Integrable Teichm\"uller spaces for analysis on Weil-Petersson curves

TL;DR

This paper characterizes integrable Teichmüller spaces using Besov spaces, establishing a biholomorphic correspondence and analyzing Cauchy transforms on Weil-Petersson curves, advancing the understanding of complex structures in Teichmüller theory.

Contribution

It introduces a real-analytic equivalence between $T_p$ and the $p$-Besov space, and links Cauchy transforms to holomorphic maps in the context of Weil-Petersson curves.

Findings

Characterization of $T_p$ via $p$-Besov space membership.

Biholomorphic correspondence between $T_p$ and $p$-Besov space.

Holomorphic dependence of Cauchy transforms on embeddings.

Abstract

The integrableTeichm\"uller space $T_p$ for $p \geq 1$ is defined by the $p$-integrability of Beltrami coefficients. We characterize a quasisymmetric homeomorphism $h$ in $T_p$ by the condition that $\log h'$ belongs to the real $p$-Besov space, with a certain modification applied in the case $p=1$. This is done as part of the arguments for establishing a biholomorphic correspondence $\Lambda$ from the product of $T_p$ for simultaneous uniformization of $p$-Weil-Petersson curves into the $p$-Besov space. In particular, this proves the real-analytic equivalence between $T_p$ and the real $p$-Besov space. Moreover, the Cauchy transform of Besov functions on Weil-Petersson curves can be expressed by the derivative of this holomorphic map $\Lambda$, and from this, the Calder\'on theorem in this setting is straightforward. It also follows that the Cauchy transforms on $p$-Weil-Petersson curves holomorphically depend on their embeddings as they vary in the Bers coordinates.

Equations

$$\Lambda :T_p^{+}\times T_p^{-}\to {\hat{B}}_p(\mathbb{R}),\gamma ([{\mu }^{+}],[{\mu }^{-}])\mapsto \log {\gamma }^{\prime }$$

$$(P^{\pm }\varphi )(z)=\frac{1}{\pi }{\int }_{\mathbb{R}}\frac{\varphi (t)}{z-t}dt(z\in {\mathbb{H}}^{\pm })$$

$${\hat{B}}_p(\mathbb{R})=C_{h^{-}}({\hat{\mathcal{B}}}_p({\mathbb{H}}^{-}))\oplus C_{h^{+}}({\hat{\mathcal{B}}}_p({\mathbb{H}}^{+})).$$

$$(P_{\Gamma }^{\pm }\psi )(\zeta )=\frac{1}{\pi }{\int }_{\Gamma }\frac{\psi (z)}{\zeta -z}dz(\zeta \in {\Omega }^{\pm })$$

$$P_{([{\mu }^{+}],[{\mu }^{-}])}^{\pm }:{\hat{B}}_p(\mathbb{R})\to C_{h^{\pm }}({\hat{\mathcal{B}}}_p({\mathbb{H}}^{\pm })).$$

$${\hat{B}}_p(\gamma (\mathbb{R}))={\hat{\mathcal{B}}}_p({\Omega }^{+})\oplus {\hat{\mathcal{B}}}_p({\Omega }^{-}).$$

$$\Vert \mu {\Vert }_p={\left({\int }_{\mathbb{H}}|\mu (z){|}^p\frac{dxdy}{|\mathrm{Im}z{|}^2}\right)}^{1/p}<\infty .$$

$${\mathcal{A}}_p(\mathbb{H})=\{\Psi \in {\mathcal{A}}_{\infty }(\mathbb{H})\mid \Vert \Psi {\Vert }_{{\mathcal{A}}_p}={\left({\int }_{\mathbb{H}}|(\mathrm{Im}z{)}^2\Psi (z){|}^p\frac{dxdy}{|\mathrm{Im}z{|}^2}\right)}^{1/p}<\infty \}.$$

$$\Vert \mu {\Vert }_{\ast }^2=\underset{I\subset \mathbb{R}}{\sup }\frac{1}{|I|}{\int }_{\hat{I}}|\mu (z){|}^2\frac{dxdy}{|\mathrm{Im}z|}<\infty ,$$

$${\Delta }_t^1\varphi (x)=\varphi (x+t)-\varphi (x);{\Delta }_t^{m+1}\varphi (x)={\Delta }_t^m\varphi (x+t)-{\Delta }_t^m\varphi (x).$$

$$\Vert \varphi {\Vert }_{{\dot{B}}_{p,q}^s}={\left({\int }_{-\infty }^{\infty }|t{|}^{-sq}\Vert {\Delta }_t^{\lfloor s\rfloor +1}\varphi {\Vert }_{L_p}^q\frac{dt}{|t|}\right)}^{1/q}.$$

$$\Vert \varphi {\Vert }_{B_p}={\left({\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{|\varphi (x+t)-\varphi (x){|}^p}{t^2}dxdt\right)}^{1/p}$$

$$\Vert \varphi {\Vert }_{B_p^{\#}}={\left({\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{|\varphi (x+2t)-2\varphi (x+t)+\varphi (x){|}^p}{t^2}dxdt\right)}^{1/p}$$

$$\Vert \varphi {\Vert }_{\mathrm{BMO}}=\underset{I\subset \mathbb{R}}{\sup }\frac{1}{|I|}{\int }_I|\varphi (x)-{\varphi }_I|dx<\infty ,$$

$${\hat{B}}_p(\mathbb{R})=B_p^{\#}(\mathbb{R})\cap \mathrm{BMO}(\mathbb{R}),$$

$$\Vert \varphi {\Vert }_{{\hat{B}}_p}\asymp \Vert \varphi {\Vert }_{B_p}=\Vert K_{\ast }(\varphi ){\Vert }_{B_p}\asymp \Vert K_{\ast }(\varphi ){\Vert }_{{\hat{B}}_p}.$$

$$\Vert {\varphi }_{\ast }{\Vert }_{{\dot{W}}_1^1}\lesssim \Vert {\varphi }_{\ast }{\Vert }_{B_1^{\#}}+\underset{\alpha \in \mathbb{C}}{\inf }\Vert {\varphi }_{\ast }-\alpha {\Vert }_{L_1}<\infty .$$

$$\underset{\alpha \in \mathbb{C}}{\inf }\Vert {\varphi }_{\ast }-\alpha {\Vert }_{L_{\infty }}\le \Vert {\varphi }_{\ast }{\Vert }_{{\dot{W}}_1^1}$$

$$\Vert \varphi {\Vert }_{{\dot{W}}_1^1}=\Vert {\varphi }_{\ast }{\Vert }_{{\dot{W}}_1^1}\lesssim \Vert {\varphi }_{\ast }{\Vert }_{{\hat{B}}_1}\asymp \Vert \varphi {\Vert }_{{\hat{B}}_1}.$$

$$\Vert \Phi {\Vert }_{{\mathcal{B}}_p}={\left({\int }_{\mathbb{H}}|(\mathrm{Im}z){\Phi }^{\prime }(z){|}^p\frac{dxdy}{|\mathrm{Im}z{|}^2}\right)}^{1/p}$$

$$\Vert \Phi {\Vert }_{{\mathcal{B}}_p^{\#}}={\left({\int }_{\mathbb{H}}|(\mathrm{Im}z{)}^2{\Phi }^{\prime \prime }(z){|}^p\frac{dxdy}{|\mathrm{Im}z{|}^2}\right)}^{1/p}$$

$$\Vert \Phi {\Vert }_{\mathrm{BMOA}}^2=\underset{I\subset \mathbb{R}}{\sup }\frac{1}{|I|}{\int }_{\hat{I}\subset \mathbb{H}}|(\mathrm{Im}z){\Phi }^{\prime }(z){|}^2\frac{dxdy}{|\mathrm{Im}z|}<\infty ,$$

$${\hat{\mathcal{B}}}_p(\mathbb{H})={\mathcal{B}}_p^{\#}(\mathbb{H})\cap \mathrm{BMOA}(\mathbb{H}),$$

$$\Vert \Phi {\Vert }_{{\hat{\mathcal{B}}}_p}=\Vert \Phi {\Vert }_{{\mathcal{B}}_p^{\#}}+\Vert \Phi {\Vert }_{\mathrm{BMOA}}.$$

$$\mathcal{H}(\varphi )(x)=\text{p.v.}\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{\varphi (t)}{x-t}dt=\underset{\epsilon \to 0}{\lim }\frac{1}{\pi }{\int }_{|t|\ge \epsilon }\frac{\varphi (t)}{x-t}dt(x\in \mathbb{R}),$$

$$\frac{1}{x-t}+\frac{1_{\{|t|\ge 1\}}}{t}\mathrm{or}\frac{1}{x-t}+\frac{t}{1+t^2}$$

$${\hat{B}}_p(\mathbb{R})=E^{+}({\hat{\mathcal{B}}}_p({\mathbb{H}}^{+}))\oplus E^{-}({\hat{\mathcal{B}}}_p({\mathbb{H}}^{-})).$$

$$\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{\varphi (t)}{z-t}dt(z\in {\mathbb{H}}^{\pm }),$$

$${\hat{B}}_p(\mathbb{R})={\hat{\mathcal{B}}}_p({\mathbb{H}}^{+})\oplus {\hat{\mathcal{B}}}_p({\mathbb{H}}^{-}).$$

$$K(x,t)=\inf \{\Vert x_0{\Vert }_{X_0}+t\Vert x_1{\Vert }_{X_1}\mid x=x_0+x_1,x_0\in X_0,x_1\in X_1\}$$

Full text

Research supported by supported by Japan Society for the Promotion of Science (KAKENHI 23K25775 and 23K17656).

