# Parametrization of the $p$-Weil–Petersson curves: holomorphic dependence

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.

Similarly to the Bers simultaneous uniformization, the product of the
$p$-Weil–Petersson Teichmüller spaces for $p\geq 1$ provides the coordinates
for the space of $p$-Weil–Petersson embeddings $\gamma$ of the real line
$\mathbb{R}$ into the complex plane $\mathbb{C}$. We prove the biholomorphic
correspondence from this space to the $p$-Besov space of
$u=\log\gamma^{\prime}$ on $\mathbb{R}$ for $p>1$. From this fundamental
result, several consequences follow immediately which clarify the analytic
structures concerning parameter spaces of $p$-Weil–Petersson curves. In
particular, it follows that the correspondence of the Riemann mapping
parameters to the arc-length parameters keeping the images of curves is a
homeomorphism with bi-real-analytic dependence of change of parameters. This
is a counterpart to a classical theorem of Coifman and Meyer for chord-arc
curves.

###### Key words and phrases:

Weil–Petersson Teichmüller space, integrable Beltrami coefficients, Bers
simultaneous uniformization, pre-Schwarzian derivative model, analytic Besov
space, Weil–Petersson curves, arc-length parametrization

###### 2020 Mathematics Subject Classification:

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

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

##  1\. Introduction

The Weil–Petersson metric was originally introduced in the study of
Teichmüller spaces of Riemann surfaces. Besides this direction of researches
of long history, a study on the subspace $T_{2}$ of the universal Teichmüller
space $T$ that admits the Weil–Petersson metric was initiated by Cui [7], and
later the Hilbert manifold structure of $T_{2}\subset T$ and the curvatures of
the Weil–Petersson metric were further investigated by Takhtajan and Teo [37].
Based on these fundamental work, Shen and his coauthors have developed
complex-analytic theories of the subspace $T_{2}$, which is nowadays called
the Weil–Petersson Teichmüller space. As Beltrami coefficients representing
elements of $T_{2}$ are square integrable with respect to the hyperbolic
metric, this is also called the integrable Teichmüller space.

The theme of this paper is in complex analytic aspects of the Weil–Petersson
Teichmüller space. In the series of papers, Shen [30] first characterized the
Weil–Petersson class $W_{2}$, which consists of quasisymmetric homeomorphisms
representing elements of $T_{2}$, without using quasiconformal extension. It
was given in terms of the fractional dimensional Sobolev space
$H^{1/2}_{\mathbb{R}}$ of real-valued functions. Then, Shen and Tang [31]
regarded $H^{1/2}_{\mathbb{R}}$ as a new parameter space for $T_{2}$ which is
real-analytically equivalent to the original complex Hilbert structure. In
this work, they considered the Weil–Petersson class $W_{2}$ on the real line
$\mathbb{R}$ and applied the arguments of chord-arc curves induced by BMO
functions in Semmes [29]. Further, Shen and Wu [33] considered the
Weil–Petersson curves in the complex plane $\mathbb{C}$, which is the complex
generalization of the elements in $W_{2}$, and proved that the Riemann
mappings onto the domains defined by Weil–Petersson curves move continuously.

Recently, Bishop [2] has accomplished a comprehensive study on the
Weil–Petersson curves. He collected about twenty characterizations of
Weil–Petersson curves from various viewpoints of analysis and geometry
including the complex analytic methods as we mentioned above. For instance, a
planar geometric characterization of a bounded Weil–Petersson curve $\Gamma$
is given by the rate at which the perimeter of the inscribed $2^{n}$-polygon
for $\Gamma$ converges to that of $\Gamma$. Other characterizations are given
by certain measurement of coarse smoothness for closed rectifiable curves
$\gamma$, the Möbius energy defined on $\gamma$ similarly to knots, hyperbolic
geometry of convex cores spanned by $\gamma$, the curvatures of minimal
surfaces with the boundary $\gamma$, and so on. One can find other related
work in the references therein including Shanon and Mumford [27] on 2D-shape
mapping, and Wang [39] arising from the SLE theory.

In this present paper, we give a foundation of the parametrization of the
space of Weil–Petersson curves in the framework of the quasiconformal
Teichmüller theory. We represent this space as the product of the
Weil–Petersson Teichmüller spaces in three ways and prove analytic and
topological correspondences of those factors. In this fashion, we can
understand the structure of the space of Weil–Petersson curves clearly and
easily. In particular, we can reprove and extend several known results that
have important applications as immediate consequences from our fundamental
theorems. We also develop those arguments in the generalization to
$p$-Weil–Petersson curves for $p>1$.

For $p\geq 1$, let ${\mathcal{M}}_{p}(\mathbb{U})$ be the set of Beltrami
coefficients that are $p$-integrable with respect to the hyperbolic metric on
the upper half-plane $\mathbb{U}\subset\mathbb{C}$. The $p$-Weil–Petersson
Teichmüller space $T_{p}(\mathbb{U})$ is the quotient space of
${\mathcal{M}}_{p}(\mathbb{U})$ by the Teichmüller equivalence. Precise
definitions are in Section 2 . It has been proved that $T_{p}(\mathbb{U})$
possesses a complex Banach manifold structure via the Bers embedding. On the
lower half-plane $\mathbb{L}\subset\mathbb{C}$, the corresponding spaces
${\mathcal{M}}_{p}(\mathbb{L})$ and $T_{p}(\mathbb{L})$ are defined. The
$p$-Weil–Petersson class $W_{p}$ is the set of quasisymmetric homeomorphisms
$f:\mathbb{R}\to\mathbb{R}$ that extend quasiconformally to $\mathbb{U}$ (and
to $\mathbb{L}$) with their complex dilatations in
${\mathcal{M}}_{p}(\mathbb{U})$ (and in ${\mathcal{M}}_{p}(\mathbb{L})$).
Then, the element $f$ in $W_{p}$ for $p>1$ can be characterized by the
property that $f$ is locally absolutely continuous and $\log f^{\prime}$
belongs to the $p$-Besov space $B_{p}^{\mathbb{R}}(\mathbb{R})$ of real-valued
functions, which coincides with $H^{1/2}_{\mathbb{R}}(\mathbb{R})$ for $p=2$.
In fact, in our paper [40] generalizing [31], it was proved that if a
quasisymmetric homeomorphism $f$ of $\mathbb{R}$ is locally absolutely
continuous and $\log f^{\prime}$ is in $B_{p}^{\mathbb{R}}(\mathbb{R})$, then
the variant of the Beurling–Ahlfors extension by the heat kernel introduced by
Fefferman, Kenig and Pipher [8] yields a quasiconformal homeomorphism of
$\mathbb{U}$ whose complex dilatation is in ${\mathcal{M}}_{p}(\mathbb{U})$.
We will elaborate these concepts in Section 3 as well as those introduced
next.

A $p$-Weil–Petersson embedding $\gamma:\mathbb{R}\to\mathbb{C}$ is the
restriction of a quasiconformal homeomorphism of $\mathbb{C}$ whose complex
dilatations on $\mathbb{U}$ and $\mathbb{L}$ belong to
${\mathcal{M}}_{p}(\mathbb{U})$ and ${\mathcal{M}}_{p}(\mathbb{L})$,
respectively. Due to this definition of a curve as a continuous mapping not
only its image, the space ${\rm WPC}_{p}$ of all normalized $p$-Weil–Petersson
embeddings for $p\geq 1$ can be parametrized in the same spirit of the Bers
simultaneous uniformization. Namely, ${\rm WPC}_{p}$ is identified with the
product of the $p$-Weil–Petersson Teichmüller spaces $T_{p}(\mathbb{U})\times
T_{p}(\mathbb{L})$. Although this natural viewpoint has been missed in the
literature, we emphasize in this paper that this can significantly clarify the
theory of ${\rm WPC}_{p}$. In our recent paper [42], we carry out similar
arguments for the space of chord-arc curves and succeed in obtaining several
interesting consequences.

In Theorems 3.6 and 5.1, we prove the following basic result on this
parametrization of ${\rm WPC}_{p}$. The $p$-Besov space $B_{p}(\mathbb{R})$ is
a complex Banach space of complex-valued functions which contains
$B_{p}^{\mathbb{R}}(\mathbb{R})$ as the real subspace. The crucial point is
not only to give the characterization of elements in ${\rm WPC}_{p}$ but also
to show that this correspondence is biholomorphic. This is a new result even
in the case of $p=2$.

######  Theorem 1.1.

For any $p$-Weil–Petersson embedding $\gamma:\mathbb{R}\to\mathbb{C}$, the
logarithm of its derivative $\log\gamma^{\prime}$ belongs to the $p$-Besov
space $B_{p}(\mathbb{R})$ for $p>1$. Moreover, this correspondence

$L:{\rm WPC}_{p}\cong T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})\to
B_{p}(\mathbb{R})$

is a biholomorphic homeomorphism onto the image.

The $p$-Weil–Petersson class $W_{p}$ of all normalized quasisymmetric
homeomorphisms of $\mathbb{R}$ onto itself is a real-analytic submanifold of
${\rm WPC}_{p}$ corresponding to the diagonal axis of the Bers coordinates
$T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$. All normalized $p$-Weil–Petersson
embeddings that are induced by Riemann mappings on $\mathbb{U}$ make a
complex-analytic submanifold $RM_{p}$ of ${\rm WPC}_{p}$ corresponding to
$\\{[0]\\}\times T_{p}(\mathbb{L})$. Every $\gamma\in{\rm WPC}_{p}$ is
represented uniquely as the reparametrization of $h\in RM_{p}$ by $f\in
W_{p}$. We set these correspondences as $f=\Pi(\gamma)$ and $h=\Phi(\gamma)$.
In the Bers coordinates, these maps are defined by
$\Pi([\mu_{1}],[\mu_{2}])=([\mu_{1}],[\mu_{1}])$ and
$\Phi([\mu_{1}],[\mu_{2}])=([0],[\mu_{2}]\ast[\mu_{1}]^{-1})$. Here, we see
that $\Phi$ is a continuous surjection by the topological group property of
$T_{p}$ (Theorem 6.1). Thus, we have a homeomorphism

$(\Pi,\Phi):{\rm WPC}_{p}\to W_{p}\times RM_{p}.$

This is the second product structure of ${\rm WPC}_{p}$.

Moreover, let $IW_{p}$ be the space of all $p$-Weil–Petersson embeddings that
have arc-length parametrizations. Every $\gamma\in{\rm WPC}_{p}$ is
represented uniquely as the reparametrization of its arc-length
parametrization $\gamma_{0}\in IW_{p}$ by $f\in W_{p}$. In this way, ${\rm
WPC}_{p}\cong W_{p}\times IW_{p}$. See Section 4 in more details. Since the
condition $\gamma\in IW_{p}$ is equivalent to the condition that $L(\gamma)$
is purely imaginary, we have
$IW_{p}=L^{-1}(iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ})$ and this is a real-
analytic submanifold of ${\rm WPC}_{p}$. Here,
$iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ is the intersection of the open
subset $L({\rm WPC}_{p})\subset B_{p}(\mathbb{R})$ with the real subspace
$iB_{p}^{\mathbb{R}}(\mathbb{R})$ consisting of purely imaginary functions.
Clearly, $L(W_{p})=B_{p}^{\mathbb{R}}(\mathbb{R})$. Then, there is a bijection
$J$ from $B_{p}^{\mathbb{R}}(\mathbb{R})\times
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ to $L({\rm WPC}_{p})$ such that this
product structure is compatible with $W_{p}\times IW_{p}$ on ${\rm WPC}_{p}$
under $J^{-1}\circ L$ (Lemma 4.2). Again by the topological group property of
$T_{p}$, $J$ is a homeomorphism (Theorem 6.3) and thus we have a homeomorphism

$J^{-1}\circ L:{\rm WPC}_{p}\cong W_{p}\times IW_{p}\to
B_{p}^{\mathbb{R}}(\mathbb{R})\times iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}.$

This is the third product structure of ${\rm WPC}_{p}$.

By investigating these product structures, we obtain topological
correspondence between the factor subspaces of the products. This in
particular shows that these subspaces admit different analytic structures that
are topologically equivalent to each other. We develop those arguments in
Section 6 and the results are summarized as follows.

######  Theorem 1.2.

The space ${\rm WPC}_{p}$ for $p>1$ admits two other product structures
$W_{p}\times RM_{p}$ and $W_{p}\times IW_{p}$ that are homeomorphic to ${\rm
WPC}_{p}$. The fiber structures for the projections to the second factors in
the both products are the same and each fiber consists of the family of all
normalized $p$-Weil–Petersson embeddings of the same image.

$W_{p}$$RM_{p}$${\rm WPC}_{p}$$IW_{p}$$J^{-1}\circ L$

$\
B_{p}^{\mathbb{R}}(\mathbb{R})$$iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$$B_{p}(\mathbb{R})$

Figure 1. The fiber structure of ${\rm WPC}_{p}$

More detailed real-analytic correspondence between the subspaces of ${\rm
WPC}_{p}$ are also obtained. Especially, the dependence of the Riemann
mappings on the arc-length parameters of $p$-Weil–Petersson curves is an
interesting problem, and this is considered in Corollary 5.2 and Theorem 5.4.
The latter theorem contains the following new achievement on the real-analytic
dependence of the following reparametrization map.

######  Theorem 1.3.

The reparametrization map from the arc-length parametrization to the Riemann
mapping parametrization defined by

$\lambda=L\circ\Pi\circ
L^{-1}|_{iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}}:iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}\to
B_{p}^{\mathbb{R}}(\mathbb{R})$

is a real-analytic homeomorphism onto an open contractible domain of
$B_{p}^{\mathbb{R}}(\mathbb{R})$ for $p>1$ whose inverse is also real-
analytic.

This is the Weil–Petersson curve version of the original result for chord-arc
curves by Coifman and Meyer [6]. Their researches provided an important topic
to this field. See Semmes [29, Section 6] and Wu [44]. Our formulation of the
space of $p$-Weil–Petersson curves can make these arguments transparent, and
in particular, the real-analycity of $\lambda$ already follows from our
arguments immediately. The essential step in the original work is the
investigation of the inverse correspondence, but once we know that $\lambda$
is real-analytic, we can apply their result to see that $\lambda^{-1}$ is also
real-analytic. These are demonstrated in Section 5.

##  2\. The $p$-Weil–Petersson Teichmüller space and the $p$-Besov space

A measurable function $\mu$ on $\mathbb{U}$ is called a Beltrami coefficient
if $\|\mu\|_{\infty}<1$. By the solution of the Beltrami equation, there
exists a quasiconformal homeomorphism $F$ of $\mathbb{U}$ onto itself whose
complex dilatation $\mu_{F}=\bar{\partial}F/\partial F$ coincides with $\mu$
uniquely up to the post-composition of affine transformations of $\mathbb{U}$.
The definition on the lower half-plane $\mathbb{L}$ can be similarly done for
this and all other concepts appearing hereafter.

