=\frac{1}{2\pi}\int_{0}^{2\pi}b(u)du=0$. Hence we
can write $h_{t}(e^{iu})=e^{i\phi_{t}(u)}$, where

$\begin{split}\phi_{t}(u)&=\int_{0}^{u}e^{tb(v)-tc(t)}dv=u+\int_{0}^{u}[e^{tb(v)-tc(t)}-1]dv\\\
&=u+t\int_{0}^{u}b(v)dv+o(t).\\\ \end{split}$

Set $B(u)=\int_{0}^{u}b(v)dv$. Then

$h_{t}(e^{iu})=e^{i\phi_{t}(u)}=e^{iu}+tie^{iu}B(u)+o(t).$ (12)

For $h_{t}\in SQS(\mathbb{S})$, set $F_{h_{t}}(z,w)=0$. That is

$F(t,z,w)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{h_{t}(e^{iu})-w}{1-\bar{w}h_{t}(e^{iu})}\frac{1-|z|^{2}}{|z-e^{iu}|^{2}}du=0.$

Write $w=E(h_{t})(z)=f_{t}(z)$ and denote by
$\mu_{t}=(f_{t})_{\bar{z}}/(f_{t})_{z}$ the complex dilatation of $f_{t}$.

######  Theorem 2.

The Douady-Earle extension operator $e:\,h\mapsto\mu$ from $SQS(\mathbb{S})$
onto its image in $\mathcal{M}(\mathbb{D})$ is Gâteaux differentiable at the
origin, and the differentiate at the origin of $e$ in the direction $b\in
BMO_{\mathbb{R}}(\mathbb{S})$ is

$de(0;b)=-\frac{(1-|z|^{2})^{2}}{2\pi
i}\int_{0}^{2\pi}\frac{3e^{2iu}B(u)}{(1-\bar{z}e^{iu})^{4}}du.$

Remark: We shall specify now what we mean by Gâteaux differentiability at the
origin of the Douady-Earle extension operator.

According to the above argument, the origin $u=0$ of the Banach space
$BMO_{\mathbb{R}}(\mathbb{S})$ corresponds to the origin $h=Id$ of the space
$SQS(\mathbb{S})$. Then the corresponding complex dilatation $\mu$ is equal to
$0$.

Let $t>0$ be small. For any $b\in BMO_{\mathbb{R}}(\mathbb{S})$, the strongly
quasisymmetric homeomorphism $h_{t}$ can be defined as before. The complex
dilatation $\mu_{t}$ denotes the image of $h_{t}$ under the Douady-Earle
extension operator.

By saying that the operator $e:\,h\mapsto\mu$ from $SQS(\mathbb{S})$ onto its
image in $\mathcal{M}(\mathbb{D})$ is Gâteaux differentiable at the origin, we
mean (by definition) that $\frac{d\mu_{t}}{dt}|_{t=0}$ exists, and it is
linear and continuous in $b$.

###### Proof.

We compute $\mu_{t}$ using the implicit function theorem and the formula
$F(t,z,f_{t}(z))=0$. We get (writing $F_{z}$ for $F_{z}(t,z,w)$, etc.) the
system

$F_{\bar{z}}+F_{\bar{w}}(\overline{f_{t}})_{\bar{z}}+F_{w}(f_{t})_{\bar{z}}=0,\qquad\bar{F}_{\bar{z}}+\bar{F}_{w}(f_{t})_{\bar{z}}+\bar{F}_{\bar{w}}(\overline{f_{t}})_{\bar{z}}=0,$

whose solution is

$(f_{t})_{\bar{z}}=\frac{\bar{F}_{\bar{z}}F_{\bar{w}}-F_{\bar{z}}\bar{F}_{\bar{w}}}{|F_{w}|^{2}-|F_{\bar{w}}|^{2}},\qquad(f_{t})_{z}=\frac{\bar{F}_{z}F_{\bar{w}}-F_{z}\bar{F}_{\bar{w}}}{|F_{w}|^{2}-|F_{\bar{w}}|^{2}},$

and finally

$\mu_{t}=\frac{(f_{t})_{\bar{z}}}{(f_{t})_{z}}=\frac{\overline{(F_{z})}F_{\bar{w}}-F_{\bar{z}}\overline{(F_{w})}}{\overline{(F_{\bar{z}})}F_{\bar{w}}-F_{z}\overline{(F_{w})}}\triangleq\frac{N}{D}.$