1. Introduction

The integrable Teichmüller space has recently garnered interest as a parameter space for Weil–Petersson curves (see [4]). We have made attempts to extend the theory, including generalizing the exponent of integrability and incorporating discussions from harmonic analysis into the complex-analytic Teichmüller space theory. In this paper, we explore a direction of research that more clearly organizes the existing arguments through the method of simultaneous uniformization of curves from the perspective of quasiconformal Teichmüller theory.

Integrable Teichmüller spaces $T_{p}$ have been formulated as subspaces of the universal Teichmüller space $T$, extending the definition of the Weil–Petersson metric on the Teichmüller space of compact Riemann surfaces. This forms the basis for the complex analytic theory of Weil–Petersson curves. Notable studies include Cui’s research on the complex structure of the integrable Teichmüller space [12], Takhtajan and Teo’s study of the curvature of the Weil–Petersson metric [36], and Shen’s research on quasiconformal Teichmüller space theory [30].

The universal Teichmüller space $T$ can be defined based on the complex dilatation $\mu$ of quasiconformal homeomorphisms, while the integrable Teichmüller space $T_{p}$ is defined by imposing $L_{p}$-integrability on $\mu$ with respect to the hyperbolic metric. Originally, the theory was for the case $p=2$, but research on generalizing the exponent is also progressing. Extensions to $p\geq 2$ were made by Tang and Shen [35], and to $p>1$ by Wei and Matsuzaki [38]. Furthermore, in [40, 41], the complex structure through the Schwarzian derivative has been given to $T_{p}$ for all $p\geq 1$. This paper also focuses on the methods for characterizing quasisymmetric mappings of $T_{1}$ through pre-Schwarzian derivatives.

The universal Teichmüller space $T$ is identified with the set of all quasisymmetric homeomorphisms $h:\mathbb{R}\to\mathbb{R}$ under appropriate normalization. However, they are not necessarily absolutely continuous. To conduct a finer analysis, better regularity is required. Teichmüller spaces consisting of locally absolutely continuous quasisymmetric homeomorphisms are collectively called absolutely continuous Teichmüller spaces. The BMO Teichmüller space, introduced by Astala and Zinsmeister [3], is a typical example that includes various absolutely continuous Teichmüller spaces such as $T_{p}$, and has been well-organized through research by Shen and Wei [32]. Current research trends include investigating problems in real-analysis within the framework of Teichmüller space theory and vice versa, which can also be applied to the study of integrable Teichmüller spaces $T_{p}$.

In the theory of absolutely continuous Teichmüller spaces, we analyze the function spaces to which $\log h^{\prime}$ for a quasisymmetric homeomorphism $h$ belong. In this paper, we first consider this problem for $T_{p}$. A special type of the Besov space $\widehat{B}_{p}(\mathbb{R})$, which is a kind of fractional dimensional Sobolev space on $\mathbb{R}$, serves as this target space. By generalizing the previous results for $p>1$, we establish the following characterization of $T_{p}$ for all $p\geq 1$. Here, ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ stands for the real subspace of $\widehat{B}_{p}(\mathbb{R})$ consisting of all real-valued functions.

Theorem 1.1**.**

For $p\geq 1$, a quasisymmetric homeomorphism $h:\mathbb{R}\to\mathbb{R}$ is an element of $T_{p}$ if and only if $h$ is locally absolutely continuous and $\log h^{\prime}$ belongs to ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$.

There are several difficulties in proving this result in the case $p=1$ compared with the case $p>1$: (i) The Besov space $\widehat{B}_{p}(\mathbb{R})$ for $p\geq 1$ should be more involved than $B_{p}(\mathbb{R})$ for $p>1$; (ii) The boundedness of the composition operator $C_{h}$ on $\widehat{B}_{p}(\mathbb{R})$, defined by $C_{h}(\phi)=\phi\circ h$, is newly presented in the case $p=1$ by using a different interpolation of Banach spaces from the case $p>1$; (iii) To prove the ‘if’ part of the statement for $p=1$, the canonical method of quasiconformal extension of $h$ does not work because of the complicated definition of $\widehat{B}_{p}(\mathbb{R})$.

Section 3 addresses (i) by defining $\widehat{B}_{p}(\mathbb{R})$, and Sections 5 and 6 address (ii) by showing the boundedness of $C_{h}$. The ‘only-if’ part of Theorem 1.1 is proved in Theorem 6.2. Due to difficulty (iii), the ‘if’ part is proved implicitly by showing that the correspondence $h\mapsto\log h^{\prime}$ is surjective from $T_{p}$ onto ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$. The proof is completed in Section 8.

A basic method underlying the entire argument for Theorem 1.1 is conformal welding together with pre-Bers embedding and simultaneous uniformization, which are cross-sections of the same nature of Teichmüller spaces. By the conformal welding, any quasisymmetric homeomorphism $h$ of $\mathbb{R}$ can be represented as the difference of the boundary extensions of conformal homeomorphisms of the upper half-plane $\mathbb{H}^{+}$ and the lower half-plane $\mathbb{H}^{-}$. Moreover, if $h\in T_{p}$, then such conformal homeomorphisms extend quasiconformally to $\mathbb{C}$ with their complex dilatations in $M_{p}(\mathbb{H}^{\pm})$, the spaces of $p$-integrable Beltrami coefficients.

Conversely, the conformal homeomorphisms $F^{\mu}$ of $\mathbb{H}^{-}$ for $\mu\in M_{p}(\mathbb{H}^{+})$ represent the elements of $T_{p}^{+}$, the Teichmüller space defined by $M_{p}(\mathbb{H}^{+})$. Here, $\log(F^{\mu})^{\prime}$ belongs to the analytic Besov space $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$, the boundary extension of whose elements are in $\widehat{B}_{p}(\mathbb{R})$. Then, we obtain the pre-Bers embedding $\beta:T_{p}^{+}\to\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$. We note that for the Bers embedding given by the Schwarzian derivative of $F^{\mu}$, there is no difficulty in setting the appropriate complex Banach space of holomorphic functions in which $T_{p}$ is embedded, but for the pre-Bers embedding, the definition of $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$ for $p=1$ is more involved like $\widehat{B}_{p}(\mathbb{R})$. This is conducted in Section 4.

The restriction $\gamma=G|_{\mathbb{R}}$ of a quasiconformal self-homeomorphism $G$ to $\mathbb{R}$ is called a quasisymmetric embedding. As a special quasisymmetric embedding, if the complex dilatations of $G$ on $\mathbb{H}^{\pm}$ are in $M_{p}(\mathbb{H}^{\pm})$, we call $\gamma:\mathbb{R}\to\mathbb{C}$ a $p$-Weil–Petersson embedding. A Weil–Petersson curve is its image. The idea of the simultaneous uniformization of quasi-Fuchsian groups also works for quasicircles (images of quasisymmetric embeddings), and thus in the same way the space of all $p$-Weil–Petersson embeddings is parametrized by the product of the Teichmüller spaces $T_{p}^{+}\times T_{p}^{-}$, which we call the Bers coordinates. Then, by generalizing the ‘if’ part of Theorem 1.1, we see that $\log\gamma^{\prime}$ belongs to $\widehat{B}_{p}(\mathbb{R})$ for a $p$-Weil–Petersson embedding $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ determined by $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$. In Section 7, we prepare these arguments for the next two sections. Naturally, this correspondence

[TABLE]

is a holomorphic injection.