For $p\geq 1$, let $\mathcal{M}_{p}(\mathbb{U})$ be the set of Beltrami
coefficients $\mu$ satisfying

$\|\mu\|_{p}^{p}=\int_{\mathbb{U}}\frac{|\mu(z)|^{p}}{y^{2}}dxdy<\infty.$

By the norm $\|\cdot\|_{\infty}+\|\cdot\|_{p}$, $\mathcal{M}_{p}(\mathbb{U})$
is a domain of the corresponding Banach space.

The $p$-Weil–Petersson Teichmüller space $T_{p}(\mathbb{U})$ on the upper
half-plane $\mathbb{U}$ is defined to be the set of all Teichmüller
equivalence classes $[\mu]$ for $\mu\in\mathcal{M}_{p}(\mathbb{U})$. Here,
$\mu$ and $\mu^{\prime}$ are equivalent if the quasiconformal homeomorphisms
$F_{\mu}$ and $F_{\mu^{\prime}}$ of $\mathbb{U}$ onto itself determined by
$\mu$ and $\mu^{\prime}$ have the same boundary extension to $\mathbb{R}$ up
to the post-composition with affine transformations of $\mathbb{R}$. The
quotient map $\pi:\mathcal{M}_{p}(\mathbb{U})\to T_{p}(\mathbb{U})$ taking the
equivalence class is called the Teichmüller projection. The unique complex
manifold structure of $T_{p}(\mathbb{U})$ for $p\geq 1$ such that the
Teichmüller projection is holomorphic with a local holomorphic right inverse
is introduced via the Bers embedding (see [45, Theorem 4.4], [36, Theorem 2.1]
and [43, Theorem 7.1]).

The boundary extension to $\mathbb{R}$ of a quasiconformal homeomorphism
$F:\mathbb{U}\to\mathbb{U}$ is called a quasisymmetric homeomorphism. If such
an $f:\mathbb{R}\to\mathbb{R}$ is the extension of $F$ whose complex
dilatation is in $\mathcal{M}_{p}(\mathbb{U})$, we say that $f$ is a
$p$-Weil–Petersson class homeomorphism. Then, the above definition of
$T_{p}(\mathbb{U})$ is equivalent to saying that $T_{p}(\mathbb{U})$ is the
set of all $p$-Weil–Petersson class homeomorphisms modulo affine
transformations of $\mathbb{R}$.

A $p$-Weil–Petersson class homeomorphism for $p>1$ can be intrinsically
defined as an increasing homeomorphism $f:\mathbb{R}\to\mathbb{R}$ such that
$f$ is locally absolutely continuous and $\log f^{\prime}$ belongs to
$B_{p}^{\mathbb{R}}(\mathbb{R})$ defined below (see [30, Theorem 1.1], [36,
Theorem 1.2], [32, Theorems 1.2 and 1.3] and [43, Theorem 6.5]). Partially
this will be seen in Lemma 3.3.

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

$\|u\|^{p}_{B_{p}}=\frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}\\!\int_{-\infty}^{\infty}\frac{|u(t)-u(s)|^{p}}{|t-s|^{2}}dsdt<\infty.$

It is easy to see that if $u\in B_{p}(\mathbb{R})$ then $|u|,\ {\rm Re}\,u,\
{\rm Im}\,u\in B_{p}(\mathbb{R})$. We can regard $B_{p}(\mathbb{R})$ as a
complex Banach space with norm $\|\cdot\|_{B_{p}}$ by taking the quotient of
constant functions. In other words, we regard $B_{p}(\mathbb{R})$ as a
homogeneous Besov space, which is often denoted by $\dot{B}_{p}(\mathbb{R})$
in the literature. The real Banach subspace of $B_{p}(\mathbb{R})$ consisting
of the elements represented by real-valued functions is denoted by
$B_{p}^{\mathbb{R}}(\mathbb{R})$. We note that when $p=1$, $B_{p}(\mathbb{R})$
degenerates into the space of constant functions (see [16, Exercise 17.14]).

Concerning the variable change operator on this space, the following result
was shown in [4, Theorem 12 and Remark 5] (see also [38, Theorem 2.2] and [3,
Theorem 1.3 and Section 3.4]).

######  Proposition 2.1.

Let $p>1$. The increasing homeomorphism $h$ from $\mathbb{R}$ onto itself is
quasisymmetric if and only if the variable change operator $P_{h}:u\mapsto
u\circ h$ gives an isomorphism of the $p$-Besov space $B_{p}(\mathbb{R})$ onto
itself, that is, $P_{h}$ and $(P_{h})^{-1}$ are bounded linear operators.

######  Remark.

In the case of $p=2$, it is known that the operator norm $\|P_{h}\|$ of the
variable change operator $P_{h}:B_{p}(\mathbb{R})\to B_{p}(\mathbb{R})$
depends only on the doubling constant of $h$ (or equivalently, the
quasisymmetric constant of $h$ or the Teichmüller distance $d_{\infty}(h,{\rm
id})$). See [21, Theorem 3.1]. On the contrary, in the case of $p\neq 2$, the
estimate of the operator norm seems more difficult. See [4, Remark 4]. In a
special case where $\log h^{\prime}$ belongs to $B_{p}(\mathbb{R})$, a certain
dependence of $\|P_{h}\|$ on $h$ will be shown later in Lemma 6.9.

Next, we consider analytic function spaces. Let $\mathcal{B}(\mathbb{U})$
denote the Bloch space of functions $\varphi$ holomorphic on $\mathbb{U}$ with
semi-norm

$\|\varphi\|_{\mathcal{B}}=\sup_{z\in\mathbb{U}}|\varphi^{\prime}(z)|y.$

Let $\mathcal{B}_{p}(\mathbb{U})$ denote the analytic $p$-Besov space for
$p>1$ (or $p$-Dirichlet space) of holomorphic functions $\varphi$ on
$\mathbb{U}$ with semi-norm

$\|\varphi\|_{\mathcal{B}_{p}}=\left(\frac{1}{\pi}\iint_{\mathbb{U}}|\varphi^{\prime}(z)|^{p}y^{p-2}dxdy\right)^{\frac{1}{p}}.$

Then,
$\mathcal{B}_{p}(\mathbb{U})\subset\mathcal{B}_{q}(\mathbb{U})\subset\mathcal{B}(\mathbb{U})$
for $1<p\leq q$, and the inclusion maps are continuous. By considering
functions in these spaces modulo additive constants, which we always do
hereafter, the semi-norms become norms and the spaces become complex Banach
spaces.

These spaces can be defined on the unit disk $\mathbb{D}$ in the same way as
on $\mathbb{U}$, and any conformal homeomorphism $\mathbb{U}\to\mathbb{D}$
induces an isometric isomorphism between the corresponding spaces. For
example, we take the Cayley transformation $T(z)=(z-i)/(z+i)$, which maps
$\overline{\mathbb{U}}$ onto $\overline{\mathbb{D}}$, and define the push-
forward operator $T_{*}:\varphi\mapsto\varphi\circ T^{-1}$ for functions
$\varphi$ on $\mathbb{U}$. Then,

$\frac{1}{\pi}\iint_{\mathbb{D}}|(T_{*}\varphi)^{\prime}(w)|^{p}\left(\frac{1-|w|^{2}}{2}\right)^{p-2}dudv=\frac{1}{\pi}\iint_{\mathbb{U}}|\varphi^{\prime}(z)|^{p}y^{p-2}dxdy,$

and by defining ${\mathcal{B}}_{p}(\mathbb{D})$ as the space of holomorphic
functions on $\mathbb{D}$ with the norm in the left side integral of the above
equation finite, ${\mathcal{B}}_{p}(\mathbb{D})$ and
${\mathcal{B}}_{p}(\mathbb{U})$ are isometric.

We also see that the Cayley transformation $T$ induces an isometric
isomorphism between $B_{p}(\mathbb{S})$ and $B_{p}(\mathbb{R})$. Here,
$B_{p}(\mathbb{S})$ can be also defined in a natural way. Indeed,

$\begin{split}&\quad\frac{1}{4\pi^{2}}\int_{\mathbb{S}}\int_{\mathbb{S}}\frac{|(T_{*}\varphi)(w_{1})-(T_{*}\varphi)(w_{2})|^{p}}{|w_{1}-w_{2}|^{2}}|dw_{1}||dw_{2}|\\\
&=\frac{1}{4\pi^{2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|\varphi(z_{1})-\varphi(z_{2})|^{p}}{|T(z_{1})-T(z_{2})|^{2}}|T^{\prime}(z_{1})||T^{\prime}(z_{2})||dz_{1}||dz_{2}|\\\
&=\frac{1}{4\pi^{2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|\varphi(z_{1})-\varphi(z_{2})|^{p}}{|z_{1}-z_{2}|^{2}}|dz_{1}||dz_{2}|,\end{split}$

where we used the identity

$\frac{|T(z_{1})-T(z_{2})|^{2}}{|z_{1}-z_{2}|^{2}}=|T^{\prime}(z_{1})||T^{\prime}(z_{2})|$

for a Möbius transformation $T$.

We will use a fact that each function $\varphi\in\mathcal{B}_{p}(\mathbb{U})$
has boundary values almost everywhere on $\mathbb{R}$, and this boundary
function $b(\varphi)$ belongs to the $p$-Besov space $B_{p}(\mathbb{R})$. As
we have seen above, the results on the pairs
$({\mathcal{B}}_{p}(\mathbb{U}),B_{p}(\mathbb{R}))$ and
$({\mathcal{B}}_{p}(\mathbb{D}),B_{p}(\mathbb{S}))$ correspond under $T_{*}$;
we will consider this problem for
$({\mathcal{B}}_{p}(\mathbb{D}),B_{p}(\mathbb{S}))$.

The boundary function $b(\phi)$ is given by the non-tangential limit of
$\phi\in\mathcal{B}_{p}(\mathbb{D})$. The existence of the non-tangential
limit, and moreover, the reproduction of $\phi$ from $b(\phi)$ by the Poisson
integral have been proved (see [46, Lemma 10.13]).

For $p=2$, the fact that $b(\phi)\in B_{2}(\mathbb{S})$ for
$\phi\in\mathcal{B}_{2}(\mathbb{D})$ is well-known as the Douglas formula for
the Dirichlet integral:

$\|\phi\|_{\mathcal{B}_{2}}=\int_{\mathbb{D}}|\phi^{\prime}(z)|^{2}\frac{dxdy}{\pi}=\int_{\mathbb{S}}\\!\int_{\mathbb{S}}\frac{|b(\phi)(z)-b(\phi)(w)|^{2}}{|z-w|^{2}}\frac{|dz|}{2\pi}\frac{|dw|}{2\pi}=\|b(\phi)\|_{B_{2}}.$

See [1, Theorem 2-5] for example. The statement for the general case is as
follows (see [46, pp.131, 301]).

######  Lemma 2.2.

The boundary function $b(\phi)$ of $\phi\in\mathcal{B}_{p}(\mathbb{D})$
belongs to $B_{p}(\mathbb{S})$ for $p>1$. The boundary extension operator
$b:\mathcal{B}_{p}(\mathbb{D})\to B_{p}(\mathbb{S})$ is a bounded linear
isomorphism onto the image.

A proof for the non-homogeneous Besov space on $\mathbb{R}$ (and on
$\mathbb{R}^{n}$) can be found in [34, Section V.5]. A more explicit proof is
in [22, Theorems 2.1 and 5.1]. These are referred to by [26, p.505].

The inverse map of the boundary extension $b$ can be extended to
$B_{p}(\mathbb{S})$ as the Riesz–Szegö projection. The boundedness of this
projection is also known (see [23, Section 2.3]). This implies the boundedness
of the conjugate operator. On the real line $\mathbb{R}$, this is the Hilbert
transformation $\mathcal{H}$, which is defined by

$({\mathcal{H}}u)(x)=\lim_{\varepsilon\to
0}\frac{1}{\pi}\int_{|x-t|>\varepsilon}\frac{u(t)}{x-t}dt$

(see [11, Section III.1]).

######  Lemma 2.3.

The Hilbert transformation $\mathcal{H}$ gives a bounded linear surjective
isomorphism $\mathcal{H}:B_{p}(\mathbb{R})\to B_{p}(\mathbb{R})$ such that
$\mathcal{H}^{2}=-I$ for $p>1$.

This also follows from the results on more general operators, for example, in
[15, Théorème A] and [12, Proposition 4.7].

The $p$-Besov space $B_{p}(\mathbb{R})$ is closely related to BMO functions
defined below. A locally integrable complex-valued function $u$ on
$\mathbb{R}$ is of BMO if

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

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

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

and the set of all such functions is denoted by ${\rm VMO}(\mathbb{R})$.

The following relation between $B_{p}(\mathbb{R})$ and ${\rm VMO}(\mathbb{R})$
is known. See [33, Section 3] and [40, Propositions 2.2 and 2.3].

######  Proposition 2.4.

$(1)$ If $u\in B_{p}(\mathbb{R})$ then $u\in{\rm VMO}(\mathbb{R})$. Moreover,
$\|u\|_{*}\leq\|u\|_{B_{p}}$. $(2)$ If $u\in B_{p}^{\mathbb{R}}(\mathbb{R})$
then $e^{u}$ is an $A_{\infty}$-weight.

Here, we say that a non-negative locally integrable function $\omega\geq 0$ is
an $A_{\infty}$-weight if there exists a constant $C_{\infty}(\omega)\geq 1$
such that

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

for every bounded interval $I\subset\mathbb{R}$. If $\omega$ is an
$A_{\infty}$-weight, then $\log\omega$ is a BMO function (see [9, Corollary
IV.2.19]).

Finally, we introduce classes of Beltrami coefficients on $\mathbb{U}$
including $\mathcal{M}_{p}(\mathbb{U})$. Let $\lambda$ be a positive Borel
measure on the upper half-plane $\mathbb{U}$. We say that $\lambda$ is a
Carleson measure if

$\|\lambda\|_{c}=\sup_{I\subset\mathbb{R}}\frac{\lambda(I\times(0,|I|])}{|I|}<\infty,$

where the supremum is taken over all bounded closed interval
$I\subset\mathbb{R}$ and $I\times(0,|I|]\subset\mathbb{U}$ is the Carleson
box. The set of all Carleson measures on $\mathbb{U}$ is denoted by ${\rm
CM}(\mathbb{U})$. A Carleson measure $\lambda\in{\rm CM}(\mathbb{U})$ is
called vanishing if

$\lim_{|I|\to 0}\frac{\lambda(I\times(0,|I|])}{|I|}=0.$

The set of all vanishing Carleson measures on $\mathbb{U}$ is denoted by ${\rm
CM}_{0}(\mathbb{U})$.