Therefore,

$\frac{d\mu_{t}}{dt}|_{t=0}=(\frac{1}{D}\dot{N}-\frac{N}{D^{2}}\dot{D})|_{t=0}.$

Recall that

$F(t,z,w)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{h_{t}(e^{iu})-w}{1-\bar{w}h_{t}(e^{iu})}\frac{1-|z|^{2}}{|z-e^{iu}|^{2}}du.$

Then we have

$F_{\bar{z}}(t,z,w)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{h_{t}(e^{iu})-w}{1-\bar{w}h_{t}(e^{iu})}\frac{e^{-iu}}{(\bar{z}-e^{-iu})^{2}}du,$
(13)

and

$F_{z}(t,z,w)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{h_{t}(e^{iu})-w}{1-\bar{w}h_{t}(e^{iu})}\frac{e^{iu}}{(z-e^{iu})^{2}}du,$
(14)

and

$F_{\bar{w}}(t,z,w)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{h_{t}(e^{iu})(h_{t}(e^{iu})-w)}{(1-\bar{w}h_{t}(e^{iu}))^{2}}\frac{1-|z|^{2}}{|z-e^{iu}|^{2}}du,$
(15)

and

$F_{w}(t,z,w)=-\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1-|z|^{2}}{(1-\bar{w}h_{t}(e^{iu}))|z-e^{iu}|^{2}}du.$
(16)

Noting that $w=E(h)(z)=z$ if $h(z)=z$, we conclude that

$F_{\bar{z}}(t,z,w)|_{t=0}=F_{\bar{z}}(0,z,z)=0,\qquad
F_{z}(t,z,w)|_{t=0}=F_{z}(0,z,z)=\frac{1}{1-|z|^{2}},$ (17)

and

$F_{\bar{w}}(t,z,w)|_{t=0}=F_{\bar{w}}(0,z,z)=0,\qquad
F_{w}(t,z,w)|_{t=0}=F_{w}(0,z,z)=-\frac{1}{1-|z|^{2}}.$ (18)

Then

$N|_{t=0}=[\overline{(F_{z})}F_{\bar{w}}-F_{\bar{z}}\overline{(F_{w})}]|_{t=0}=0.$

Thus we have

$\frac{d\mu_{t}}{dt}|_{t=0}=\frac{1}{D}\dot{N}|_{t=0}=(1-|z|^{2})^{-1}(\frac{d}{dt}F_{\bar{w}}(t,z,w)+\frac{d}{dt}F_{\bar{z}}(t,z,w))|_{t=0}.$
(19)

We next compute $\frac{d}{dt}F_{\bar{w}}(t,z,w)|_{t=0}$ and
$\frac{d}{dt}F_{\bar{z}}(t,z,w)|_{t=0}$. Let us start by computing
$\frac{\partial f_{t}}{\partial t}(z)|_{t=0}$. Differentiating
$F(t,z,f_{t}(z))=0$ and $\overline{F(t,z,f_{t}(z))}=0$ with respect to $t$
using the implicit function theorem we get the system

$\begin{split}&F_{t}(0,z,z)+F_{w}(0,z,z)\frac{\partial f_{t}}{\partial
t}(z)|_{t=0}+F_{\bar{w}}(0,z,z)\overline{\frac{\partial f_{t}}{\partial
t}(z)|_{t=0}}=0,\\\
&\overline{F_{t}(0,z,z)}+\overline{F_{\bar{w}}(0,z,z)}\frac{\partial
f_{t}}{\partial t}(z)|_{t=0}+\overline{F_{w}(0,z,z)}\overline{\frac{\partial
f_{t}}{\partial t}(z)|_{t=0}}=0,\\\ \end{split}$

whose solution is

$\frac{\partial f_{t}}{\partial
t}(z)|_{t=0}=\frac{F_{\bar{w}}(0,z,z)\overline{(F_{t}(0,z,z))}-\overline{(F_{w}(0,z,z))}F_{t}(0,z,z)}{|F_{w}(0,z,z)|^{2}-|F_{\bar{w}}(0,z,z)|^{2}}.$