By the pre-Bers embedding $\beta^{\pm}:T_{p}^{\pm}\to\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\mp})$, the tangent space of the Bers coordinates $T_{p}^{+}\times T_{p}^{-}$ may be regarded as $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})\oplus\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$. On the other hand, the Szegö projections

[TABLE]

in $\widehat{B}_{p}(\mathbb{R})$ are surjective onto $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})$, and under the identification of $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})$ with the closed subspaces of $\widehat{B}_{p}(\mathbb{R})$ by taking the boundary extension to $\mathbb{R}$, they yield the associated direct sum decomposition $\widehat{B}_{p}(\mathbb{R})=\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})\oplus\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$.

The derivative $d_{([0],[0])}\Lambda$ at the origin of $T_{p}^{+}\times T_{p}^{-}$ is the identity map $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})\oplus\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})\to\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})\oplus\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$. From this, we see that the image of $\Lambda$ contains ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$, which proves the ‘if’ part of Theorem 1.1. The derivative $d_{([\mu^{+}],[\mu^{-}])}\Lambda$ at any point of $T_{p}^{+}\times T_{p}^{-}$ maps $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$ to $C_{h^{-}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-}))$ and $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$ to $C_{h^{+}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))$, where $C_{h^{\pm}}$ are the composition operators induced by the quasisymmetric homeomorphisms $h^{\pm}=[\mu^{\pm}]$ in $T_{p}^{\pm}$. Then, we have the surjectivity of the derivative of $\Lambda$ by showing that

[TABLE]

As a consequence, we have the following crucial property of $\Lambda$. This is demonstrated in Theorem 9.1 and Corollary 9.5 of Section 9.

Theorem 1.2**.**

The map $\Lambda:T_{p}^{+}\times T_{p}^{-}\to\widehat{B}_{p}(\mathbb{R})$ is a biholomorphic homeomorphism onto its image for $p\geq 1$. Moreover, the composition of $\Lambda$ with the canonical symmetric embedding $T_{p}\to T_{p}^{+}\times T_{p}^{-}$ yields a real-analytic diffeomorphism of $T_{p}$ onto ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$.

Therefore, the integrable Teichmüller space $T_{p}$ for every $p\geq 1$ can be identified with the real Banach space ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ with respect to the real-analytic structure.

This biholomorphic homeomorphism $\Lambda$ also works effectively for the investigation of certain problems in real analysis concerning Weil–Petersson curves. For an overview of real analytic studies of bi-Lipschitz embeddings of $\mathbb{R}$ onto chord-arc curves, Coifman and Meyer [10] and Semmes [28] provide comprehensive explanations, which motivate our work. The former focuses on such embeddings using harmonic analytic methods, while the latter utilizes theories of quasiconformal mappings. We extend these studies to the context of Weil–Petersson embeddings.

Our strategy is that by the biholomorphic homeomorphism $\Lambda$ from the Bers coordinates $T_{p}^{+}\times T_{p}^{-}$ of the $p$-Weil–Petersson embeddings to the Besov space $\widehat{B}_{p}(\mathbb{R})$, we relate the derivative of $\Lambda$ at $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$ to the Cauchy projection with respect to the Weil–Petersson curve $\Gamma=\gamma(\mathbb{R})$ for $\gamma=\gamma([\mu^{+}],[\mu^{-}])$. Let $\Omega^{\pm}$ be the domains in $\mathbb{C}$ separated by $\Gamma$. The Cauchy integrals

[TABLE]

of an integrable function $\psi$ on $\Gamma$ yield holomorphic functions on $\Omega^{\pm}$. We consider the Besov space $\widehat{B}_{p}(\gamma(\mathbb{R}))$ on $\Gamma$ as the push-forward of $\widehat{B}_{p}(\mathbb{R})$ by $\gamma$ and the analytic Besov spaces $\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$ on $\Omega^{\pm}$ as the push-forward of $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})$ by the normalized Riemann mappings $\mathbb{H}^{\pm}\to\Omega^{\pm}$. Then, the Cauchy integrals induce the projections $P_{\Gamma}^{\pm}:\widehat{B}_{p}(\gamma(\mathbb{R}))\to\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$. By taking the boundary extension, we assume that $\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$ are closed subspaces of $\widehat{B}_{p}(\gamma(\mathbb{R}))$.

On the other hand, the isomorphic derivative $d_{([\mu^{+}],[\mu^{-}])}\Lambda$ of the biholomorphic homeomorphism $\Lambda$ conjugates the projections of the tangent space of $T_{p}^{+}\times T_{p}^{-}$ at $([\mu^{+}],[\mu^{-}])$ onto both factors with the projections regarding the direct sum decomposition $\widehat{B}_{p}(\mathbb{R})=C_{h^{-}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-}))\oplus C_{h^{+}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))$, which are

[TABLE]

Then, a novel aspect of our argument lies in the assertion that the Cauchy projections ${P}^{\pm}_{\Gamma}$ are the conjugates of $P_{([\mu^{+}],[\mu^{-}])}^{\pm}$ by the push-forward $\gamma_{*}$. By this relationship, we can reduce the properties of the Cauchy projections ${P}^{\pm}_{\Gamma}$ for the embedding $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ to the derivative $d_{([\mu^{+}],[\mu^{-}])}\Lambda$ of the biholomorphic homeomorphism.

One of the consequences from this argument is as follows, which is given in Theorem 10.3 and Corollary 10.4 of Section 10. This problem originates in Cardelón’s work.

Theorem 1.3**.**

In the Besov space $\widehat{B}_{p}(\gamma(\mathbb{R}))$ on the $p$-Weil–Petersson curve $\Gamma=\gamma(\mathbb{R})$ of $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ for $p\geq 1$, the Cauchy projections $P^{\pm}_{\Gamma}$ coincide with $\gamma_{*}\circ P^{\pm}_{([\mu^{+}],[\mu^{-}])}\circ\gamma_{*}^{-1}$, which are bounded linear projections onto $\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$ associated with the topological direct sum decomposition

[TABLE]

A significant property of the Cauchy projections in the analysis on the curves is, however, its analytic dependence and the estimate of the operator norm as the curves vary (see [10, 28]). In our method, it is clear that $\|P^{\pm}_{\Gamma}\|$ can be estimated in terms of $\|C_{h^{\pm}}\|$ for the composition operators $C_{h^{\pm}}$. In addition, since $P^{\pm}_{\Gamma}$ is related to the derivative of the biholomorphic homeomorphism $\Lambda$, its holomorphic dependence is also clear.

Theorem 1.4**.**

For $p\geq 1$, the conjugates of the Cauchy projections ${P^{\pm}}_{([\mu^{+}],[\mu^{-}])}$ acting on $\widehat{B}_{p}(\mathbb{R})$ depend holomorphically on $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$ with respect to the operator norm topology.

An equivalent statement for the conjugate of the Cauchy transform ${\mathcal{H}}_{\Gamma}=-i(P_{\Gamma}^{+}-P_{\Gamma}^{-})$ is given in Theorem 11.1 of Section 11. We also exhibit an important application of this result after Coifman and Meyer [11].

2. Integrable Teichmüller spaces

Within the universal Teichmüller space, integrable Teichmüller spaces are defined by the integrability of Beltrami coefficients. One of the fundamental problems in integrable Teichmüller spaces is characterizing these spaces through quasisymmetric homeomorphisms, which are the boundary mappings of quasiconformal homeomorphisms. In this section, to introduce this problem, we lay the foundations of integrable Teichmüller spaces.

An orientation-preserving homeomorphism $h:\mathbb{R}\to\mathbb{R}$ is called a quasisymmetric homeomorphism if it satisfies the following doubling condition: for any bounded interval $I\subset\mathbb{R}$, there exists a constant $M>1$ (called a doubling constant) such that $|h(I)|\leq M|h(\frac{1}{2}I)|$, where $|\cdot|$ denotes the Lebesgue measure on $\mathbb{R}$ and $\frac{1}{2}I$ represents the interval with the same center as $I$ but half the length. An orientation-preserving homeomorphism $H$ defined on a planar region is called a quasiconformal homeomorphism if $H$ has locally integrable partial derivatives $H_{z},H_{\bar{z}}$ in the distribution sense, and the complex dilatation $\mu(z)=H_{\bar{z}}/H_{z}$ satisfies $\|\mu\|_{\infty}<1$.