For a Beltrami coefficient $\mu$ on $\mathbb{U}$, we define a positive Borel
measure $\lambda_{\mu}$ that is absolutely continuous with respect to the
Lebesgue measure and satisfies

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

Using this, a norm of $\mu$ is defined by
$\|\mu\|_{c}=\|\lambda_{\mu}\|_{c}^{1/2}$. Let ${\mathcal{M}}_{c}(\mathbb{U})$
be the set of all Beltrami coefficients on $\mathbb{U}$ with
$\|\mu\|_{c}<\infty$, which is a domain of the Banach space with norm
$\|\cdot\|_{c}+\|\cdot\|_{\infty}$. The following claim implies the inclusion
${\mathcal{M}}_{p}(\mathbb{U})\subset{\mathcal{M}}_{c}(\mathbb{U})$.

######  Proposition 2.5.

If $\mu\in{\mathcal{M}}_{p}(\mathbb{U})$ for $p\geq 1$, then
$\lambda_{\mu}\in{\rm CM}_{0}(\mathbb{U})$. Moreover, for $p\geq 2$,
$\|\mu\|_{c}\leq C_{p}\|\mu\|_{p}$ for some constant $C_{p}>0$ depending only
on $p$.

###### Proof.

For the first statement, we may assume that $p\geq 2$ because
${\mathcal{M}}_{p}(\mathbb{U})\subset{\mathcal{M}}_{2}(\mathbb{U})$ if $p<2$.
Let $p^{\prime}=p/2\geq 1$ and take $q^{\prime}\geq 1$ satisfying
$1/p^{\prime}+1/q^{\prime}=1$. When $p^{\prime}=1$, the inequality becomes
simpler. For any bounded interval $I\subset\mathbb{R}$, we have

$\displaystyle\quad\frac{1}{|I|}\int_{0}^{|I|}\\!\int_{I}\frac{|\mu(z)|^{2}}{y}dxdy=\frac{1}{|I|}\int_{0}^{|I|}\\!\int_{I}\frac{|\mu(z)|^{2}}{y^{2/p^{\prime}}}\cdot
y^{\frac{2}{p^{\prime}}-1}dxdy$
$\displaystyle\leq\left(\int_{0}^{|I|}\\!\int_{I}\frac{|\mu(z)|^{p}}{y^{2}}dxdy\right)^{2/p}\left(|I|^{1-q^{\prime}}\int_{0}^{|I|}y^{-\left(1-\frac{1}{p^{\prime}-1}\right)}dy\right)^{1/q^{\prime}}.$

The first factor in the last line of the above inequality is bounded by
$\|\mu\|_{p}^{2}$ and tends to $0$ uniformly as $|I|\to 0$. The second factor
is equal to $(p^{\prime}-1)^{-1/q^{\prime}}$ ($=1$ when $p^{\prime}=1$), which
we define as $C_{p}^{2}$. Taking the square root shows the statement. ∎

##  3\. The Bers coordinates of the space of $p$-Weil–Petersson curves

In this section, we introduce the Bers coordinates for the space of
$p$-Weil–Petersson embeddings $\mathbb{R}\to\mathbb{C}$ and show the
holomorphic correspondence to the $p$-Besov space for $p>1$.

######  Definition.

A continuous embedding $\gamma:\mathbb{R}\to\mathbb{C}$ passing through
$\infty$ is called a $p$-Weil–Petersson embedding for $p\geq 1$ if there is a
quasiconformal homeomorphism $G:\mathbb{C}\to\mathbb{C}$ such that
$G|_{\mathbb{R}}=\gamma$ and its complex dilatation
$\mu_{G}=\bar{\partial}G/\partial G$ on $\mathbb{U}$ belongs to
$\mathcal{M}_{p}(\mathbb{U})$ and $\mu_{G}$ on $\mathbb{L}$ belongs to
$\mathcal{M}_{p}(\mathbb{L})$. We call such $G$ a $p$-Weil–Petersson
quasiconformal homeomorphism associated with $\gamma$.

The image of $\gamma:\mathbb{R}\to\mathbb{C}$ as above is called a
$p$-Weil–Petersson curve. To consider the space of $p$-Weil–Petersson curves,
we add their parametrizations and regard them as $p$-Weil–Petersson
embeddings. Special types of $p$-Weil–Petersson embeddings $\gamma$ are as
follows. If such an embedding $\gamma$ maps $\mathbb{R}$ onto itself, this is
nothing but a $p$-Weil–Petersson class homeomorphism. If $\gamma$ extends
conformally to $\mathbb{U}$, we call it the Riemann mapping parametrization of
a $p$-Weil–Petersson curve.

We can define a BMO embedding $\gamma:\mathbb{R}\to\mathbb{C}$ by replacing
the above $\mathcal{M}_{p}(\mathbb{U})$ and $\mathcal{M}_{p}(\mathbb{L})$ with
$\mathcal{M}_{c}(\mathbb{U})$ and $\mathcal{M}_{c}(\mathbb{L})$. By
Proposition 2.5, we see that any $p$-Weil–Petersson embedding $\gamma$ for
$p\geq 1$ is a BMO embedding. Hence, we can utilize the following known
properties of BMO embeddings (see [42, Proposition 3.3, Theorem 3.6]) for
$p$-Weil–Petersson embeddings.

######  Proposition 3.1.

A BMO-embedding $\gamma:\mathbb{R}\to\mathbb{C}$ has its derivative
$\gamma^{\prime}$ almost everywhere on $\mathbb{R}$ and
$\log\gamma^{\prime}\in{\rm BMO}(\mathbb{R})$. Moreover, if
$|\gamma^{\prime}|$ is an $A_{\infty}$-weight on $\mathbb{R}$, then $\gamma$
is locally absolutely continuous and the image $\gamma(\mathbb{R})$ is a
chord-arc curve.

A locally rectifiable Jordan curve $\Gamma$ passing through $\infty$ is called
a chord-arc curve if there is a constant $K\geq 1$ such that the length of the
arc between $a,b\in\Gamma$ is bounded by $K|a-b|$.

######  Lemma 3.2.

If $h:\mathbb{R}\to\mathbb{C}$ is a Riemann mapping parametrization of a
$p$-Weil–Petersson curve for $p>1$, then $h$ is locally absolutely continuous
with $\log h^{\prime}\in B_{p}(\mathbb{R})$.

###### Proof.

By definition, $h$ extends to a conformal homeomorphism $H$ on $\mathbb{U}$.
By Theorem 7.1 in Appendix, we have $\log
H^{\prime}\in\mathcal{B}_{p}(\mathbb{U})$. Then, the non-tangential limit
$b(\log H^{\prime})$ belongs to $B_{p}(\mathbb{R})$ by Lemma 2.2. On the other
hand, $\log h^{\prime}$ coincides with $b(\log H^{\prime})$ (a.e.) since $H$
has the quasiconformal extension to $\mathbb{C}$ (see [25, Theorem 5.5]).
Thus, we have $\log h^{\prime}\in B_{p}(\mathbb{R})$. In particular,
$|h^{\prime}|\in A_{\infty}$ by Proposition 2.4. Then, we conclude by
Proposition 3.1 that $h$ is locally absolutely continuous on $\mathbb{R}$. ∎

The following result is given in [30, p.1056] for $p=2$, and in [36, p.669]
for $p\geq 2$. The generalization to $p>1$ is possible as we do in [43,
Theorem 6.5].

######  Lemma 3.3.

If $f:\mathbb{R}\to\mathbb{R}$ is a $p$-Weil–Petersson class homeomorphism for
$p>1$, then $f$ is locally absolutely continuous with $\log f^{\prime}\in
B_{p}^{\mathbb{R}}(\mathbb{R})$.

###### Proof.

By the well-known conformal sewing principle (see [14, Section III.1.4] and
[37, p.11]), there exists a pair of quasiconformal homeomorphisms $H$ and
$H_{*}$ on the whole plane $\mathbb{C}$ such that $H$ is conformal on
$\mathbb{U}$, $H^{*}$ is conformal on $\mathbb{L}$, and $f=h_{*}^{-1}\circ h$
for $h=H|_{\mathbb{R}}$ and $h_{*}=H_{*}|_{\mathbb{R}}$ on $\mathbb{R}$.
Moreover, we can choose these $H$ and $H_{*}$ so that the complex dilatation
of $H|_{\mathbb{L}}$ is in $\mathcal{M}_{p}(\mathbb{L})$ and the complex
dilatation of $H_{*}|_{\mathbb{U}}$ is in $\mathcal{M}_{p}(\mathbb{U})$. This
is a crucial step and its argument is in [43]. We note that to obtain the
appropriate mapping $H_{*}$, we have to show that the inverse $f^{-1}$ has a
quasiconformal extension whose complex dilatation belongs to
$\mathcal{M}_{p}(\mathbb{U})$. This is a part of the property that $T_{p}$ has
the group structure, which will be explained in Section 6.

By Lemma 3.2, we have $h$ and $h_{*}$ are locally absolutely continuous with
$\log h^{\prime}\in B_{p}(\mathbb{R})$ and $\log h_{*}^{\prime}\in
B_{p}(\mathbb{R})$. From $h_{*}\circ f=h$ on $\mathbb{R}$, we see that the
increasing homeomorphism $f$ maps a set of null measure to a set of null
measure because $h$ is locally absolutely continuous and
$|h_{*}^{\prime}(x)|>0$ almost everywhere on $\mathbb{R}$. Hence, $f$ is
locally absolutely continuous. Taking the derivatives of the both sides of the
above equality, we have

$P_{f}(\log h_{*}^{\prime})+\log f^{\prime}=\log h^{\prime}.$

By Proposition 2.1, we see that $P_{f}(\log h_{*}^{\prime})\in
B_{p}(\mathbb{R})$. Hence, $\log f^{\prime}\in B_{p}(\mathbb{R})$. ∎

The combination of the above two lemmas yields the general property on
$p$-Weil–Petersson embeddings.

######  Theorem 3.4.

A $p$-Weil–Petersson embedding $\gamma:\mathbb{R}\to\mathbb{C}$ is locally
absolutely continuous and $\log\gamma^{\prime}$ belongs to $B_{p}(\mathbb{R})$
for $p>1$.

###### Proof.

Let $G:\mathbb{C}\to\mathbb{C}$ be a $p$-Weil–Petersson quasiconformal
homeomorphism associated with $\gamma$ such that
$\mu_{1}=\mu_{G}|_{\mathbb{U}}\in\mathcal{M}_{p}(\mathbb{U})$ and
$\mu_{2}=\mu_{G}|_{\mathbb{L}}\in\mathcal{M}_{p}(\mathbb{L})$. We take a
quasiconformal homeomorphism $F:\mathbb{C}\to\mathbb{C}$ whose complex
dilatation is $\mu_{1}(z)$ for $z\in\mathbb{U}$ and
$\overline{\mu_{1}(\bar{z})}$ for $z\in\mathbb{L}$, which maps $\mathbb{R}$
onto itself. By Lemma 3.3, $f=F|_{\mathbb{R}}$ is locally absolutely
continuous and $\log f^{\prime}$ belongs to $B_{p}^{\mathbb{R}}(\mathbb{R})$.
Next, we take a quasiconformal homeomorphism $H:\mathbb{C}\to\mathbb{C}$ that
is conformal on $\mathbb{U}$ and whose complex dilatation on $\mathbb{L}$ is
the push-forward $F_{*}\mu_{2}$ of $\mu_{2}$ by $F$. Namely, the complex
dilatation of $H\circ F$ is $\mu_{2}$. Then, $H\circ F$ coincides with $G$ up
to an affine transformation of $\mathbb{C}$, and hence, we may assume that
$H\circ F=G$.

We may replace $F|_{\mathbb{L}}$ with a bi-Lipschitz diffeomorphism under the
hyperbolic metric whose complex dilatation $\tilde{\mu}_{1}$ belongs to
$\mathcal{M}_{p}(\mathbb{L})$ (see [7, Theorem 6], [45, Theorem 2.4], and [43,
Lemma 5.1]). The complex dilatation of $H|_{\mathbb{L}}$ is explicitly given
by

$F_{*}\mu_{2}(\zeta)=\frac{\mu_{2}(z)-\tilde{\mu}_{1}(z)}{1-\overline{\tilde{\mu}_{1}(z)}\mu_{2}(z)}\cdot\frac{\overline{F_{z}}}{F_{z}}$

for $\zeta=F(z)\in\mathbb{L}$. Using the property that $F$ is a bi-Lipschitz
diffeomorphism, we see from this formula that $F_{*}\mu_{2}$ also belongs to
$\mathcal{M}_{p}(\mathbb{L})$. Then by Lemma 3.2, $h=H|_{\mathbb{R}}$ is
locally absolutely continuous with $\log h^{\prime}\in B_{p}(\mathbb{R})$.

By $h\circ f=\gamma$, we see that $\gamma$ is also locally absolutely
continuous, and taking the derivative, we have

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

By $\log f^{\prime}\in B_{p}(\mathbb{R})$ and $\log h^{\prime}\in
B_{p}(\mathbb{R})$ combined with Proposition 2.1, we obtain that
$\log\gamma^{\prime}\in B_{p}(\mathbb{R})$. ∎

We impose the normalization $\gamma(0)=0$ and $\gamma(1)=1$ (and
$\gamma(\infty)=\infty$) on a $p$-Weil–Petersson embedding $\gamma$. Let ${\rm
WPC}_{p}$ be the set of all such normalized $p$-Weil–Petersson embeddings for
$p\geq 1$. We also denote the subset of ${\rm WPC}_{p}$ consisting of all
normalized $p$-Weil–Petersson class homeomorphisms by $W_{p}$, and the subset
consisting of all normalized Riemann mapping parametrizations of
$p$-Weil–Petersson curves by $RM_{p}$. For
$\mu_{1}\in\mathcal{M}_{p}(\mathbb{U})$ and
$\mu_{2}\in\mathcal{M}_{p}(\mathbb{L})$, we denote by $G=G(\mu_{1},\mu_{2})$
the normalized $p$-Weil–Petersson quasiconformal homeomorphism of $\mathbb{C}$
($G(0)=0$, $G(1)=1$, and $G(\infty)=\infty$) with
$\mu_{G}|_{\mathbb{U}}=\mu_{1}$ and $\mu_{G}|_{\mathbb{L}}=\mu_{2}$. We define
a map

$\widetilde{\iota}:\mathcal{M}_{p}(\mathbb{U})\times\mathcal{M}_{p}(\mathbb{L})\to{\rm
WPC}_{p}$

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

######  Proposition 3.5.

The space ${\rm WPC}_{p}$ of all normalized $p$-Weil–Petersson embeddings is
identified with $T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$ for $p\geq 1$.
More precisely, $\widetilde{\iota}$ splits into a well-defined bijection

$\iota:T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})\to{\rm WPC}_{p}$

by the product of the Teichmüller projections
$\widetilde{\pi}:\mathcal{M}_{p}(\mathbb{U})\times\mathcal{M}_{p}(\mathbb{L})\to
T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$ such that
$\widetilde{\iota}=\iota\circ\widetilde{\pi}$.