By (12) we have

$F_{t}(0,z,z)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{(1-z\bar{z})^{2}ie^{2iu}B(u)}{(e^{iu}-z)(1-\bar{z}e^{iu})^{3}}du.$
(20)

It follows from (18) and (20) that

$\frac{\partial f_{t}}{\partial
t}(z)|_{t=0}=\frac{(1-z\bar{z})^{3}}{2\pi}\int_{0}^{2\pi}\frac{ie^{2iu}B(u)}{(e^{iu}-z)(1-\bar{z}e^{iu})^{3}}du.$
(21)

The derivative at $t=0$ of the function $F_{\bar{w}}(t,z,w)$ is

$\frac{d}{dt}F_{\bar{w}}(t,z,w)|_{t=0}=F_{\bar{w}t}(0,z,z)+F_{\bar{w}w}(0,z,z)\frac{\partial
f_{t}}{\partial t}(z)|_{t=0}+F_{\bar{w}\bar{w}}(0,z,z)\overline{\frac{\partial
f_{t}}{\partial t}(z)|_{t=0}}.$

By means of (15) we have

$F_{\bar{w}t}(0,z,z)=\frac{1-|z|^{2}}{2\pi}\int_{0}^{2\pi}\frac{ie^{3iu}B(u)}{(1-\bar{z}e^{iu})^{3}}(\frac{1}{e^{iu}-z}+\frac{2\bar{z}}{1-\bar{z}e^{iu}}+e^{-iu})du,$

and

$F_{\bar{w}w}(0,z,z)=-\frac{z}{(1-|z|^{2})^{2}},\qquad
F_{\bar{w}\bar{w}}(0,z,z)=0.$

Then

$\begin{split}\frac{d}{dt}F_{\bar{w}}(t,z,w)|_{t=0}&=-\frac{1-|z|^{2}}{2\pi
i}\int_{0}^{2\pi}\frac{e^{3iu}B(u)}{(1-\bar{z}e^{iu})^{3}}(\frac{1}{e^{iu}-z}+\frac{2\bar{z}}{1-\bar{z}e^{iu}}+e^{-iu})du\\\
&+z(1-|z|^{2})\frac{1}{2\pi
i}\int_{0}^{2\pi}\frac{e^{2iu}B(u)}{(e^{iu}-z)(1-\bar{z}e^{iu})^{3}}du.\\\
\end{split}$ (22)

Similarly,

$\frac{d}{dt}F_{\bar{z}}(t,z,w)|_{t=0}=F_{\bar{z}t}(0,z,z)+F_{\bar{z}w}(0,z,z)\frac{\partial
f_{t}}{\partial t}(z)|_{t=0}+F_{\bar{z}\bar{w}}(0,z,z)\overline{\frac{\partial
f_{t}}{\partial t}(z)|_{t=0}}.$

By (13) we have

$F_{\bar{z}t}(0,z,z)=\frac{1-|z|^{2}}{2\pi}\int_{0}^{2\pi}\frac{ie^{2iu}B(u)}{(1-\bar{z}e^{iu})^{4}}du,$

and

$F_{\bar{z}w}(0,z,z)=0,\qquad F_{\bar{z}\bar{w}}(0,z,z)=0.$

Therefore,

$\frac{d}{dt}F_{\bar{z}}(t,z,w)|_{t=0}=-\frac{1-|z|^{2}}{2\pi
i}\int_{0}^{2\pi}\frac{e^{2iu}B(u)}{(1-\bar{z}e^{iu})^{4}}du.$ (23)

It follows from (19) (22) and (23) that

$\frac{d\mu_{t}}{dt}|_{t=0}=-\frac{(1-|z|^{2})^{2}}{2\pi
i}\int_{0}^{2\pi}\frac{3e^{2iu}B(u)}{(1-\bar{z}e^{iu})^{4}}du.$ (24)

It is easy to see $\frac{d\mu_{t}}{dt}|_{t=0}$ is linear in $b$.