Any quasiconformal homeomorphism $H:\mathbb{H}\to\mathbb{H}$ ($\mathbb{H}$ is either the upper half-plane $\mathbb{H}^{+}$ or the lower half-plane $\mathbb{H}^{-}$) can be continuously extended to a quasisymmetric homeomorphism $h:\mathbb{R}\to\mathbb{R}$. Conversely, any quasisymmetric homeomorphism $h$ is a continuous extension of such a quasiconformal homeomorphism $H$.

A complex-valued measurable function $\mu$ on $\mathbb{H}$ that satisfies $\|\mu\|_{\infty}<1$ is called a Beltrami coefficient. The space of all Beltrami coefficients on $\mathbb{H}$ is denoted by $M(\mathbb{H})$. For a Beltrami coefficient $\mu\in M(\mathbb{H})$, there uniquely exists a quasiconformal self-homeomorphism $H(\mu)$ of $\mathbb{H}$ with the complex dilatation $\mu$ that satisfies the normalization condition (which means fixing $0,1,\infty$ in all cases also for other mappings appearing in this paper) by the measurable Riemann mapping theorem. Therefore, the entirety of such quasiconformal homeomorphisms $H(\mu)$ can be identified with $M(\mathbb{H})$.

The set of all quasisymmetric homeomorphisms $h:\mathbb{R}\to\mathbb{R}$ that satisfy the normalization conditions is called the universal Teichmüller space, which is denoted by $T$. The correspondence through boundary extension from quasiconformal homeomorphisms $H(\mu)$ of $\mathbb{H}$ to quasisymmetric homeomorphisms $h(\mu)$ of $\mathbb{R}$ defines the Teichmüller projection $\pi:M(\mathbb{H})\to T$. The equivalence class $[\mu]$ of Beltrami coefficients $\mu$ relative to $\pi$ is called a Teichmüller equivalence class.

The topology of $T$ is induced from $M(\mathbb{H})$. More precisely, it is the quotient topology determined by $\pi$. In another way, it can be given as the underlying topology of the Teichmüller distance defined by the $L_{\infty}$-norm on $M(\mathbb{H})$. It is well-known that they are equivalent.

The complex structure and the metric structure of $T$ are induced through the embedding of $T$ into a certain complex Banach space. For $\mu\in M(\mathbb{H}^{+})$, let $F^{\mu}$ be a quasiconformal self-homeomorphism of $\mathbb{C}$ satisfying the normalization condition, having the complex dilatation $\mu$ in the upper half-plane $\mathbb{H}^{+}$, and being conformal in the lower half-plane $\mathbb{H}^{-}$. For Beltrami coefficients $\mu$ and $\nu$, we see that $\pi(\mu)=\pi(\nu)$ if and only if $F^{\mu}|_{\mathbb{H}^{-}}=F^{\nu}|_{\mathbb{H}^{-}}$. By taking the Schwarzian derivative $S_{F}=(\log F^{\prime})^{\prime\prime}-\frac{1}{2}((\log F^{\prime})^{\prime})^{2}$ of $F=F^{\mu}|_{\mathbb{H}^{-}}$, $T$ can be topologically embedded into the complex Banach space of ${\mathcal{A}}_{\infty}(\mathbb{H})$ of holomorphic functions $\Psi$ on $\mathbb{H}=\mathbb{H}^{-}$ with the hyperbolic $L_{\infty}$-norm $\|\Psi\|_{\mathcal{A}_{\infty}}=\sup_{z\in\mathbb{H}}\,|({\rm Im}\,z)^{2}\Psi(z)|$. This mapping $\alpha:T\to{\mathcal{A}}_{\infty}(\mathbb{H})$ well defined by $[\mu]\mapsto S_{F}$ is called the Bers embedding. The image $\alpha(T)$ is a bounded domain in ${\mathcal{A}}_{\infty}(\mathbb{H})$, which is identified with $T$.

Moreover, $T$ has the structure of a group with the composition of normalized quasisymmetric homeomorphisms as its operation. For $h(\mu)=[\mu]$ and $h(\nu)=[\nu]$, the composition $h(\mu)\circ h(\nu)$ is denoted by $[\mu]\ast[\nu]$ and the inverse $h(\mu)^{-1}$ is denoted by $[\mu]^{-1}$. In fact, the composition of normalized quasiconformal self-homeomorphisms $H(\mu)$ and $H(\nu)$ of $\mathbb{H}$ gives the group structure $\mu\ast\nu$ on $M(\mathbb{H})$ and the Teichmüller projection $\pi$ is a homomorphism as $\pi(\mu\ast\nu)=[\mu]\ast[\nu]$. The right translation $r_{[\nu]}:T\to T$ for any $[\nu]\in T$ is defined as $[\mu]\mapsto[\mu]\ast[\nu]$, providing a biholomorphic automorphism of $T$. However, $T$ is not a topological group.

For literature on quasiconformal mappings and the universal Teichmüller space, useful references include [1, 19]. Proofs of the aforementioned facts can be found therein.

We introduce the integrable Teichmüller space $T_{p}$ in $T$.

Definition 2.1**.**

For $p\geq 1$, a Beltrami coefficient $\mu\in M(\mathbb{H})$ is $p$-integrable with respect to the hyperbolic metric if it satisfies

[TABLE]

The set of all such $p$-integrable Beltrami coefficients is denoted by $M_{p}(\mathbb{H})$. The $p$-integrable Teichmüller space (also referred to as Weil–Petersson Teichmüller space) is defined as $T_{p}=\pi(M_{p}(\mathbb{H}))$ for the Teichmüller projection $\pi$.

It follows immediately from this definition that $T_{p}\subset T_{q}$ if $p<q$.

Since the universal Teichmüller space $T$ is defined as the set of all normalized quasisymmetric homeomorphisms, the subset $T_{p}\subset T$ is also regarded as a certain subset of normalized quasisymmetric homeomorphisms. We consider this characterization later.

The topology of the integrable Teichmüller space $T_{p}$ is induced from $M_{p}(\mathbb{H})$ with norm $\|\mu\|_{p}+\|\mu\|_{\infty}$, which is the quotient topology by $\pi$. This can be also given as the underlying topology of the Weil–Petersson distance defined by the $L_{p}$-norm on $M_{p}(\mathbb{H})$, which are known to be equivalent (see [42, Proposition 5.2]).

The complex structure and the metric structure of $T_{p}$ for $p\geq 1$ are also defined through the Bers embedding $\alpha:T\to A_{\infty}(\mathbb{H})$. The image of $T_{p}$ under $\alpha$ is contained in the complex Banach space of holomorphic functions

[TABLE]

We note that the inclusion $\mathcal{A}_{p}(\mathbb{H})\hookrightarrow\mathcal{A}_{\infty}(\mathbb{H})$ is continuous, and $\mathcal{A}_{p}(\mathbb{H})\hookrightarrow\mathcal{A}_{q}(\mathbb{H})$ is continuous if $p<q$. Moreover, $\alpha$ gives a topological embedding of $T_{p}$ respecting these stronger topologies whose image is a connected open subset of $\mathcal{A}_{p}(\mathbb{H})$. In this way, the complex Banach structure is provided for $T_{p}$. These facts for all $p\geq 1$ are shown in [40, 42]. Concerning the Weil–Petersson metric defined through the Bers embedding, one can refer to [42, Section 5].

The BMO Teichmüller space $T_{B}$ mentioned in the introduction (see [3, 32]) is identified with the set of all normalized strongly quasisymmetric homeomorphisms. Here, an orientation-preserving homeomorphism $h:\mathbb{R}\to\mathbb{R}$ is called strongly quasisymmetric if there exist positive constants $C,\delta>0$ such that for any bounded interval $I\subset\mathbb{R}$ and any measurable subset $E\subset I$, it holds that $|h(E)|/|h(I)|\leq C(|E|/|I|)^{\delta}$. This condition is equivalent to saying that $h$ is locally absolutely continuous and $h^{\prime}$ is an $A_{\infty}$-weight (see [9]).