We call $T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$ the Bers coordinates of
${\rm WPC}_{p}$. Any normalized $p$-Weil–Petersson embedding $\gamma$ is
represented by a pair $([\mu_{1}],[\mu_{2}])$ of the Teichmüller equivalence
classes of $\mu_{1}\in\mathcal{M}(\mathbb{U})$ and
$\mu_{2}\in\mathcal{M}(\mathbb{L})$ via $G(\mu_{1},\mu_{2})$.

We may provide complex Banach manifold structures for $T_{p}(\mathbb{U})$ and
$T_{p}(\mathbb{L})$ by using the pre-Schwarzian derivative models as in
Theorem 7.2 of Appendix. Namely, $T_{p}(\mathbb{U})$ is identified with the
domain $\mathcal{T}_{p}(\mathbb{L})$ of the analytic $p$-Besov space
${\mathcal{B}}_{p}(\mathbb{L})$, and $T_{p}(\mathbb{L})$ is identified with
the domain $\mathcal{T}_{p}(\mathbb{U})$ of ${\mathcal{B}}_{p}(\mathbb{U})$
for $p>1$:

$\displaystyle T_{p}(\mathbb{U})$
$\displaystyle\cong\mathcal{T}_{p}(\mathbb{L})=\\{{\mathcal{L}}_{G(\mu,0)}\in{\mathcal{B}}_{p}(\mathbb{L})\mid\mu\in\mathcal{M}_{p}(\mathbb{U})\\};$
$\displaystyle T_{p}(\mathbb{L})$
$\displaystyle\cong\mathcal{T}_{p}(\mathbb{U})=\\{{\mathcal{L}}_{G(0,\mu)}\in{\mathcal{B}}_{p}(\mathbb{U})\mid\mu\in\mathcal{M}_{p}(\mathbb{L})\\}.$

Then, by Proposition 3.5, we may also regard ${\rm WPC}_{p}$ as a domain of
${\mathcal{B}}_{p}(\mathbb{L})\times{\mathcal{B}}_{p}(\mathbb{U})$ for $p>1$.

By Theorem 3.4, we can consider a map $L:{\rm WPC}_{p}\to B_{p}(\mathbb{R})$
defined by $L(\gamma)=\log\gamma^{\prime}$. Then, with respect to the complex
structure of ${\rm WPC}_{p}$ given as above, we see the following.

######  Theorem 3.6.

The map $L:{\rm WPC}_{p}\to B_{p}(\mathbb{R})$ is holomorphic for $p>1$.

###### Proof.

We will prove that $L$ is holomorphic at any point
$\gamma=G(\mu_{1},\mu_{2})|_{\mathbb{R}}$ in ${\rm WPC}_{p}$. Since ${\rm
WPC}_{p}$ can be regarded as a domain of the product
${\mathcal{B}}_{p}(\mathbb{L})\times{\mathcal{B}}_{p}(\mathbb{U})$ of the
Banach spaces, the Hartogs theorem for Banach spaces (see [5, Theorem 14.27]
and [20, Theorem 36.8]) implies that we have only to prove that $L$ is
separately holomorphic. Thus, by fixing $[\mu_{1}]\in T_{p}(\mathbb{U})$, we
will show that $\log(G(\mu_{1},\mu)|_{\mathbb{R}})^{\prime}\in
B_{p}(\mathbb{R})$ depends holomorphically on $[\mu]\in T_{p}(\mathbb{L})$.
The other case is similarly treated.

By the proof of Theorem 3.4, we have

$\log(G(\mu_{1},\mu)|_{\mathbb{R}})^{\prime}=\log\gamma^{\prime}=\log
h^{\prime}\circ f+\log f^{\prime}.$

We may choose a bi-Lipschitz diffeomorphism $F:\mathbb{L}\to\mathbb{L}$ that
is the extension of $f:\mathbb{R}\to\mathbb{R}$ and whose complex dilatation
still belongs to $\mathcal{M}_{p}(\mathbb{L})$ as before (see [43, Lemma
5.1]). Let $h:\mathbb{R}\to\mathbb{C}$ be the restriction of the
quasiconformal homeomorphism $H_{F_{*}\mu}$ of $\mathbb{C}$ that is conformal
on $\mathbb{U}$ and has the complex dilatation $F_{*}\mu$ on $\mathbb{L}$.
Since $F$ is a bi-Lipschitz diffeomorphism, we see that $F_{*}$ acts on
${\mathcal{M}}_{p}(\mathbb{L})$ as a biholomorphic automorphism, and its
action projects down to $T_{p}(\mathbb{L})$ also as a biholomorphic
automorphism. For $p\geq 2$, this is shown in [37, Chap.1, Corollary 2.12] and
[45, Proposition 5.3], and the same proof is valid for $p\geq 1$ once we know
that $F_{*}\mu\in{\mathcal{M}}_{p}(\mathbb{L})$ for every
$\mu\in{\mathcal{M}}_{p}(\mathbb{L})$ (see [43, Proposition 5.3]). Local
boundedness of $F_{*}$ is enough to show the holomorphy of $F_{*}$, which is
explained in [41, Proposition 3.1] in a similar setting. Hence,
${\mathcal{L}}_{H_{F_{*}\mu}|_{\mathbb{U}}}=\log(H_{F_{*}\mu}|_{\mathbb{U}})^{\prime}\in{\mathcal{B}}_{p}(\mathbb{U})$
depends on $[\mu]\in T_{p}(\mathbb{L})$ holomorphically as we see in the proof
of Theorem 7.2 in Appendix.

By Lemma 2.2, we see that the boundary extension
$b:{\mathcal{B}}_{p}(\mathbb{U})\to B_{p}(\mathbb{R})$ is a bounded linear
operator for $p>1$. Moreover, by Proposition 2.1, the variable change operator
$P_{f}:B_{p}(\mathbb{R})\to B_{p}(\mathbb{R})$ induced by $f$ is also a
bounded linear operator. Therefore,

$\log h^{\prime}\circ f=P_{f}\circ
b({\mathcal{L}}_{H_{F_{*}\mu}|_{\mathbb{U}}})\in B_{p}(\mathbb{R})$

in particular depends on $[\mu]\in T_{p}(\mathbb{L})$ holomorphically, and so
does $\log(G(\mu_{1},\mu)|_{\mathbb{R}})^{\prime}$. ∎

##  4\. Conformal welding and curve theoretic coordinates

We introduce other coordinates of ${\rm WPC}_{p}$ and $L({\rm WPC}_{p})$, and
investigate their relationship. To this end, we utilize the canonical
automorphisms of ${\rm WPC}_{p}$.

For $\nu\in{\mathcal{M}}_{p}(\mathbb{U})$, the same symbol $\nu$ still denotes
the complex dilatation $\overline{\nu(\bar{z})}$ for $z\in\mathbb{L}$ in
${\mathcal{M}}_{p}(\mathbb{L})$. This also gives the identification of
$T_{p}(\mathbb{U})$ and $T_{p}(\mathbb{L})$, which is often denoted by $T_{p}$
hereafter. For any $[\nu]\in T_{p}$, we define the right translation of ${\rm
WPC}_{p}$ for $p\geq 1$ by

$\widetilde{R}_{[\nu]}:([\mu_{1}],[\mu_{2}])\mapsto([\mu_{1}]\ast[\nu],[\mu_{2}]\ast[\nu]),$

where $R_{[\nu]}([\mu])=[\mu]\ast[\nu]$ is the composition of elements in
$T_{p}$ that is given by the Teichmüller class of the complex dilatation of
$F^{\mu}\circ F^{\nu}$ for the normalized $p$-Weil–Petersson class
homeomorphisms $F^{\mu}$ and $F^{\nu}$ of $\mathbb{U}$ (or $\mathbb{L}$) onto
itself with the given complex dilatations. It is known that the right
translation $R_{[\nu]}$ is a biholomorphic automorphism of $T_{p}$ for $p\geq
1$ as is seen in the proof of Theorem 3.6. Hence, $\widetilde{R}_{[\nu]}$
gives a biholomorphic automorphism of ${\rm WPC}_{p}$.

First, we consider the conformal welding coordinates of ${\rm WPC}_{p}$ for
$p\geq 1$. Under the Bers coordinates ${\rm WPC}_{p}\cong
T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$, the subspace $W_{p}\subset{\rm
WPC}_{p}$ is identified with the diagonal locus

$\\{([\mu],[\mu])\in T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})\mid[\mu]\in
T_{p}\\}.$

Since this is the fixed point locus of the anti-holomorphic involution
$([\mu_{1}],[\mu_{2}])\mapsto([\mu_{2}],[\mu_{1}])$, we see that $W_{p}$ is a
real-analytic submanifold of ${\rm WPC}_{p}$. Moreover, the subspace
$RM_{p}\subset{\rm WPC}_{p}$ is identified with the second coordinate axis

$\\{([0],[\mu])\in T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})\mid[\mu]\in
T_{p}\\},$

which is a complex-analytic submanifold of ${\rm WPC}_{p}$.

We define the projections to these submanifolds

$\Pi:{\rm WPC}_{p}\to W_{p},\qquad\Phi:{\rm WPC}_{p}\to RM_{p}$

by $\Pi([\mu_{1}],[\mu_{2}])=([\mu_{1}],[\mu_{1}])$ and
$\Phi([\mu_{1}],[\mu_{2}])=([0],[\mu_{2}]\ast[\mu_{1}]^{-1})$ in the Bers
coordinates, where $[\mu]^{-1}$ is the inverse of an element in $T_{p}$ that
is given by the Teichmüller class of the complex dilatation $\mu^{-1}$ of
$(F^{\mu})^{-1}$. Then, every $\gamma\in{\rm WPC}_{p}$ is decomposed uniquely
into $\gamma=\Phi(\gamma)\circ\Pi(\gamma)$. This corresponds to the
decomposition $\gamma=h\circ f$ in the proof of Theorem 3.4. Clearly, $\Pi$ is
real-analytic. We see that $\Phi$ is continuous later by Theorem 6.1. The
biholomorphic automorphism $\widetilde{R}_{[\nu]}$ of ${\rm WPC}_{p}$ for
$[\nu]\in T_{p}$ satisfies that $\Phi\circ\widetilde{R}_{[\nu]}=\Phi$.

The projections $\Pi$ and $\Phi$ defines another product structure
$W_{p}\times RM_{p}$ on ${\rm WPC}_{p}$ for $p\geq 1$. Namely, we have a
bijection

$(\Pi,\Phi):{\rm WPC}_{p}\to W_{p}\times RM_{p}.$

Once we see that $\Phi$ is continuous, $(\Pi,\Phi)$ is a homeomorphism. This
is the coordinate change of ${\rm WPC}_{p}$ from the Bers coordinates to the
one we may call the conformal welding coordinates. Since $W_{p}$ and $RM_{p}$
are both identified with $T_{p}$, by marking $T_{p}$ with $W_{p}\cong
T_{p}^{W}$ and $RM_{p}\cong T_{p}^{RM}$, the coordinate change is expressed as

$T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})\to T_{p}^{W}\times
T_{p}^{RM}:\quad([\mu_{1}],[\mu_{2}])\mapsto([\mu_{1}],[\mu_{2}]\ast[\mu_{1}]^{-1}).$

Next, we consider the curve theoretical coordinates of the space of
$p$-Weil–Petersson embeddings by using the image $L({\rm WPC}_{p})$ in
$B_{p}(\mathbb{R})$ for $p>1$. We see that $L$ is injective because $\gamma$
can be reproduced from $w=\log\gamma^{\prime}\in L({\rm WPC}_{p})$ by

$\gamma(x)=\int_{0}^{x}e^{w(t)}dt.$

We have assumed that $B_{p}(\mathbb{R})$ is the Banach space of all
equivalence classes of complex-valued functions modulo additive constants, and
can also regard it as the set of representatives $w$ satisfying the
normalization condition $\int_{0}^{1}e^{w(t)}dt=1$. For $w\in L({\rm
WPC}_{p})$, this is always possible by adding some complex constant to $w$.

Let $u\in B_{p}^{\mathbb{R}}(\mathbb{R})$ and let
$\gamma_{u}:\mathbb{R}\to\mathbb{R}$ be the $p$-Weil–Petersson class
homeomorphism in $W_{p}$ defined by $\gamma_{u}(x)=\int_{0}^{x}e^{u(t)}dt$.
Then, the variable change operator $P_{\gamma_{u}}:B_{p}(\mathbb{R})\to
B_{p}(\mathbb{R})$ is given by $w\mapsto w\circ\gamma_{u}$ for $w\in
B_{p}(\mathbb{R})$, which is a linear isomorphism of the Banach space
$B_{p}(\mathbb{R})$ onto itself by Proposition 2.1. Moreover, we define
$Q_{u}(w)=P_{\gamma_{u}}(w)+u$ for $w\in B_{p}(\mathbb{R})$, which is an
affine isomorphism of $B_{p}(\mathbb{R})$ onto itself. Since $P_{\gamma_{u}}$
preserves $B_{p}^{\mathbb{R}}(\mathbb{R})$, we see that $Q_{u}$ maps
$u_{0}+iB_{p}^{\mathbb{R}}(\mathbb{R})$ onto
$u_{1}+iB_{p}^{\mathbb{R}}(\mathbb{R})$ for some $u_{1}\in
B_{p}^{\mathbb{R}}(\mathbb{R})$ depending on $u_{0}\in
B_{p}^{\mathbb{R}}(\mathbb{R})$, where $iB_{p}^{\mathbb{R}}(\mathbb{R})$
denotes the real subspace of $B_{p}(\mathbb{R})$ consisting of all purely
imaginary functions modulo complex-valued constant functions.

We see that the affine isomorphism $Q_{u}$ of $B_{p}(\mathbb{R})$ keeps the
subset $L({\rm WPC}_{p})$ invariant. We often use the correspondence between
$u\in B_{p}^{\mathbb{R}}(\mathbb{R})$ and $[\nu]\in T_{p}$ through
$\gamma_{u}\in W_{p}\cong T_{p}$.

######  Proposition 4.1.

The right translation $\widetilde{R}_{[\nu]}$ satisfies

$L\circ\widetilde{R}_{[\nu]}=Q_{L([\nu],[\nu])}\circ L$

on ${\rm WPC}_{p}$ for every $[\nu]\in T_{p}$. Hence, the affine isomorphism
$Q_{u}$ of $B_{p}(\mathbb{R})$ for any $u\in B_{p}^{\mathbb{R}}(\mathbb{R})$
maps $L({\rm WPC}_{p})$ onto itself.