Finally we need to show continuity, i.e. that

$\frac{|\frac{d\mu_{t}}{dt}|_{t=0}|^{2}}{1-|z|^{2}}dxdy\in CM(\mathbb{D})$
(25)

with norm controlled by $\|b\|_{BMO}^{2}$. Since each $b\in
BMO_{\mathbb{R}}(\mathbb{S})$ with $\int_{0}^{2\pi}b(x)dx=0$ is a real-valued
periodic function with period $2\pi$, the Fourier series of the function
$b(x)$ is given by

$b(x)=\sum_{n>0}b_{n}e^{inx}+\sum_{n>0}b_{-n}e^{-inx}$

where

$b_{n}=\frac{1}{2\pi}\int_{0}^{2\pi}b(x)e^{-inx}dx,\qquad
b_{-n}=\frac{1}{2\pi}\int_{0}^{2\pi}b(x)e^{inx}dx.$

Since $b$ is real-valued, we have $b_{-n}=\overline{b_{n}}$. Then

$b(x)=\sum_{n>0}b_{n}e^{inx}+\overline{\sum_{n>0}b_{n}e^{inx}}.$

It follows from the relation $B(u)=\int_{0}^{u}b(x)dx$ that $B(u)\in
BMO_{\mathbb{R}}(\mathbb{S})$ and

$B(u)=-i\sum_{n>0}\frac{1}{n}b_{n}e^{inu}+\overline{(-i\sum_{n>0}\frac{1}{n}b_{n}e^{inu})}.$

Set $F(e^{iu})=-i\sum_{n>0}\frac{1}{n}b_{n}e^{inu}$. Then
$B(u)=F(e^{iu})+\overline{F(e^{iu})}$.

To prove (25), write

$\begin{split}\int_{0}^{2\pi}\frac{e^{2iu}B(u)}{(1-\bar{z}e^{iu})^{4}}du&=\int_{0}^{2\pi}\frac{e^{2iu}F(e^{iu})}{(1-\bar{z}e^{iu})^{4}}du+\int_{0}^{2\pi}\frac{e^{2iu}\overline{F(e^{iu})}}{(1-\bar{z}e^{iu})^{4}}du\\\
&=I_{1}+I_{2}.\\\ \end{split}$

The first term on the right hand side $I_{1}$ equals to $0$ by Cauchy Theorem.
By Cauchy integral formula we have

$I_{2}=\overline{\int_{0}^{2\pi}\frac{e^{-2iu}F(e^{iu})}{(1-ze^{-iu})^{4}}du}=\overline{-i\int_{\mathbb{S}}\frac{\zeta
F(\zeta)}{(\zeta-z)^{4}}d\zeta}=\overline{\frac{\pi}{3}((zF(z))^{{}^{\prime}})^{{}^{\prime\prime}}}.$

Noting that

$(zF(z))^{{}^{\prime}}=F(z)+zF^{{}^{\prime}}(z)=(-i\sum_{n>0}\frac{1}{n}b_{n}z^{n})+(-i\sum_{n>0}b_{n}z^{n})\in
BMOA(\mathbb{D}).$

By a result of Fefferman and Stein [9], since $z\mapsto zF^{\prime}(z)+F(z)$
is in $BMOA(\mathbb{D})$,

$\frac{|\frac{d\mu_{t}}{dt}|_{t=0}|^{2}}{1-|z|^{2}}dxdy=\frac{1}{4}|((zF(z))^{{}^{\prime}})^{{}^{\prime\prime}}|^{2}(1-|z|^{2})^{3}dxdy\in
CM(\mathbb{D})$

with norm controlled by $\|b\|_{BMO}^{2}$. ∎

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