The BMO Teichmüller space $T_{B}$ is also represented by Beltrami coefficients. Namely, we define $\mu\in M(\mathbb{H})$ to be in a subset $M_{B}(\mathbb{H})$ if it satisfies the Carleson measure condition

[TABLE]

where the supremum is taken over all bounded intervals $I$ on $\mathbb{R}$ and $\widehat{I}$ is the square in $\mathbb{H}$ over $I$. Then, $T_{B}=\pi(M_{B}(\mathbb{H}))$. It is known that $T_{p}$ is contained in $T_{B}$ for every $p\geq 1$ (see [38, Proposition 2.2]). Hence, any quasisymmetric homeomorphism $h\in T_{p}$ is strongly quasisymmetric, and in particular, it is locally absolutely continuous. We consider the problem of determining the function space on $\mathbb{R}$ to which $\log h^{\prime}$ belongs.

3. Besov Spaces

We set up the function space on $\mathbb{R}$ to which $\log h^{\prime}$ belongs for a quasisymmetric homeomorphism $h\in T_{p}$. For $m\in\mathbb{N}$ and $t\in\mathbb{R}$, the $m$-th order difference of a function $\phi$ on $\mathbb{R}$ is defined as

[TABLE]

For $s\in\mathbb{R}$ and $0<p,\,q\leq\infty$, a semi-norm of a locally integrable complex-valued function $\phi$ on $\mathbb{R}$ is given by

[TABLE]

The set of those $\phi$ with $\|\phi\|_{\dot{B}^{s}_{p,q}}<\infty$ is referred to as the homogeneous Besov space.

This is the classical definition of Besov spaces, but in relation to integrable Teichmüller spaces, only the critical case $p=q=1/s$ is considered. Therefore, we provide the following specific definition.

Definition 3.1**.**

For $p>1$, a locally integrable complex-valued function $\phi$ on $\mathbb{R}$ belongs to the Besov space $B_{p}(\mathbb{R})$ if

[TABLE]

is finite. Moreover, for $p\geq 1$,

[TABLE]

defines the set $B_{p}^{\#}(\mathbb{R})$ consisting of those $\phi$ for which this quantity is finite.

Remark 3.1**.**

For $p=1$, functions $\phi$ such that $\|\phi\|_{B_{p}}<\infty$ are restricted to constant functions. See [7] as noted in [20, Exercise 17.14].

We see that $B_{p}(\mathbb{R})\hookrightarrow B_{q}(\mathbb{R})$ and $B_{p}^{\#}(\mathbb{R})\hookrightarrow B_{q}^{\#}(\mathbb{R})$ are continuous for $p<q$ by the Sobolev embedding theorem (see [27, Theorem 2.14]).

A locally integrable complex-valued function $\phi$ on $\mathbb{R}$ is said to be BMO if

[TABLE]

where the supremum is taken over all bounded intervals $I$ on $\mathbb{R}$, and $\phi_{I}$ denotes the integral mean of $\phi$ over $I$. The set of all BMO functions is denoted by ${\rm BMO}(\mathbb{R})$. It is easy to see that $\|\phi\|_{\rm BMO}\leq\|\phi\|_{L_{\infty}}$. It is known that $\log h^{\prime}$ for a strongly quasisymmetric homeomorphism $h$ belongs to ${\rm BMO}(\mathbb{R})$ since $h^{\prime}$ is an $A_{\infty}$-weight.

Definition 3.2**.**

For $p\geq 1$, we define the function space $\widehat{B}_{p}(\mathbb{R})$ as

[TABLE]

with the semi-norm $\|\phi\|_{{\widehat{B}}_{p}}=\|\phi\|_{B_{p}^{\#}}+\|\phi\|_{\rm BMO}$.

For $p>1$, the semi-norms $\|\phi\|_{B_{p}}$ and $\|\phi\|_{{\widehat{B}}_{p}}$ are equivalent. Indeed, $\|\phi\|_{B_{p}}\asymp\|\phi\|_{B_{p}^{\#}}$ is well-known (see [20, Proposition 17.17]) and $\|\phi\|_{\rm BMO}\leq\|\phi\|_{B_{p}}$ can be found in [38, Proposition 2.2].

The reason for adding the BMO norm is twofold: firstly, to restrict indeterminacy to only a constant function difference, and secondly, to ensure that when defining Besov norms for functions on the unit circle $\mathbb{S}$ in the same manner as below, they correspond isomorphically through the Cayley transformation.

By ignoring differences in constant functions, $\|\cdot\|_{B_{p}}$ and $\|\cdot\|_{{\widehat{B}}_{p}}$ can be treated as norms, allowing $B_{p}(\mathbb{R})$ and $\widehat{B}_{p}(\mathbb{R})$ to be regarded as complex Banach spaces.

On the unit circle $\mathbb{S}$, we define the spaces $B_{p}(\mathbb{S})$, $B_{p}^{\#}(\mathbb{S})$, and $\widehat{B}_{p}(\mathbb{S})$ of integrable functions in the same way. For the Cayley transformation $K(x)=(x-i)/(x+i)$ which maps $\widehat{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$ onto $\mathbb{S}$ with $K(\infty)=1$, let $K_{*}(\phi)=\phi_{*}$ be the push-forward of a function $\phi$ on $\mathbb{R}$ to that on $\mathbb{S}$; $K_{*}(\phi)=\phi\circ K^{-1}=\phi_{*}$. Then, $K_{*}$ defines the isometric isomorphism from $B_{p}(\mathbb{R})$ onto $B_{p}(\mathbb{S})$ for $p>1$ (see [41, p.8]). We also note that $\|K_{*}(\phi)\|_{\rm BMO}\asymp\|\phi\|_{\rm BMO}$ (see [14, Corollary VI.1.3]). Moreover, we see that $K_{*}$ gives a Banach isomorphism from $\widehat{B}_{p}(\mathbb{R})$ onto $\widehat{B}_{p}(\mathbb{S})$ for $p>1$. Indeed,

[TABLE]

In the case $p=1$, we can obtain the same result, but the proof is more complicated and carried out later at the end of Section 4.

Proposition 3.1**.**

$K_{*}:\widehat{B}_{p}(\mathbb{R})\to\widehat{B}_{p}(\mathbb{S})$* is a Banach isomorphism for $p\geq 1$.*

The homogeneous Sobolev space ${\dot{W}}^{1}_{1}(\mathbb{R})$ consists of locally integrable complex valued functions $\phi$ on $\mathbb{R}$ such that the distributional derivative $\phi^{\prime}$ is integrable on $\mathbb{R}$. The semi-norm is given by $\|\phi\|_{{\dot{W}}^{1}_{1}}=\|\phi^{\prime}\|_{L_{1}}$. We have $\|K_{*}(\phi)\|_{{\dot{W}}^{1}_{1}}=\|\phi\|_{{\dot{W}}^{1}_{1}}$ by the change of variables. The following result is used later.

Lemma 3.2**.**

The semi-norm $\|\phi\|_{\widehat{B}_{1}}=\|\phi\|_{B_{1}^{\#}}+\|\phi\|_{\rm BMO}$ on $\widehat{B}_{1}(\mathbb{R})$ is equivalent to $\|\phi\|_{B_{1}^{\#}}+\|\phi\|_{{\dot{W}}^{1}_{1}}$. In particular, $\widehat{B}_{1}(\mathbb{R})=B_{1}^{\#}(\mathbb{R})\cap{\dot{W}}^{1}_{1}(\mathbb{R})$. The same norm estimate holds true for $\widehat{B}_{1}(\mathbb{S})$, but $\widehat{B}_{1}(\mathbb{S})=B_{1}^{\#}(\mathbb{S})$ in the unit circle case.

Proof.

We first prove the statement for $\widehat{B}_{1}(\mathbb{S})$, and then transfer it to $\widehat{B}_{1}(\mathbb{R})$. On the unit circle $\mathbb{S}$, $B_{1}^{\#}(\mathbb{S})$ is contained in ${\dot{W}}^{1}_{1}(\mathbb{S})$. Indeed, if $\phi_{*}\in B_{1}^{\#}(\mathbb{S})$, then $\phi_{*}-\alpha$ also belongs to $L_{1}(\mathbb{S})$ for every $\alpha\in\mathbb{C}$ by compactness of $\mathbb{S}$, and it follows from [20, Theorem 17.66] that

[TABLE]

In addition, this is bounded by $\|\phi_{*}\|_{B_{1}^{\#}}+2\pi\|\phi_{*}\|_{\rm BMO}\lesssim\|\phi_{*}\|_{\widehat{B}_{1}}$. Therefore, $\|\phi_{*}\|_{B_{1}^{\#}}+\|\phi_{*}\|_{{\dot{W}}^{1}_{1}}\lesssim\|\phi_{*}\|_{\widehat{B}_{1}}$ is satisfied. The converse estimate follows from $\|\phi_{*}\|_{\rm BMO}\leq\|\phi_{*}\|_{L_{\infty}}$ and

[TABLE]

which is in the specificity of one-dimensional spaces. Thus, $\|\phi_{*}\|_{B_{1}^{\#}}+\|\phi_{*}\|_{{\dot{W}}^{1}_{1}}\asymp\|\phi_{*}\|_{\widehat{B}_{1}}$.