###### Proof.

For any $([\mu_{1}],[\mu_{2}])\in{\rm WPC}_{p}$, we have

$\begin{split}L\circ\widetilde{R}_{[\nu]}([\mu_{1}],[\mu_{2}])&=L([\mu_{1}]\ast[\nu],[\mu_{2}]\ast[\nu])\\\
&=\log\gamma^{\prime}_{L([\mu_{1}]\ast[\nu],[\mu_{2}]\ast[\nu])}=\log(\gamma_{L([\mu_{1}],[\mu_{2}])}\circ\gamma_{L([\nu],[\nu])})^{\prime}\\\
&=\log\gamma^{\prime}_{L([\mu_{1}],[\mu_{2}])}\circ\gamma_{L([\nu],[\nu])}+\log\gamma^{\prime}_{L([\nu],[\nu])}\\\
&=L([\mu_{1}],[\mu_{2}])\circ\gamma_{L([\nu],[\nu])}+L([\nu],[\nu])=Q_{L([\nu],[\nu])}\circ
L([\mu_{1}],[\mu_{2}])\end{split}$

as required. For any $u\in B_{p}^{\mathbb{R}}(\mathbb{R})$, we choose
$[\nu]\in T_{p}$ corresponding to $\gamma_{u}\in W_{p}$. Namely,
$L([\nu],[\nu])=u$. Then, we have

$Q_{u}(L({\rm WPC}_{p}))=Q_{L([\nu],[\nu])}\circ L({\rm
WPC}_{p})=L\circ\widetilde{R}_{[\nu]}({\rm WPC}_{p})=L({\rm WPC}_{p}),$

which proves the second statement. ∎

We define the following subset of $L({\rm WPC}_{p})$ corresponding to the arc-
length parametrization:

$iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}=iB_{p}^{\mathbb{R}}(\mathbb{R})\cap
L({\rm WPC}_{p}).$

Let $iv\in iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$. Then,
$\gamma_{iv}(x)=\int_{0}^{x}e^{iv(t)}dt$ is a $p$-Weil–Petersson embedding of
arc-length parametrization. Precisely speaking, by the normalization,
$\gamma_{iv}$ is parametrized by the multiple of its arc-length by a positive
constant. We can regard $iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ as a
parameter space of such $p$-Weil–Petersson embeddings for $p>1$. All
$p$-Weil–Petersson embeddings are obtained by the reparametrization of their
arc-length parametrizations as follows.

######  Lemma 4.2.

Let $u\in B_{p}^{\mathbb{R}}(\mathbb{R})$ and $iv\in
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$. Then, $\gamma_{Q_{u}(iv)}(x)$ is
obtained from the $p$-Weil–Petersson embedding $\gamma_{iv}(x^{\prime})$ of
arc-length parametrization by the change of parameter
$x^{\prime}=\gamma_{u}(x)$, which is also a $p$-Weil–Petersson embedding.
Conversely, every $p$-Weil–Petersson embedding is obtained in this way. Hence,
the map

$J:B_{p}^{\mathbb{R}}(\mathbb{R})\times
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}\to L({\rm WPC}_{p})\subset
B_{p}(\mathbb{R})$

defined by $J(u,iv)=Q_{u}(iv)=u+iP_{\gamma_{u}}(v)$ is bijective.

###### Proof.

Since $Q_{u}(iv)=u+iP_{\gamma_{u}}(v)=u+iv\circ\gamma_{u}$, we have

$\begin{split}\gamma_{Q_{u}(iv)}(x)=\int_{0}^{x}e^{u(t)}e^{iv\circ\gamma_{u}(t)}dt=\int_{0}^{x}\gamma^{\prime}_{u}(t)e^{iv\circ\gamma_{u}(t)}dt=\int_{0}^{\gamma_{u}(x)}e^{iv(s)}ds=\gamma_{iv}(\gamma_{u}(x))\end{split}$

by $s=\gamma_{u}(t)$. Proposition 4.1 implies that the reparametrization of a
$p$-Weil–Petersson embedding is also a $p$-Weil–Petersson embedding.
Conversely, let $\gamma_{u+iv^{\prime}}$ be any $p$-Weil–Petersson embedding
for $u+iv^{\prime}\in L({\rm WPC}_{p})$. Then, by choosing $v\in
B_{p}^{\mathbb{R}}(\mathbb{R})$ satisfying $P_{\gamma_{u}}(v)=v^{\prime}$, we
see that $\gamma_{u+iv^{\prime}}$ is obtained from $\gamma_{iv}$ by the change
of the parameter. Then, $\gamma_{iv}$ is a $p$-Weil–Petersson embedding, and
hence $iv\in iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$. ∎

We will see later in Theorem 6.3 that the above bijection $J$ is in fact a
homeomorphism.

Now, we have two product structures $W_{p}\times RM_{p}$ and
$B_{p}^{\mathbb{R}}(\mathbb{R})\times iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$
on ${\rm WPC}_{p}$ for $p>1$. There is a close relation between these
structures through $L$ and $J$. Each fiber of the projection $\Phi$ consists
of a family of embeddings with the same image, and hence their arc-length
parametrizations are the same. This observation leads the following.

######  Proposition 4.3.

For any $iv\in iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$, let $\gamma=\Phi\circ
L^{-1}\circ J(0,iv)\in RM_{p}$. Then, the fiber $\Phi^{-1}(\gamma)$ coincides
with $L^{-1}\circ J(B_{p}^{\mathbb{R}}(\mathbb{R})\times\\{iv\\})$, which is
the family of normalized $p$-Weil–Petersson embeddings with the same image
$\gamma(\mathbb{R})$.

##  5\. Biholomorphic correspondence

We prove the main theorem in this section as follows. In this section, we
always assume that $p>1$.

######  Theorem 5.1.

The holomorphic map $L:{\rm WPC}_{p}\to B_{p}(\mathbb{R})$ is a biholomorphic
homeomorphism onto its image. In particular, $L({\rm WPC}_{p})$ is an open
contractible domain of $B_{p}(\mathbb{R})$ which contains
$B_{p}^{\mathbb{R}}(\mathbb{R})$.

###### Proof.

In virtue of Theorem 3.6, in order to prove that $L$ is biholomorphic, it
suffices to show that $L$ has a local holomorphic inverse at any point $w\in
L({\rm WPC}_{p})\subset B_{p}(\mathbb{R})$. This in particular shows that
$L({\rm WPC}_{p})$ is open. It is proved in [33, Theorem 6.1] based on the
arguments in [29] that if $w=iv\in iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$
for $p=2$, then there is a neighborhood $V_{iv}\subset L({\rm WPC}_{p})$ of
$iv$ and a holomorphic map $\Psi_{iv}:V_{iv}\to{\rm WPC}_{p}$ such that
$L\circ\Psi_{iv}={\rm id}|_{V_{iv}}$. For general $p>1$, the proof is
essentially the same.

If $w=u+iv^{\prime}$ is an arbitrary point in $L({\rm WPC}_{p})$, then we
utilize $Q_{u}$, and find $iv\in iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ with
$Q_{u}(iv)=u+iv^{\prime}$ by Lemma 4.2. Since $Q_{u}$ is a biholomorphic
automorphism of $L({\rm WPC}_{p})$ by Proposition 4.1, we see that
$\widetilde{R}_{[\nu]}\circ\Psi_{iv}\circ Q_{u}^{-1}$ is holomorphic on
$Q_{u}(V_{iv})$ for $[\nu]\in T_{p}$ corresponding to $\gamma_{u}\in W_{p}$.
Then, Proposition 4.1 implies that

$L\circ\widetilde{R}_{[\nu]}\circ\Psi_{iv}\circ
Q_{u}^{-1}=Q_{L([\nu],[\nu])}\circ L\circ\Psi_{iv}\circ
Q_{u}^{-1}=Q_{L([\nu],[\nu])}\circ Q_{u}^{-1}=\rm id$

on $Q_{u}(V_{iv})$. Hence, $\widetilde{R}_{[\nu]}\circ\Psi_{iv}\circ
Q_{u}^{-1}$ is a local holomorphic inverse of $L$ on $Q_{u}(V_{iv})$.

We know that the $p$-Weil–Petersson Teichmüller space $T_{p}$ is contractible
by [7, Theorem 6] for $p=2$, by [45, Proposition 3.5] for $p\geq 2$, and by
[40, Corollary 1.4] for general $p>1$. Hence, the product
$T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$ is contractible, and so is $L({\rm
WPC}_{p})$. ∎

######  Remark.

The space $L({\rm WPC}_{p})$ is denoted by $\widehat{\mathcal{T}}_{e}$ in [33,
Theorem 2.5] in the case of $p=2$ and proved that it is a contractible open
domain in $H^{1/2}(\mathbb{R})=B_{2}(\mathbb{R})$. It is also shown in [33,
Theorem 2.2] that $iB_{2}^{\mathbb{R}}(\mathbb{R})^{\circ}$, the parameter
space for Weil–Petersson curves with arc-length parametrization, coincides
with an open subset of $iB_{2}^{\mathbb{R}}(\mathbb{R})$ consisting of all
elements corresponding to chord-arc curves with arc-length parametrization.
This result can be generalized to any $p>1$.

Let $IW_{p}\subset{\rm WPC}_{p}$ denote the subset of all arc-length
parametrizations of normalized $p$-Weil–Petersson embeddings. Namely,

$IW_{p}=L^{-1}(iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}).$

As $iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ is a real-analytic submanifold of
the domain $L({\rm WPC}_{p})$ in the complex Banach space $B_{p}(\mathbb{R})$,
$IW_{p}$ is a real-analytic submanifold of the complex manifold ${\rm
WPC}_{p}$. By $W_{p}=L^{-1}(B_{p}^{\mathbb{R}}(\mathbb{R}))$, we again see
that $W_{p}$ is a real-analytic submanifold of ${\rm WPC}_{p}$.

######  Corollary 5.2.

$(1)$ $W_{p}$ is a real-analytic submanifold of ${\rm WPC}_{p}$ that is the
diagonal of $T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$, and $L|_{W_{p}}$ is a
real-analytic homeomorphism onto $B_{p}^{\mathbb{R}}(\mathbb{R})$ whose
inverse is also real-analytic. $(2)$ $IW_{p}$ is a real-analytic submanifold
of ${\rm WPC}_{p}$, and $L|_{IW_{p}}$ is a real-analytic homeomorphism onto
$iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ whose inverse is also real-analytic.

The real analytic property of $L|_{W_{p}}$ has been shown in [31, Theorem 2.3]
in the case of $p=2$ by a different method. Part (1) of the above corollary
shows that $W_{p}$ is equipped with both the complex-analytic structure of
$T_{p}$ and the real-analytic structure of $B_{p}^{\mathbb{R}}(\mathbb{R})$,
which are real-analytically equivalent. Later in Theorem 6.6, we will see that
$IW_{p}$ is canonically equipped with the complex-analytic structure of
$T_{p}$ which is topologically equivalent to the real-analytic structure of
$iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$.

In [40, Theorem 4.4], we construct a holomorphic map
$\Lambda:U(B^{\mathbb{R}}_{p}(\mathbb{R}))\to\mathcal{M}_{p}(\mathbb{U})$ on
some neighborhood $U(B^{\mathbb{R}}_{p}(\mathbb{R}))$ of the real-valued
subspace $B^{\mathbb{R}}_{p}(\mathbb{R})$ for $p>1$. In the same way, we have
the correspondence to the complex dilatations on $\mathbb{L}$. Thus, we can
extend $\Lambda$ to a holomorphic map

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

This induces the inverse of $L$ on the neighborhood
$U(B^{\mathbb{R}}_{p}(\mathbb{R}))$ as is shown in [40, Theorem 4.5].

######  Theorem 5.3.

The neighborhood $U(B^{\mathbb{R}}_{p}(\mathbb{R}))$ is contained in $L({\rm
WPC}_{p})$, and the holomorphic map
$\widetilde{\pi}\circ\widetilde{\Lambda}:U(B^{\mathbb{R}}_{p}(\mathbb{R}))\to{\rm
WPC}_{p}$ is the inverse of $L$ on $U(B^{\mathbb{R}}_{p}(\mathbb{R}))$.

We compare the arc-length parametrizations $IW_{p}$ with the Riemann mapping
parametrizations $RM_{p}$. Both are the sets of all representatives of
$p$-Weil–Petersson curves, which follows from Proposition 4.3. Hence, there is
a canonical bijection between $IW_{p}$ and $RM_{p}$ giving the change of the
representatives, namely, keeping the images of the corresponding embeddings
the same. For the projection $\Phi:{\rm WPC}_{p}\to RM_{p}$, this bijection is
nothing but its restriction $\Phi|_{IW_{p}}:IW_{p}\to RM_{p}$. We will see
that $\Phi|_{IW_{p}}$ is a homeomorphism by Proposition 6.5 in the next
section.

Here, we consider the other projection $\Pi$ restricted to $IW_{p}$, which has
been studied with great interest in the literature. For any $\gamma\in
IW_{p}$, $\Pi(\gamma)\in W_{p}$ is defined by the $p$-Weil–Petersson class
homeomorphism inducing the parameter change from $\gamma$ to $\Phi(\gamma)\in
RM_{P}$. We will prove the bi-real-analytic property of this mapping. For the
space of chord-arc curves, this property for the corresponding map was proved
in [6, Theorem 1] by operator theoretical arguments. The first statement of
the following theorem that $\lambda$ is real-analytic in the case of $p=2$ is
in [33, Theorem 7.1].

######  Theorem 5.4.

The map $\Pi|_{IW_{p}}:IW_{p}\to W_{p}$ is real-analytic. Hence,

$\lambda=L\circ\Pi\circ
L^{-1}|_{iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}}:iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}\to
B_{p}^{\mathbb{R}}(\mathbb{R})$

is also real-analytic. Moreover, $\lambda$ is injective and the inverse
$\lambda^{-1}$ is real-analytic. Namely, $\lambda$ is a real-analytic
homeomorphism onto an open subset of $B_{p}^{\mathbb{R}}(\mathbb{R})$ whose
inverse is also real-analytic. This is true for $\Pi|_{IW_{p}}$.

###### Proof.

By Corollary 5.2, $IW_{p}$ is a real-analytic submanifold of ${\rm WPC}_{p}$.
Hence, the restriction $\Pi|_{IW_{p}}$ of the projection $\Pi:{\rm WPC}_{p}\to
W_{p}$ is real-analytic. Since $L$ is biholomorphic by Theorem 5.1, the
conjugate map $\lambda$ is real-analytic.