The corresponding result for $\widehat{B}_{1}(\mathbb{R})$ is true due to the isomorphism $K_{*}:\widehat{B}_{1}(\mathbb{R})\to\widehat{B}_{1}(\mathbb{S})$ by Proposition 3.1 as well as $\|K_{*}(\phi)\|_{{\dot{W}}^{1}_{1}}=\|\phi\|_{{\dot{W}}^{1}_{1}}$. Indeed, these imply

[TABLE]

The converse estimate $\|\phi\|_{\rm BMO}\leq\|\phi\|_{{\dot{W}}^{1}_{1}}$ is obtained in the same reason as above. ∎

4. The Hilbert transform and the Szegö projection

We introduce holomorphic functions on $\mathbb{H}$ whose boundary extension to $\mathbb{R}$ is in the Besov space on $\mathbb{R}$, which are called analytic Besov functions. Conversely, we see that any Besov function on $\mathbb{R}$ can be represented as a sum of the boundary extensions of analytic Besov functions on the upper half-plane $\mathbb{H}^{+}$ and the lower half-plane $\mathbb{H}^{-}$.

Definition 4.1**.**

For $p>1$, we define the analytic Besov space $\mathcal{B}_{p}(\mathbb{H})$ as the set of holomorphic functions $\Phi$ on $\mathbb{H}$ for which the semi-norm

[TABLE]

is finite. Moreover, for $p\geq 1$,

[TABLE]

defines the set $\mathcal{B}_{p}^{\#}(\mathbb{R})$ consisting of those $\Phi$ for which this quantity is finite.

Remark 4.1**.**

For $p=1$, functions $\Phi$ such that $\|\Phi\|_{\mathcal{B}_{p}}<\infty$ are restricted to constant functions. See [44, p.132].

We see that $\mathcal{B}_{p}(\mathbb{H})\hookrightarrow\mathcal{B}_{q}(\mathbb{H})$ and $\mathcal{B}_{p}^{\#}(\mathbb{H})\hookrightarrow\mathcal{B}_{q}^{\#}(\mathbb{H})$ are continuous for $p<q$ (see [42, Propositions 2.1, 2.2]).

A holomorphic function $\Phi$ on $\mathbb{H}$ is BMOA if it satisfies the Carleson measure condition

[TABLE]

where the supremum is taken over all bounded intervals $I$ on $\mathbb{R}$, and $\widehat{I}$ denotes the square in $\mathbb{H}$ with $I$ as its base. The set of all BMOA functions is denoted by $\mathrm{BMOA}(\mathbb{H})$.

Definition 4.2**.**

For $p\geq 1$, we define the function space $\widehat{\mathcal{B}}_{p}(\mathbb{H})$ as

[TABLE]

with the semi-norm

[TABLE]

For $p>1$, the semi-norms $\|\Phi\|_{\mathcal{B}_{p}}$ and $\|\Phi\|_{\widehat{\mathcal{B}}_{p}}$ are equivalent. This is because $\|\Phi\|_{\mathcal{B}_{p}^{\#}}\lesssim\|\Phi\|_{\mathcal{B}_{p}}$ and $\|\Phi\|_{\mathrm{BMOA}}\lesssim\|\Phi\|_{\mathcal{B}_{p}}$ (see [42, Section 2]). The rationale for adding the BMOA norm is similar to that in the case of the Besov space norm.

Ignoring differences in constant functions allows $\|\cdot\|_{\mathcal{B}_{p}}$ and $\|\cdot\|_{\widehat{\mathcal{B}}_{p}}$ to be treated as norms, making it possible to regard $\mathcal{B}_{p}(\mathbb{H})$ and $\widehat{\mathcal{B}}_{p}(\mathbb{H})$ as complex Banach spaces.

Here is the relationship between integrable Beltrami coefficients and analytic Besov spaces (see [42, Theorem 3.11]):

Lemma 4.1**.**

Let $p\geq 1$. For a quasiconformal self-homeomorphism $F^{\mu}$ of $\mathbb{C}$, which has complex dilatation $\mu$ in $\mathbb{H}^{+}$ and is conformal in $\mathbb{H}^{-}$, the condition $\mu\in M_{p}(\mathbb{H}^{+})$ is equivalent to $\log(F^{\mu}|_{\mathbb{H}^{-}})^{\prime}\in\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$.

Next, we consider the boundary extension of holomorphic functions in the analytic Besov space $\widehat{\mathcal{B}}_{p}(\mathbb{H})$ for $p\geq 1$. On the half-plane $\mathbb{H}$ and its boundary $\mathbb{R}$, we will see that the trace of $\widehat{\mathcal{B}}_{p}(\mathbb{H})$ on $\mathbb{R}$ forms a closed subspace of the Besov space $\widehat{B}_{p}(\mathbb{R})$. The essential issue is the correspondence between ${\mathcal{B}}_{p}^{\#}(\mathbb{H})$ and $B_{p}^{\#}(\mathbb{R})$ (or between ${\mathcal{B}}_{p}(\mathbb{H})$ and $B_{p}(\mathbb{R})$ when $p>1$).

Proposition 4.2**.**

There are the following relations between the function spaces ${\mathcal{B}}_{p}^{\#}(\mathbb{H})$ and $B_{p}^{\#}(\mathbb{R})$ for $p\geq 1$. The same results hold true between ${\mathcal{B}}^{\#}_{p}(\mathbb{D})$ and ${B}^{\#}_{p}(\mathbb{S})$.

1. (1)

Every holomorphic function $\Phi\in{\mathcal{B}}_{p}^{\#}(\mathbb{H})$ has non-tangential limits almost everywhere on $\mathbb{R}$. This defines the boundary function $\phi$ of $\Phi$ that belongs to $B_{p}^{\#}(\mathbb{R})$; 2. (2)

Conversely, for a measurable function $\phi\in B_{p}^{\#}(\mathbb{R})$ that is extendable to a holomorphic function on $\mathbb{H}$, the Poisson integral $\Phi$ of $\phi$ belongs to ${\mathcal{B}}_{p}^{\#}(\mathbb{H})$; 3. (3)

In the above correspondence between $\Phi\in{\mathcal{B}}_{p}^{\#}(\mathbb{H})$ and $\phi\in B_{p}^{\#}(\mathbb{R})$, their norms are uniformly comparable as $\|\Phi\|_{{\mathcal{B}}_{p}^{\#}}\asymp\|\phi\|_{B_{p}^{\#}}$.

There are several references verifying these claims. Among them, we cite [34, §V.5.1] for discussions from real analysis in the case of $\mathbb{H}$, and [25, Theorems 2.1 and 5.1] from complex analysis in the case of $\mathbb{D}$. Concerning statement (2), the Poisson integral of any $\phi\in{B}_{p}^{\#}(\mathbb{R})$ not necessarily holomorphically extendable can be also characterized by the integrability of its partial derivatives generalizing the definition of ${\mathcal{B}}_{p}^{\#}(\mathbb{H})$.

The other factors in the norms for $\widehat{\mathcal{B}}_{p}(\mathbb{H})$ and $\widehat{B}_{p}(\mathbb{R})$ correspond neatly under the boundary extension $\Phi\mapsto\phi$ because $\|\Phi\|_{\rm BMOA}\asymp\|\phi\|_{\rm BMO}$ (see [44, Theorem 9.19]). For $p=1$, we may also use $\|\Phi\|_{\dot{H}_{1}^{1}}=\|\phi\|_{{\dot{W}}^{1}_{1}}$ in account of Lemma 3.2 and the corresponding claim [42, Proposition 2.5] for $\widehat{\mathcal{B}}_{1}(\mathbb{H})$ in terms of the Hardy space norm $\|\Phi\|_{\dot{H}_{1}^{1}}=\|\Phi^{\prime}\|_{{H}_{1}}$.

Consequently, we obtain the following:

Proposition 4.3**.**

Any $\Phi\in\widehat{\mathcal{B}}_{p}(\mathbb{H})$ has non-tangential limits almost everywhere on $\mathbb{R}$, and its boundary function belongs to $\widehat{B}_{p}(\mathbb{R})$. This defines the trace operator $E:\widehat{\mathcal{B}}_{p}(\mathbb{H})\to\widehat{B}_{p}(\mathbb{R})$, which is a Banach isomorphism onto its image.

Considering the both half-planes $\mathbb{H}^{+}$ and $\mathbb{H}^{-}$, we obtain the closed subspaces $E^{+}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))$ and $E^{-}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-}))$ in $\widehat{B}_{p}(\mathbb{R})$. The functions in $E^{+}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))$ and $E^{-}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-}))$ correspond by complex conjugation. By the identification under the Banach isomorphism $E^{\pm}:\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})\to\widehat{B}_{p}(\mathbb{R})$, we may regard $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})$ are closed subspaces of $\widehat{B}_{p}(\mathbb{R})$.

Conversely, the projection associated with $E$ from the Besov space $\widehat{B}_{p}(\mathbb{R})$ to the analytic Besov space $\widehat{\mathcal{B}}_{p}(\mathbb{H})\cong E(\widehat{\mathcal{B}}_{p}(\mathbb{H}))$ is specifically provided by using the following map.

Definition 4.3**.**

For a rapidly decreasing function $\phi$ in the Schwartz class on $\mathbb{R}$, we define the singular integral

[TABLE]

and extend it to a linear operator on $\widehat{B}_{p}(\mathbb{R})$ to be called the Hilbert transform.

There are several ways of extending $\mathcal{H}$ to a linear operator on $\widehat{B}_{p}(\mathbb{R})$. Especially on ${\rm BMO}(\mathbb{R})$, we may use the regularized kernel

[TABLE]

for $\mathcal{H}$ instead of $(x-t)^{-1}$. This defines the extension up to an additive constant. It is also possible to extend $\mathcal{H}$ as the operator on certain distributions (see [24, Chapter 4]).

It is well-known that $\mathcal{H}$ gives a Banach automorphism of ${\rm BMO}(\mathbb{R})$ satisfying ${\mathcal{H}}\circ{\mathcal{H}}=-I$ (see [14, Chapter VI] and [17]). The Besov space $B_{p}^{\#}(\mathbb{R})$ for $p\geq 1$ possesses the same property with $\mathcal{H}$ (see [16, Proposition 4.7]). Then, we obtain the following:

Proposition 4.4**.**

The Hilbert transform $\mathcal{H}$ is a Banach automorphism of $\widehat{B}_{p}(\mathbb{R})$ satisfying ${\mathcal{H}}\circ{\mathcal{H}}=-I$ for $p\geq 1$.

Let $P^{\pm}=\frac{1}{2}(I\pm i\mathcal{H})$, which we call the Riesz projections. On the unit circle $\mathbb{S}$, the positive Riesz projection $P^{+}$ can be alternatively defined by the positive part of the Fourier transform of a function $\phi_{*}$ on $\mathbb{S}$, and the negative Riesz projection $P^{-}$ by the negative part.

We can apply the Riesz projections $P^{\pm}$ to $\widehat{B}_{p}(\mathbb{R})$ by virtue of Proposition 4.4. We note that $P^{+}+P^{-}=I$ and $P^{+}\circ P^{-}=P^{-}\circ P^{+}=O$ by the definition and the property ${\mathcal{H}}\circ{\mathcal{H}}=-I$. Moreover, the images of $P^{\pm}$ coincide with $E^{\pm}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm}))$, which are the closed subspaces of $\widehat{B}_{p}(\mathbb{R})$ consisting of all elements that extend to holomorphic functions on $\mathbb{H}^{\pm}$ by the Poisson integral.

Proposition 4.5**.**

The Riesz projections $P^{\pm}$ in the Besov space $\widehat{B}_{p}(\mathbb{R})$ for $p\geq 1$ are bounded linear projections onto the closed subspaces $E^{\pm}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))$. They yield the topological direct sum decomposition

[TABLE]

Remark 4.2**.**

The boundedness of the Riesz projections $P^{\pm}$ on $B_{p}^{\#}(\mathbb{S})$ for $p\geq 1$ was verified in [26, p.449] in a way of using the definition of $P^{\pm}$ by the Fourier transform and the characterization of $B_{p}^{\#}(\mathbb{S})$ in terms of approximation by polynomials. From this, we see that $P^{\pm}$ is bounded on $\widehat{B}_{p}(\mathbb{S})$, and hence so is the Hilbert transform $\mathcal{H}=-i(2P^{+}-I)=i(2P^{-}-I)$.

Definition 4.4**.**

Holomorphic functions of the upper and the lower half-planes $\mathbb{H}^{\pm}$ defined by the Cauchy integrals of $\phi$,

[TABLE]

are called the Szegö projections of $\phi$.

We see that the Szegö projections give the bounded linear maps $\widehat{B}_{p}(\mathbb{R})\to\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})$ whose composition with the trace operators $E^{\pm}:\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})\to\widehat{B}_{p}(\mathbb{R})$ coincide with the Riesz projections $P^{\pm}$ (by the Plemelj formula in the special case). In the sequel, we do not distinguish them, denote both of them by $P^{\pm}$, and call the Szegö projections. Moreover, we regard $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})$ as the subspaces of $\widehat{B}_{p}(\mathbb{R})$ by omitting $E^{\pm}$ and represent the direct sum decomposition of $\widehat{B}_{p}(\mathbb{R})$ in Proposition 4.5 by

[TABLE]

Finally in this section, we apply the above results to:

Proof of Proposition 3.1.

By the decomposition of $\widehat{B}_{p}(\mathbb{R})$ as in Proposition 4.5, it suffices to prove that $K_{*}:\widehat{B}_{p}(\mathbb{R})\to\widehat{B}_{p}(\mathbb{S})$ is a Banach isomorphism restricted to $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})$. This has been verified by [42, Theorem 2.4], and thus we obtain the required result. ∎

5. Composition Operators and interpolation

The composition operator $C_{h}$ defined for a quasisymmetric homeomorphism $h:\mathbb{R}\to\mathbb{R}$ is introduced as acting on the space of functions $\phi$ on $\mathbb{R}$ by $C_{h}(\phi)=\phi\circ h$. Then, we prove that $C_{h}$ is a bounded linear operator on $\widehat{B}_{p}(\mathbb{R})$.

Proposition 5.1**.**

For $p>1$, the composition operator $C_{h}$ induced by any quasisymmetric homeomorphism $h$ gives a Banach automorphism of ${B}_{p}(\mathbb{R})$ such that the operator norm $\|C_{h}\|$ is bounded by a constant depending only on the doubling (quasisymmetry) constant of $h$. The converse also holds.

The proof relies on interpolation of Banach spaces, as elucidated in [6, Theorem 1.3] and [5, Theorem 12]. For $p>1$, the Besov space ${B}_{p}(\mathbb{R})$ can be regarded as the trace of a certain function space on $\mathbb{H}$, interpolated with the $2$-dimensional BMO function space ${\rm BMO}(\mathbb{H})$. The crucial role of a quasiconformal self-homeomorphism of $\mathbb{H}$ in this argument is that its composition operator acts boundedly on ${\rm BMO}(\mathbb{H})$. This proof does not extend to $p=1$.

For $p=1$, we can obtain the following claim instead, which is enough for characterizing the quasisymmetric homeomorphisms in $T_{1}$.

Theorem 5.2**.**

Suppose that a quasisymmetric homeomorphism $h$ is locally absolutely continuous and satisfies $\log h^{\prime}\in{\dot{W}}^{1}_{1}(\mathbb{R})$. Then, the composition operator $C_{h}$ gives a Banach automorphism of $\widehat{B}_{1}(\mathbb{R})$ such that the operator norm $\|C_{h}\|$ is bounded by a constant depending only on $\|\log h^{\prime}\|_{{\dot{W}}^{1}_{1}}$ and tending $1$ as $\|\log h^{\prime}\|_{{\dot{W}}^{1}_{1}}\to 0$.