We will prove the real-analycity of the inverse of $\lambda$. To this end, we
use the corresponding result for the space of chord-arc curves with the
notation being the same as in [42]. The definitions of the corresponding
subspaces can be found in this paper, but they are not so essential in the
arguments below. Based on Propositions 2.5 and 3.1, we see that the space
${\rm CA}$ of all BMO embeddings with chord-arc images contains ${\rm
WPC}_{p}$. Then, there are the inclusion relations of subspaces

$W_{p}\subset{\rm SQS},\ IW_{p}\subset{\rm ICA},\ RM_{p}\subset{\rm
RM}^{\circ}$

in ${\rm WPC}_{p}\subset{\rm CA}$. We also have the inclusion relations of
subspaces

$B_{p}^{\mathbb{R}}(\mathbb{R})\subset{\rm BMO}_{\mathbb{R}}^{*}(\mathbb{R}),\
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}\subset i{\rm
BMO}_{\mathbb{R}}(\mathbb{R})^{\circ}$

in $B_{p}(\mathbb{R})\subset{\rm BMO}(\mathbb{R})$ by Proposition 2.4, where
${\rm BMO}_{\mathbb{R}}^{*}(\mathbb{R})$ stands for the space of all real-
valued BMO functions $u$ with $e^{u}$ being an $A_{\infty}$-weight.

The corresponding map to $\lambda$ between these larger spaces is denoted by
$\tilde{\lambda}:i{\rm BMO}_{\mathbb{R}}(\mathbb{R})^{\circ}\to{\rm
BMO}_{\mathbb{R}}^{*}(\mathbb{R})$. Then,
$\lambda=\tilde{\lambda}|_{iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}}$. It is
known that $\tilde{\lambda}$ is a real-analytic homeomorphism onto an open
subset of ${\rm BMO}_{\mathbb{R}}^{*}(\mathbb{R})$ whose inverse is also real-
analytic (see [28, Theorem 5]).

First of all, we prove the injectivity of $\lambda$. This is the same as the
case of the space of chord-arc curves. Every $\gamma_{0}\in IW_{p}$ is
decomposed uniquely into $\gamma_{0}=h\circ f$ for $h\in RM_{p}$ and $f\in
W_{p}$. Taking the logarithm of the derivative of this equation, we have

$\log\gamma_{0}^{\prime}=\log h^{\prime}\circ f+\log f^{\prime}.$

Since $\log\gamma_{0}^{\prime}=iv$ is purely imaginary and $\log f^{\prime}$
is real, the real and the imaginary parts of this equation become

$0={\rm Re}\log h^{\prime}\circ f+\log f^{\prime}\quad{\rm and}\quad v={\rm
Im}\log h^{\prime}\circ f.$

Moreover, since $\log h^{\prime}$ is the boundary extension of the holomorphic
function $\log H^{\prime}$ for the Riemann mapping $H$ on $\mathbb{U}$, ${\rm
Re}\log h^{\prime}$ and ${\rm Im}\log h^{\prime}$ are related by the Hilbert
transformation $\mathcal{H}$ on $\mathbb{R}$:

${\rm Im}\log h^{\prime}={\mathcal{H}}({\rm Re}\log h^{\prime}).$

Then, the combination of these equations yields that

$-P_{f}\circ{\mathcal{H}}\circ P_{f}^{-1}(\log f^{\prime})=v.$ ($\ast$)

This shows that $v$ is determined by $f$ and thus
$\lambda:\log\gamma_{0}^{\prime}\mapsto\log f^{\prime}$ is injective.

######  Claim 1.

Suppose that we have the decomposition
$\tilde{\gamma}_{0}=\tilde{h}\circ\tilde{f}$ of $\tilde{\gamma}_{0}\in{\rm
ICA}$ by $\tilde{h}\in{\rm RM}^{\circ}$ and $\tilde{f}\in{\rm SQS}$. In this
situation, if $\tilde{f}$ belongs to $W_{p}$, then we obtain $\tilde{h}\in
RM_{p}$ and $\tilde{\gamma}_{0}\in IW_{p}$.

###### Proof.

The formula corresponding to ($\ast$ ‣ 5) reads as

$-P_{\tilde{f}}\circ{\mathcal{H}}\circ
P_{\tilde{f}}^{-1}(\log\tilde{f}^{\prime})=\tilde{v}.$ ($**$)

Here, $\tilde{f}\in W_{p}$ implies $\log\tilde{f}^{\prime}\in
B_{p}^{\mathbb{R}}(\mathbb{R})$. Moreover, since $P_{\tilde{f}}$ preserves
$B_{p}^{\mathbb{R}}(\mathbb{R})$ by Proposition 2.1, we have
$P_{\tilde{f}}^{-1}(\log\tilde{f}^{\prime})\in
B_{p}^{\mathbb{R}}(\mathbb{R})$. By Lemma 2.3, the Hilbert transformation
$\mathcal{H}$ maps $B_{p}^{\mathbb{R}}(\mathbb{R})$ to
$B_{p}^{\mathbb{R}}(\mathbb{R})$. This implies ${\mathcal{H}}\circ
P_{\tilde{f}}^{-1}(\log\tilde{f}^{\prime})\in B_{p}^{\mathbb{R}}(\mathbb{R})$.
By applying $P_{\tilde{f}}$ again, we see that the left side of ($**$ ‣ 5) is
in $B_{p}^{\mathbb{R}}(\mathbb{R})$, and hence $\tilde{v}\in
B_{p}^{\mathbb{R}}(\mathbb{R})$. Since
$\log\tilde{\gamma}_{0}^{\prime}=i\tilde{v}$, we have $\tilde{\gamma}_{0}\in
IW_{p}$ and thus $\tilde{h}=\tilde{\gamma}_{0}\circ\tilde{f}^{-1}\in RM_{p}$.
∎

By the conjugation of $L$, this claim is equivalent to saying that if
$\tilde{\lambda}(w)\in B_{p}^{\mathbb{R}}(\mathbb{R})$ for $w\in i{\rm
BMO}_{\mathbb{R}}(\mathbb{R})^{\circ}$ then $w\in
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$.

We move to the investigation of the derivatives of $\lambda$ and
$\tilde{\lambda}$. We note the following two facts: (1) As $\lambda$ is real-
analytic, the derivative $d_{w}\lambda:iB_{p}^{\mathbb{R}}(\mathbb{R})\to
B_{p}^{\mathbb{R}}(\mathbb{R})$ is a bounded linear operator at every point
$w$ of the domain of $\lambda$; (2) As $\tilde{\lambda}$ is real-analytic and
$\tilde{\lambda}^{-1}$ is also real-analytic, the derivative
$d_{\tilde{w}}\tilde{\lambda}:i{\rm BMO}_{\mathbb{R}}(\mathbb{R})\to{\rm
BMO}_{\mathbb{R}}(\mathbb{R})$ is a surjective bounded linear isomorphism at
every point $\tilde{w}$ of the domain of $\tilde{\lambda}$.

######  Claim 2.

$d_{w}\tilde{\lambda}|_{iB_{P}^{\mathbb{R}}(\mathbb{R})}=d_{w}\lambda$ at
every point $w$ of the domain $iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ of
$\lambda$.

###### Proof.

Take any $iv\in iB_{P}^{\mathbb{R}}(\mathbb{R})$, and set $d_{w}\lambda(iv)=u$
and $d_{w}\tilde{\lambda}(iv)=\tilde{u}$. Then,

$\lim_{t\to
0}\left\|\frac{\lambda(w+tiv)-\lambda(w)}{t}-u\right\|_{B_{p}}=0;\quad\lim_{t\to
0}\left\|\frac{\tilde{\lambda}(w+tiv)-\tilde{\lambda}(w)}{t}-\tilde{u}\right\|_{*}=0.$

Since $w+tiv\in iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ for all
$t\in\mathbb{R}$ sufficiently close to $0$, we have
$\tilde{\lambda}(w+tiv)=\lambda(w+tiv)$ as well as
$\tilde{\lambda}(w)=\lambda(w)$. Combined with the estimate of the norms
$\|\cdot\|_{*}\leq\|\cdot\|_{B_{p}}$ by Proposition 2.4, these two limits
imply $u=\tilde{u}$. Hence, $d_{w}\tilde{\lambda}(iv)=d_{w}\lambda(iv)$ for
every $iv\in iB_{P}^{\mathbb{R}}(\mathbb{R})$, that is,
$d_{w}\tilde{\lambda}|_{iB_{P}^{\mathbb{R}}(\mathbb{R})}=d_{w}\lambda$. ∎

Property (2) as above implies that $d_{\tilde{w}}\tilde{\lambda}$ is injective
at every $\tilde{w}$. Then, $d_{w}\lambda$ is also injective at every $w\in
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ by Claim 2. Next, we will show that
$d_{w}\lambda$ is surjective. After this, we see from the open mapping theorem
that $d_{w}\lambda:iB_{p}^{\mathbb{R}}(\mathbb{R})\to
B_{p}^{\mathbb{R}}(\mathbb{R})$ is a bounded linear isomorphism. Under this
condition, the inverse mapping theorem implies that $\lambda^{-1}$ is real-
analytic in some neighborhood of any point in the image
$\lambda(iB_{P}^{\mathbb{R}}(\mathbb{R})^{\circ})$, and thus $\lambda^{-1}$ is
globally real-analytic on $\lambda(iB_{P}^{\mathbb{R}}(\mathbb{R})^{\circ})$
which is an open subset of $B_{P}^{\mathbb{R}}(\mathbb{R})$.

The remaining task is to show that $d_{w}\lambda$ is surjective at every $w\in
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$. We take any tangent vector $u\in
B_{p}^{\mathbb{R}}(\mathbb{R})$ at $\lambda(w)$, and consider a segment
$\\{\lambda(w)+tu\\}\subset B_{p}^{\mathbb{R}}(\mathbb{R})$ with $t$ in a
sufficiently small interval $[0,\varepsilon]$. Since
$\lambda(w)\in\tilde{\lambda}(i{\rm BMO}_{\mathbb{R}}(\mathbb{R})^{\circ})$
and $\tilde{\lambda}(i{\rm BMO}_{\mathbb{R}}(\mathbb{R})^{\circ})$ is open, we
may assume that $\\{\lambda(w)+tu\\}\subset\tilde{\lambda}(i{\rm
BMO}_{\mathbb{R}}(\mathbb{R})^{\circ})$. Then, the inverse image
$\tilde{\lambda}^{-1}\\{\lambda(w)+tu\\}$ of the segment is a real-analytic
curve $\beta(t)$ in $i{\rm BMO}_{\mathbb{R}}(\mathbb{R})^{\circ}$ starting at
$w=\beta(0)$. The tangent vector $iv=\frac{d}{dt}\beta(t)|_{t=0}$ of
$\beta(t)$ at $t=0$ satisfies $d_{w}\tilde{\lambda}(iv)=u$. On the other hand,
since $\\{\lambda(w)+t\xi\\}$ is contained in
$B_{p}^{\mathbb{R}}(\mathbb{R})$, Claim 1 implies that $\beta(t)$ is contained
in $iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$. Hence, $iv\in
iB_{p}^{\mathbb{R}}(\mathbb{R})$. Then by Claim 2, we have
$d_{w}\lambda(iv)=u$. This shows that $d_{w}\lambda$ is surjective. ∎

In this Weil–Petersson curve version of the Coifman–Meyer theorem, we can also
ask a question about the characterization of the domains
$iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ and
$\lambda(iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ})$ which are contractible and
real-analytically equivalent to each other. The contractibility will be seen
by Proposition 6.5.

##  6\. The topological group structure and its applications

For further investigation, we show and use the following fact.

######  Theorem 6.1.

The $p$-Weil–Petersson Teichmüller space $T_{p}$ for $p\geq 1$ is a
topological group under the operation $\ast$.

For $p=2$, this was proved in [37, Chap.1, Theorem 3.8]. A similar argument to
this case using the quasiconformal extension and estimating the integral of
the complex dilatation also works for any $p\geq 1$. For Theorem 6.1, we show
the following basic result by using bi-Lipschitz quasiconformal extension.

######  Lemma 6.2.

If $[\mu]$ and $[\nu]$ converge to $[\rm 0]$ in $T_{p}$ for $p\geq 1$, then
$[\mu]\ast[\nu]\to[\rm 0]$ and $[\nu]^{-1}\to[\rm 0]$.

###### Proof.

Let $F:\mathbb{U}\to\mathbb{U}$ be a quasiconformal homeomorphism with its
complex dilatation $\mu\in\mathcal{M}_{p}(\mathbb{U})$ in the equivalence
class $[\mu]$. We may choose $\mu$ so that $\|\mu\|_{p}\to 0$ and
$\|\mu\|_{\infty}\to 0$ as $[\mu]\to[\rm 0]$. Let $H:\mathbb{U}\to\mathbb{U}$
be a bi-Lipschitz diffeomorphism with its complex dilatation
$\nu\in\mathcal{M}_{p}(\mathbb{U})$ in the equivalence class $[\nu]$. The
existence of such an extension is guaranteed by [43, Lemma 5.1]. This also
implies that $\|\nu\|_{p}\to 0$ and $\|\nu\|_{\infty}\to 0$ as $[\nu]\to[\rm
0]$ and that the bi-Lipschitz constant $L\geq 1$ of $H$ is uniformly bounded
throughout when $[\nu]$ tends to $[\rm 0]$. We use the chain rule for the
complex dilatation:

$\mu\ast\nu(z)=\frac{(\mu\circ
H(z))\cdot\frac{\overline{H_{z}}}{H_{z}}+\nu(z)}{1+\mu\circ
H(z)\cdot{\bar{\nu}}(z)\frac{\overline{H_{z}}}{H_{z}}};\quad\nu^{-1}(H(z))=-\nu(z)\frac{\overline{H_{z}}}{H_{z}}.$

For the composition, we estimate the integral as

$\begin{split}\int_{\mathbb{U}}|\mu\ast\nu(z)|^{p}\frac{dxdy}{y^{2}}\leq\frac{2^{p-1}}{(1-\|\mu\|_{\infty}\|\nu\|_{\infty})^{p}}\left(\int_{\mathbb{U}}|\mu\circ
H(z)|^{p}\frac{dxdy}{y^{2}}+\int_{\mathbb{U}}|\nu(z)|^{p}\frac{dxdy}{y^{2}}\right).\end{split}$

Since $H^{-1}$ is Lipschitz with the constant $L$ in the hyperbolic metric, we
have

$\begin{split}\int_{\mathbb{U}}|\mu\circ
H(z)|^{p}\frac{dxdy}{y^{2}}=\int_{\mathbb{U}}|\mu(\zeta)|^{p}\frac{d\xi
d\eta}{J_{H}(z)y^{2}}\leq KL^{2}\int_{\mathbb{U}}|\mu(\zeta)|^{p}\frac{d\xi
d\eta}{\eta^{2}}\end{split}$

for the Jacobian determinant $J_{H}$ of $H$ and the maximal dilatation $K\geq
1$ of $H$. Thus,

$\|\mu\ast\nu\|_{p}\leq
2^{\frac{p-1}{p}}\frac{(KL^{2}\|\mu\|^{p}_{p}+\|\nu\|^{p}_{p})^{1/p}}{1-\|\mu\|_{\infty}\|\nu\|_{\infty}};\quad\|\mu\ast\nu\|_{\infty}\leq\frac{\|\mu\|_{\infty}+\|\nu\|_{\infty}}{1-\|\mu\|_{\infty}\|\nu\|_{\infty}}.$

This implies that $[\mu]\ast[\nu]\to[\rm 0]$ as $[\mu]\to[\rm 0]$ and
$[\nu]\to[\rm 0]$.

For the inverse operation, we obtain similarly

$\int_{\mathbb{U}}|\nu^{-1}(z)|^{p}\frac{dxdy}{y^{2}}\leq\int_{\mathbb{U}}|\nu\circ
H^{-1}(z)|^{p}\frac{dxdy}{y^{2}}\leq
KL^{2}\int_{\mathbb{U}}|\nu(\zeta)|^{p}\frac{d\xi d\eta}{\eta^{2}}$

since $H$ is Lipschitz with the constant $L$. Thus,

$\|\nu^{-1}\|_{p}\leq(KL^{2})^{1/p}\|\nu\|_{p};\quad\|\nu^{-1}\|_{\infty}=\|\nu\|_{\infty}.$

This implies that $[\nu]^{-1}\to[\rm 0]$ as $[\nu]\to[\rm 0]$. ∎

###### Proof of Theorem 6.1.

Lemma 6.2 implies that $T_{p}$ is a partial topological group. We have already
seen that that the right translation $R_{[\nu]}$ for any $[\nu]\in T_{p}$ is
continuous (in fact, biholomorphic) at the beginning of Section 4. Hence, in
order to show that $T_{p}$ is a topological group according to [10, Lemma
1.1], we have only to prove that the adjoint map $T_{p}\to T_{p}$ defined by
$[\mu]\ast[\nu]\ast[\mu]^{-1}$ for any fixed $[\mu]\in T_{p}$ is continuous as
$[\nu]\to[0]$. The arguments are essentially the same as in [37, Chapter 1,
Lemma 3.5]. We omit the details here. ∎

There are several consequences from Theorem 6.1. We first introduce the curve
theoretical product structure for the space ${\rm WPC}_{p}$ of normalized
$p$-Weil–Petersson curves, and verify the topological equivalence of those
real and complex Banach manifold structures on ${\rm WPC}_{p}$ and its
subspaces for $p>1$. Then, the continuity of Riemann mappings defined by
$p$-Weil–Petersson curves, and the boundedness of the image of a bounded set
under the biholomorphic homeomorphism $L$ on ${\rm WPC}_{p}$ are proved.

Theorem 6.1 in particular implies that $R_{[\nu]}([\mu])=[\mu]\ast[\nu]$ and
$(R_{[\nu]})^{-1}([\mu])=[\mu]\ast[\nu]^{-1}$ are continuous with respect to
two variables $([\mu],[\nu])\in T_{p}\times T_{p}$. Since
$\widetilde{R}_{[\nu]}$ and $Q_{L([\nu],[\nu])}$ correspond as in Proposition
4.1, this fact yields the following consequence on the bijection $J$ given in
Lemma 4.2.

######  Theorem 6.3.

The bijection $J:B_{p}^{\mathbb{R}}(\mathbb{R})\times
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}\to L({\rm WPC}_{p})$ is a
homeomorphism for $p>1$.

###### Proof.

We prove that $Q_{u}(w)$ and $Q_{u}^{-1}(w)$ are continuous for $(u,w)\in
B_{p}^{\mathbb{R}}(\mathbb{R})\times L({\rm WPC}_{p})$. By Proposition 4.1
with $L([\nu],[\nu])=u$ for some $[\nu]\in T_{p}$, we obtain that

$Q_{u}(w)=L\circ\widetilde{R}_{[\nu]}\circ L^{-1}(w).$

Then, Theorems 5.1 and 6.1 imply that this is continuous. Similarly,
$Q_{u}^{-1}(w)$ is also continuous. Since $J(u,iv)=Q_{u}(iv)$ and
$J^{-1}(u+iv)=(u,Q_{u}^{-1}(u+iv))$, both $J$ and $J^{-1}$ are continuous. ∎

######  Remark.

The continuity of $Q_{u}(w)$ also implies that the variable change operator
$P:B_{p}(\mathbb{R})\times W_{p}\to B_{p}(\mathbb{R})$ defined by
$P_{h}(w)=w\circ h$ is continuous for both $w\in B_{p}(\mathbb{R})$ and $h\in
W_{p}$. Indeed, $P_{h}(w)=Q_{\log h^{\prime}}(w)-\log h^{\prime}$. For $p=2$,
the corresponding claim to this continuity is in [33, Proposition 7.1] for
instance.

######  Corollary 6.4.

The space ${\rm WPC}_{p}$ of all normalized $p$-Weil–Petersson embeddings is
endowed with the real Banach manifold structure modeled on
$B_{p}^{\mathbb{R}}(\mathbb{R})\times iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$
by $J^{-1}\circ L$, which is topologically equivalent to its complex Banach
manifold structure $T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$.

Next, we investigate the continuity of Riemann mappings defined by
$p$-Weil–Petersson curves with respect to the topological structure of ${\rm
WPC}_{p}$. As we have seen in Corollary 6.4, the complex-analytic structure
$T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$ and the real-analytic structure
$B_{p}^{\mathbb{R}}(\mathbb{R})\times iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$
on ${\rm WPC}_{p}$ are topologically equivalent. Thus, we can use either of
these structures in order to consider the continuity.

######  Proposition 6.5.

The map $\Phi:{\rm WPC}_{p}\to RM_{p}\cong T_{p}^{RM}$ is a continuous
surjection for $p\geq 1$. Moreover, $\Phi|_{IW_{p}}:IW_{p}\to RM_{p}$ is a
homeomorphism for $p>1$.

###### Proof.

The first statement follows from Theorem 6.1. For
$(\Phi|_{IW_{p}})^{-1}:RM_{p}\to IW_{p}$, we consider its conjugate by
$J^{-1}\circ L$. This is nothing but the projection
$B_{p}^{\mathbb{R}}(\mathbb{R})\times
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}\to
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$ to the second factor by Proposition
4.3. Hence, $(\Phi|_{IW_{p}})^{-1}$ is continuous and $\Phi|_{IW_{p}}$ is a
homeomorphism. ∎

######  Remark.

It is proved in [33, Theorem 2.4] that the map
$iB_{2}^{\mathbb{R}}(\mathbb{R})^{\circ}\to T_{2}^{RM}$ is a homeomorphism,
which corresponds to our $\Phi|_{IW_{p}}$ under $IW_{2}\cong
iB_{2}^{\mathbb{R}}(\mathbb{R})^{\circ}$. In our framework, $IW_{2}$ and
$iB_{2}^{\mathbb{R}}(\mathbb{R})^{\circ}$ are real-analytically equivalent
under $L$, so this claim can be rephrased as just the continuity of $\Phi$ as
in Proposition 6.5.

Thus, we can conclude that $IW_{p}$ and $RM_{p}$, both of which can be
regarded as the space of the images of all $p$-Weil–Petersson embeddings, are
naturally endowed with the two analytic structures in the following sense.

######  Theorem 6.6.

The real-analytic submanifold $IW_{p}$ and the complex-analytic submanifold
$RM_{p}$ of ${\rm WPC}_{p}$ for $p>1$ are equipped with both the complex-
analytic structure of $T_{p}$ and the real-analytic structure of
$iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$, which are topologically equivalent.

By $J^{-1}\circ L$, we can introduce the product structure $W_{p}\times
IW_{p}$ to ${\rm WPC}_{p}$ from $B_{p}^{\mathbb{R}}(\mathbb{R})\times
iB_{p}^{\mathbb{R}}(\mathbb{R})^{\circ}$. Both the products $W_{p}\times
RM_{p}$ and $W_{p}\times IW_{p}$ are homeomorphic to ${\rm WPC}_{p}$ for
$p>1$. We summarize the correspondence of the product structures $W_{p}\times
RM_{p}$ and $W_{p}\times IW_{p}$ on ${\rm WPC}_{p}$ more precisely than
Proposition 4.3 by adding the homeomorphic property of the sections for the
projection $\Phi$. The proof is similar. The fiber structure of the projection
to the second factors in the both products are preserved since such a fiber
consists of $p$-Weil–Petersson curves of the same image.

######  Proposition 6.7.

$(1)$ For every $\gamma_{0}\in IW_{p}$, the projection $\Phi:{\rm WPC}_{p}\to
RM_{p}$ restricted to $W_{p}\times\\{\gamma_{0}\\}\subset W_{p}\times IW_{p}$
is a constant map, and hence $W_{p}\times\\{\gamma_{0}\\}$ is the fiber of
$\Phi$ over $\Phi(\gamma_{0})$. $(2)$ For every $f\in W_{p}$, $\Phi$
restricted to $\\{f\\}\times IW_{p}\subset W_{p}\times IW_{p}$ is a
homeomorphism onto $RM_{p}\cong T_{p}^{RM}$, and hence $\\{f\\}\times IW_{p}$
is the section of $\Phi$ through $f$.

We remark that in the comparison of the product structures $W_{p}\times
RM_{p}$ and $W_{p}\times IW_{p}$ on ${\rm WPC}_{p}$, the fibers to the
projection to the first factors are not preserved. The change of the
representatives $\Pi|_{IW_{p}}:IW_{p}\to W_{p}$ considered in Theorem 5.4
measures the difference between the fibers $RM_{p}$ and $IW_{p}$ over the
origin.

Finally, we add one more property to the biholomorphic mapping $L:{\rm
WPC}_{p}\to B_{p}(\mathbb{R})$ restricted to $W_{p}$, which is concerning the
correspondence of bounded subsets. Here, the boundedness in $W_{p}$ is
considered regarding a metric structure of $W_{p}\cong T_{p}^{W}$. The
invariant metric provided for $T_{p}$ is the $p$-Weil–Petersson metric (see
[7], [17], [37]), and let $d_{p}$ denote the $p$-Weil–Petersson distance in
$T_{p}$. This can be defined for $p\geq 1$.

The correspondence $f\mapsto\log f^{\prime}$ for $f\in W_{p}$ gives a real-
analytic equivalence of the $p$-Weil–Petersson Teichmüller space $T_{p}$ to
$B_{p}^{\mathbb{R}}(\mathbb{R})$ (see Corollary 5.2). Translating Lemma 6.2 to
$B_{p}^{\mathbb{R}}(\mathbb{R})$ for $p>1$, we see that $\|\log(f\circ
h)^{\prime}\|_{B_{p}}\to 0$ as $\|\log f^{\prime}\|_{B_{p}}\to 0$ and $\|\log
h^{\prime}\|_{B_{p}}\to 0$ for $f,h\in W_{p}$. Extending this consequence to a
claim for the variable change operator $P_{h}$, we obtain the following
proposition. For BMO norm, an analogues result was stated in [6, p.18]. We
have seen that $P_{h}$ is a bounded linear operator in Proposition 2.1.

######  Proposition 6.8.

For any $h\in W_{p}$, let $P_{h}:B_{p}(\mathbb{R})\to B_{p}(\mathbb{R})$ be
the bounded linear operator defined by $w\mapsto w\circ h$ for every $w\in
B_{p}(\mathbb{R})$. Then, there exist constants $\tau_{0}>0$ and $C_{0}>0$
such that the operator norm of $P_{h}$ satisfies $\|P_{h}\|\leq C_{0}$ for
every $h\in W_{p}$ with $\|\log h^{\prime}\|_{B_{p}}\leq\tau_{0}$.

###### Proof.

Lemma 6.2 implies that there exist $\delta_{0}>0$ and $\tau_{0}>0$ such that
if $\|\log f^{\prime}\|_{B_{p}}\leq\delta_{0}$ and $\|\log
h^{\prime}\|_{B_{p}}\leq\tau_{0}$, then $\|\log(f\circ
h)^{\prime}\|_{B_{p}}\leq 1$. We may choose $\tau_{0}$ so that $\tau_{0}\leq
1$. Then, for any $u=\log f^{\prime}\in B_{p}^{\mathbb{R}}(\mathbb{R})$ with
$\|u\|_{B_{p}}\leq\delta_{0}$ and $h\in W_{p}$ with $\|\log
h^{\prime}\|_{B_{p}}\leq\tau_{0}$, we have

$\|P_{h}(u)\|_{B_{p}}\leq\|\log(f\circ h)^{\prime}\|_{B_{p}}+\|\log
h^{\prime}\|_{B_{p}}\leq 2.$

For $w=u+iv\in B_{p}(\mathbb{R})$ with $\|w\|_{B_{p}}=\delta_{0}$, we apply
this estimate to $\|P_{h}(u)\|_{B_{p}}$ and $\|P_{h}(v)\|_{B_{p}}$; we obtain
that if $\|w\|_{B_{p}}=\delta_{0}$ and $\|\log
h^{\prime}\|_{B_{p}}\leq\tau_{0}$, then $\|P_{h}(w)\|_{B_{p}}\leq 4$. This
completes the proof by setting $C_{0}=4/\delta_{0}$. ∎

######  Lemma 6.9.

For any $h\in W_{p}$, the operator norm $\|P_{h}\|$ of the variable change
operator $P_{h}:B_{p}(\mathbb{R})\to B_{p}(\mathbb{R})$ depends only on the
$p$-Weil–Petersson distance $d_{p}(h,\rm id)$ for $p>1$.

###### Proof.

For the constant $\tau_{0}$ in Proposition 6.8, we choose a constant $r_{0}>0$
such that if $h\in W_{p}$ satisfies $d_{p}(h,{\rm id})\leq r_{0}$ then $\|\log
h^{\prime}\|_{B_{p}}\leq\tau_{0}$. Any element $h\in W_{p}$ can be joined to
$\rm id$ by a curve in $W_{p}$ with its length arbitrary close to
$d_{p}(h,{\rm id})$. We choose the minimal number of consecutive points

${\rm id}=h_{0},h_{1},\ldots,h_{n}=h$

on the curve such that $d_{p}(h_{i},h_{i-1})<r_{0}$ for any $i=1,\ldots,n$.
Then, the number $n$ is determined by $d_{p}(h,\rm id)$, and the invariance of
$d_{p}$ under the right translation implies that the composition $h_{i}\circ
h_{i-1}^{-1}$ satisfies $d_{p}(h_{i}\circ h_{i-1}^{-1},{\rm id})<r_{0}$, and
hence $\|\log(h_{i}\circ h_{i-1}^{-1})^{\prime}\|_{B_{p}}\leq\tau_{0}$. By
decomposing $h$ into these $n$ mappings, we have

$P_{h}=P_{h_{1}\circ h_{0}^{-1}}\circ P_{h_{2}\circ
h_{1}^{-1}}\circ\cdots\circ P_{h_{n}\circ h_{n-1}^{-1}}.$

Then, Proposition 6.8 shows that $\|P_{h}\|\leq C_{0}^{n}$. ∎

######  Remark.

For $p=2$, this claim follows from the stronger result mentioned in the remark
after Proposition 2.1. This is because the Teichmüller distance $d_{\infty}$
is bounded by a certain multiple of the $p$-Weil–Petersson distance $d_{p}$,
that is, $d_{\infty}\lesssim d_{p}$. See [18, Proposition 6.10].

######  Theorem 6.10.

For a bounded subset $W\subset W_{p}$ for $p>1$, the image $L(W)$ is also
bounded in $B_{p}^{\mathbb{R}}(\mathbb{R})$. In more details, if $h\in W_{p}$
is within distance $r$ from $\rm id$ in the $p$-Weil–Petersson distance
$d_{p}$, then $\|\log h^{\prime}\|_{B_{p}}$ is bounded by a constant depending
only on $r$.

###### Proof.

For $h\in W_{p}$ with $d_{p}(h,{\rm id})\leq r$, we decompose $h$ as in the
proof of Lemma 6.9. Then, by this lemma, we have

$\displaystyle\|\log h^{\prime}\|_{B_{p}}$ $\displaystyle=\|\log((h_{n}\circ
h_{n-1}^{-1})\circ(h_{n-1}\circ h_{n-2}^{-1})\circ\cdots\circ(h_{1}\circ
h_{0}^{-1}))^{\prime}\|_{B_{p}}$ $\displaystyle\leq\|P_{h_{1}\circ
h_{0}^{-1}}\circ P_{h_{2}\circ h_{1}^{-1}}\circ\cdots\circ P_{h_{n-1}\circ
h_{n-2}^{-1}}(\log(h_{n}\circ h_{n-1}^{-1})^{\prime})\|_{B_{p}}$
$\displaystyle\quad+\cdots+\|\log(h_{1}\circ h_{0}^{-1})^{\prime}\|_{B_{p}}$
$\displaystyle\leq C_{0}^{n-1}\tau_{0}+C_{0}^{n-2}\tau_{0}+\cdots+\tau_{0}.$

Since $n$ depends only on $r$, the statement is proved. ∎

Developing Theorem 6.10 further, we expect (a) for any bounded subset
$\widetilde{W}\subset{\rm WPC}_{p}$, the image $L(\widetilde{W})$ is bounded
in $B_{p}(\mathbb{R})$; (b) conversely, for any bounded subset $B\subset
B_{p}^{\mathbb{R}}(\mathbb{R})$, the inverse image $L^{-1}(B)$ is bounded in
$W_{p}$.

Concerning statement (a), the boundedness on ${\rm WPC}_{p}\cong
T_{p}(\mathbb{U})\times T_{p}(\mathbb{L})$ is defined by the product of the
$p$-Weil–Petersson metrics. In order to prove (a), it suffices to show the
boundedness of the image of a bounded set under the embedding
$T_{p}\to\mathcal{B}_{p}$ into the pre-Schwarzian derivative model. This is
seen by the representation of $L$ in the proof of Theorem 3.6. For the Bers
embedding $T_{p}\to A^{p}$ (see Appendix), this follows from [17, Proposition
8.4], but the embedding into $\mathcal{B}_{p}$ seems more involved.

Concerning statement (b), if we could prove that $\|\mu_{u}\|_{p}$ and
$\|\mu_{u}\|_{\infty}<1$ are dominated by $\|u\|_{B_{p}}$ for the complex
dilatation $\mu_{u}$ of a certain quasiconformal extension
$F_{u}:\mathbb{U}\to\mathbb{U}$ of $\gamma_{u}:\mathbb{R}\to\mathbb{R}$, we
would obtain the result from [17, Theorem 5.4]. However, we only have the
estimates of $\|\mu_{u}\|_{p}$ and $\|\mu_{u}\|_{\infty}$ in terms of
$\|u\|_{B_{p}}$ in the case where $F_{u}$ is the variant of the
Beurling–Ahlfors extension by the heat kernel and $\|u\|_{B_{p}}$ is
sufficiently small (see [40, Proposition 3.5, Lemma 4.2]).

##  7\. Appendix: The pre-Schwarzian derivative model on the upper half-plane

In this appendix, we provide a complex Banach manifold structure for the
$p$-Weil–Petersson Teichmüller space $T_{p}=T_{p}(\mathbb{U})$ by using pre-
Schwarzian derivatives on $\mathbb{U}$. This is well-known in the case of the
unit disk $\mathbb{D}$ for $p=2$. However, since the pre-Schwarzian derivative
is not Möbius invariant, we carefully treat the case of $\mathbb{U}$. We will
see below that there is a certain advantage of considering the pre-Schwarzian
derivative model on $\mathbb{U}$ compared with $\mathbb{D}$. The
generalization to any $p>1$ is also mentioned. We note that if we use the
Schwarzian derivative model, there is no difference between $\mathbb{U}$ and
$\mathbb{D}$ due to its Möbius invariance, and $T_{p}$ is equipped with the
complex Banach manifold structure for $p\geq 1$.

Let $A^{\infty}(\mathbb{U})$ denote the Banach space of functions $\varphi$
holomorphic on $\mathbb{U}$ with norm

$\|\varphi\|_{A^{\infty}}=\sup_{z\in\mathbb{U}}|\varphi(z)|y^{2}.$

For $p\geq 1$, we also denote by $A^{p}(\mathbb{U})$ the Banach space of
holomorphic functions $\varphi$ on $\mathbb{U}$ with norm

$\|\varphi\|_{A^{p}}=\left(\frac{1}{\pi}\iint_{\mathbb{U}}|\varphi(z)|^{p}y^{2p-2}dxdy\right)^{\frac{1}{p}}.$

For any locally univalent function $g$, the derivative of the logarithm
$\mathcal{L}_{g}$ and the Schwarzian derivative $\mathcal{S}_{g}$ are defined
respectively by

$\mathcal{L}_{g}=\log
g^{\prime},\quad\mathcal{S}_{g}=\mathcal{L}_{g}^{\prime\prime}-\frac{1}{2}(\mathcal{L}_{g}^{\prime})^{2}.$

The derivative of $\mathcal{L}_{g}$ is called the pre-Schwarzian derivative of
$g$. We will show the following result on the upper half-plane $\mathbb{U}$.
In the case of the unit disk $\mathbb{D}$, the corresponding theorem for
$p\geq 2$ was proved in [13, Theorems 1, 2]. However, since $\mathcal{L}_{g}$
is not Möbius invariant, this is not straightforward from that on
$\mathbb{D}$. As mentioned below, the case of $p=2$ was proved in [32].

######  Theorem 7.1.

Let $p>1$. Let $g:\mathbb{U}\to\mathbb{C}$ be a conformal mapping on
$\mathbb{U}$ and can be extended to the whole plane $\mathbb{C}$
quasiconformally such that $\lim_{z\to\infty}g(z)=\infty$. Then, the following
statements are equivalent $\\!:$

  1. (a)

$g$ extends to a quasiconformal homeomorphism of $\mathbb{C}$ whose complex
dilatation $\mu$ belongs to $\mathcal{M}_{p}(\mathbb{L});$

  2. (b)

$\mathcal{L}_{g}\in\mathcal{B}_{p}(\mathbb{U});$

  3. (c)

$\mathcal{S}_{g}\in A^{p}(\mathbb{U})$.

###### Proof of Theorem 7.1.

The equivalence of (b) and (c) for $p=2$ was investigated in [32, Theorem
4.4]. Essentially the same argument is valid for general $p>1$. The
implication (a) $\Rightarrow$ (c) for $p\geq 1$ is asserted in [43, Lemma
4.1], and (c) $\Rightarrow$ (a) for $p\geq 1$ is contained in [43, Theorem
7.1]. The equivalence of (a) and (c) was proved formerly by [7, Theorem 2] for
$p=2$, and by [13, Theorems 2] and [31, Theorem 2.1] for $p\geq 2$. ∎

Under these preparations, we introduce the pre-Schwarzian derivative model of
Teichmüller spaces on $\mathbb{U}$. Let $g:\mathbb{U}\to\mathbb{C}$ be a
conformal mapping on $\mathbb{U}$ satisfying the condition $g(\infty)=\infty$
(i.e. $\lim_{z\to\infty}g(z)=\infty$) that extends to a quasiconformal
homeomorphism of the whole plane $\mathbb{C}$. Then, the set $\mathscr{T}$ of
$\mathcal{S}_{g}\in A^{\infty}(\mathbb{U})$ for all such $g$ is the Schwarzian
derivative model of the universal Teichmüller space $T$, and the set
$\mathcal{T}$ of $\mathcal{L}_{g}\in{\mathcal{B}}(\mathbb{U})$ for all such
$g$ is the pre-Schwarzian derivative model of $T$. It is known that
$\mathscr{T}$ is a bounded domain in $A^{\infty}(\mathbb{U})$ identified with
$T$ (the Bers embedding), which defines a complex Banach manifold structure
for $T$ (see [14, Section III.4]).

However, if we do not impose the condition $g(\infty)=\infty$ on $g$ and
consider $\mathcal{L}_{g}$ for all those $g$, then they are classified into
uncountably many components and $\mathcal{T}$ is the one containing $0$. To
see this, we consider the conformal mapping $\tilde{g}=g\circ T^{-1}$ of
$\mathbb{D}$ pushed forward by the Cayley transformation
$T:\mathbb{U}\to\mathbb{D}$. Since

$\log\tilde{g}^{\prime}=T_{*}\mathcal{L}_{g}+\log(T^{-1})^{\prime}$

and $\log(T^{-1})^{\prime}\in\mathcal{B}(\mathbb{D})$, this defines an affine
isometry $\mathcal{B}(\mathbb{U})\to\mathcal{B}(\mathbb{D})$. Under this
isometric isomorphism, the components in ${\mathcal{B}}(\mathbb{U})$
correspond to those in ${\mathcal{B}}(\mathbb{D})$ bijectively, which are
${\mathcal{T}}_{\omega}(\mathbb{D})$ ($\omega\in\mathbb{S}$) and
${\mathcal{T}}_{\rm bdd}(\mathbb{D})$ characterized by the property that a
point $\omega\in\mathbb{S}$ or no point of $\mathbb{S}$ is mapped to $\infty$
by the conformal mapping $\tilde{g}:\mathbb{D}\to\mathbb{C}$. See [47] and
[35, Section 4]. Then, the component
$\mathcal{T}\subset\mathcal{B}(\mathbb{U})$ containing $0$ corresponds to
${\mathcal{T}}_{T(\infty)}(\mathbb{D})={\mathcal{T}}_{1}(\mathbb{D})$, which
is biholomorphically equivalent to the universal Teichmüller space
$T\cong\mathscr{T}$.

Theorem 7.1 guarantees that we can also consider the pre-Schwarzian derivative
model of the $p$-Weil–Petersson Teichmüller space $T_{p}$ for $p>1$ in the
same manner. Concerning the correspondence between the spaces on $\mathbb{U}$
and $\mathbb{D}$, there is a difference from the case of $T$, which is a fact
that $\log(T^{-1})^{\prime}$ does not belong to $\mathcal{B}_{p}(\mathbb{D})$.
Nevertheless, Theorem 7.1 asserts that there is a bijective correspondence
between $\mathcal{T}_{p}=\mathcal{T}\cap\mathcal{B}_{p}(\mathbb{U})$ and
$\mathscr{T}_{p}=\mathscr{T}\cap A^{p}(\mathbb{U})$ under the map
$\alpha:\mathcal{B}_{p}(\mathbb{U})\to A^{p}(\mathbb{U})$ given by
$\alpha(\varphi)=\varphi^{\prime\prime}-(\varphi^{\prime})^{2}/2$ from
$\mathcal{S}_{g}=\mathcal{L}_{g}^{\prime\prime}-\frac{1}{2}(\mathcal{L}_{g}^{\prime})^{2}$.
It was proved in [36, Lemma 2.3] that $\alpha$ is holomorphic in the case of
$\mathbb{D}$ for $p\geq 2$. This is also true in the case of $\mathbb{U}$ for
$p>1$. Moreover, $\mathscr{T}_{p}$ is a contractible domain in
$A^{p}(\mathbb{U})$ identified with the $p$-Weil–Petersson Teichmüller space
$T_{p}$ for $p\geq 1$, which provides the complex Banach manifold structure
for $T_{p}$ (see [7, 13, 45, 43]).

We finish our discussion by stating the following theorem.

######  Theorem 7.2.

The holomorphic map $\alpha$ restricted to $\mathcal{T}_{p}$ is a
biholomorphic homeomorphism onto $\mathscr{T}_{p}$ for $p>1$. Hence, the
complex Banach manifold structures on these two models of the
$p$-Weil–Petersson Teichmüller space $T_{p}$ are biholomorphically equivalent.

###### Proof.

We have seen that $\alpha:\mathcal{T}_{p}\to\mathscr{T}_{p}$ is a holomorphic
bijection. For a Beltrami coefficient $\mu$ in $\mathcal{M}_{p}(\mathbb{L})$,
we take a conformal map $f_{\mu}$ on $\mathbb{U}$ with
$f_{\mu}(\infty)=\infty$ that is quasiconformally extendable to $\mathbb{L}$
having the complex dilatation $\mu$. Then, the map
$\ell:\mathcal{M}_{p}(\mathbb{L})\to\mathcal{T}_{p}$ defined by
$\mu\mapsto\mathcal{L}_{f_{\mu}}$ is holomorphic. This is shown in [36,
Theorem 2.4] in the case of $\mathbb{D}^{*}$ for $p\geq 2$. In the case of
$\mathbb{L}$ for $p>1$, the same proof can be applied once we fill the step of
showing that the Bers projection
$\sigma:\mathcal{M}_{p}(\mathbb{L})\to\mathscr{T}_{p}$ given by
$\mu\mapsto\mathcal{S}_{f_{\mu}}$ is continuous. This is given in [43, Lemma
4.1].

Moreover, as the composition $\alpha\circ\ell$ coincides with $\sigma$, there
is a local holomorphic right inverse of $\alpha\circ\ell$ at any point of
$\mathscr{T}_{p}$ (see [36, Theorem 2.1], [45, Proposition 4.3], and [43,
Theorem 7.1]). It follows from this property that $\alpha$ also has a local
holomorphic right inverse at any point of $\mathscr{T}_{p}$, which shows that
$\alpha^{-1}$ is holomorphic. ∎

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