The proof of Theorem 5.2 utilizes the real interpolation of $B_{1}^{\#}(\mathbb{R})$ by ${\dot{W}}^{2}_{1}(\mathbb{R})$ and $L_{1}(\mathbb{R})$ as defined below with its property given in the next proposition. See [20, Chap.16] for interpolation of Banach spaces. For each factor of this real interpolation of $B_{1}^{\#}(\mathbb{R})$, we will show that $C_{h}$ is a bounded linear operator to itself. From this, we see that $C_{h}$ is also a bounded linear operator on $B_{1}^{\#}(\mathbb{R})$.

Definition 5.1**.**

Let $(X_{0},\|\cdot\|_{X_{0}})$ and $(X_{1},\|\cdot\|_{X_{1}})$ be Banach spaces that are continuously embedded into a common topological vector space, which we call an admissible pair of Banach spaces. For $t>0$,

[TABLE]

is defined as the norm of $x\in X_{0}+X_{1}$. Then, for $\sigma\in(0,1)$ and $q\geq 1$, the set

[TABLE]

is defined to be the real interpolation of $X_{0}$ and $X_{1}$.

The real interpolation space $(X_{0},X_{1})_{\sigma,q}$ is a Banach space with the norm $\|\cdot\|_{\sigma,q}$ (see [20, Theorem 16.5]). The boundedness of the associated operator with this interpolation is stated as follows (see [20, Theorem 16.12]).

Proposition 5.3**.**

Let $(X_{0},\|\cdot\|_{X_{0}})$ and $(X_{1},\|\cdot\|_{X_{1}})$ be an admissible pair of Banach spaces. Let $C:X_{0}+X_{1}\to X_{0}+X_{1}$ be a linear transformation such that its restrictions $C|_{X_{0}}$ and $C|_{X_{1}}$ are bounded linear transformations of $X_{0}$ and $X_{1}$, respectively. Then, for any $\sigma\in(0,1)$ and $q\geq 1$, $C|_{(X_{0},X_{1})_{\sigma,q}}$ is a bounded linear transformation of the real interpolation $(X_{0},X_{1})_{\sigma,q}$ and its operator norm satisfies

[TABLE]

Proof of Theorem 5.2.

Since $\widehat{B}_{1}(\mathbb{R})=B_{1}^{\#}(\mathbb{R})\cap{\dot{W}}^{1}_{1}(\mathbb{R})$ by Lemma 3.2, we first consider $B_{1}^{\#}(\mathbb{R})$ by representing it as the real interpolation

[TABLE]

where ${\dot{W}}^{2}_{1}(\mathbb{R})$ is the homogeneous Sobolev space with norm $\|\phi\|_{{\dot{W}}^{2}_{1}}=\|\phi^{\prime\prime}\|_{L_{1}}$ (see [20, Theorem 17.30]). Then, we take the intersection with ${\dot{W}}^{1}_{1}(\mathbb{R})$ for recovering $\widehat{B}_{1}(\mathbb{R})$.

The specificity of one-dimensional spaces is applied here: if $\log h^{\prime}\in{\dot{W}}^{1}_{1}(\mathbb{R})$, then $\log h^{\prime}$ is bounded and continuous, and so are $h^{\prime}$ and $(h^{\prime})^{-1}$. By the normalization $h(0)=0$ and $h(1)=1$, there is some $\xi\in[0,1]$ such that $h^{\prime}(\xi)=1$. This implies that $\|\log h^{\prime}\|_{L_{\infty}}\leq\|\log h^{\prime}\|_{{\dot{W}}^{1}_{1}}$. From this, we have $1/M\leq h^{\prime}(x)\leq M$ for some constant $M\geq 1$ depending only on $\|\log h^{\prime}\|_{{\dot{W}}^{1}_{1}}$ and tending $1$ as $\|\log h^{\prime}\|_{{\dot{W}}^{1}_{1}}\to 0$. Moreover, by $(\log h^{\prime})^{\prime}=h^{\prime\prime}/h^{\prime}$, we have $h\in{\dot{W}}^{2}_{1}(\mathbb{R})$ with $\|h\|_{{\dot{W}}^{2}_{1}}\leq M\|\log h^{\prime}\|_{{\dot{W}}^{1}_{1}}$.

From these conditions, we see that $C_{h}$ is bounded linear operators of $L_{1}(\mathbb{R})$, ${\dot{W}}^{1}_{1}(\mathbb{R})$, and ${\dot{W}}^{2}_{1}(\mathbb{R})$ onto themselves:

[TABLE]

[TABLE]

Here, in the last inequality above, we have used the fact that if $\phi\in{\dot{W}}^{2}_{1}(\mathbb{R})$ then $\phi^{\prime}\in L_{\infty}(\mathbb{R})$ and $\|\phi^{\prime}\|_{L_{\infty}}\leq\|\phi\|_{{\dot{W}}^{2}_{1}}=\int_{\mathbb{R}}|\phi^{\prime\prime}(t)|dt$ under assumption $\phi^{\prime}(0)=0$.

In the above argument, we also have shown that the operator norm of the composition operator $C_{h}$ on $L_{1}(\mathbb{R})$, ${\dot{W}}^{1}_{1}(\mathbb{R})$, and ${\dot{W}}^{2}_{1}(\mathbb{R})$ are all bounded in terms of $\|\log h^{\prime}\|_{{\dot{W}}^{1}_{1}}$. Therefore, by Proposition 5.3, we obtain the required statement. ∎

6. Conformal welding

In general, a quasisymmetric homeomorphism $h:\mathbb{R}\to\mathbb{R}$ can be expressed as the discrepancy between the boundary values $f_{1}=F_{1}|_{\mathbb{R}}$ and $f_{2}=F_{2}|_{\mathbb{R}}$ of two conformal homeomorphisms $F_{1}:\mathbb{H}^{-}\to\Omega^{-}$ and $F_{2}:\mathbb{H}^{+}\to\Omega^{+}$, where $\Omega^{-}$ and $\Omega^{+}$ are complementary domains in $\mathbb{C}$ with $\partial\Omega^{-}=\partial\Omega^{+}$. This expression $h=f_{2}^{-1}\circ f_{1}$ is called conformal welding. If absolute continuity is present, then $\log f_{2}^{\prime}\circ h+\log h^{\prime}=\log f_{1}^{\prime}$ is satisfied. This allows us to see that the property of $\log h^{\prime}$ is determined by that of $\log f_{1}^{\prime}$ and $\log f_{2}^{\prime}$ if the composition operator $C_{h}$ preserves it.

We proceed to the problem of finding the class of quasisymmetric homeomorphisms $h$ whose quasiconformal extensions $H(\mu)$ to $\mathbb{H}$ have complex dilatations $\mu\in M_{p}(\mathbb{H})$. First, we consider the boundary extension of a conformal homeomorphism of $\mathbb{H}$.

Lemma 6.1**.**

For $\mu\in M_{p}(\mathbb{H}^{+})$, let $F^{\mu}$ be the normalized quasiconformal self-homeomorphism of $\mathbb{C}$ that is conformal on $\mathbb{H}^{-}$ and has the complex dilatation $\mu$ on $\mathbb{H}^{+}$. Then, $f=F^{\mu}|_{\mathbb{R}}$ is locally absolutely continuous, $f^{\prime}(x)\neq 0$ almost everywhere on $\mathbb{R}$, and $\log f^{\prime}$ belongs to $\widehat{B}_{p}(\mathbb{R})$.

Proof.

By Lemma 4.1, we see that $\Phi=\log(F^{\mu}|_{\mathbb{H}^{-}})^{\prime}$ belongs to $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$. Moreover, by Proposition 4.3, the boundary extension $\phi=E(\Phi)$ is in $\widehat{B}_{p}(\mathbb{R})$, and in particular, $\phi(x)$ is finite almost everywhere on $\mathbb{R}$. Since $f(\mathbb{R})$ is a chord-arc curve, it is well-known that $f$ is locally absolutely continuous and $f^{\prime}(x)$ coincides with the angular derivative of $F^{\mu}$ at $x$ almost everywhere on $\mathbb{R}$ (see [41, Lemma 3.2]), which is non-zero almost everywhere. Hence, $\log f^{\prime}=\phi$. This completes the proof. ∎

Representing a quasisymmetric homeomorphism $h=H(\mu)|_{\mathbb{R}}$ with $\mu\in M_{p}(\mathbb{H})$ by conformal welding, we obtain the following characterization of it in one direction. The other direction will be shown later. The boundedness of the composition operator $C_{h}$ is crucial in this proof.

Theorem 6.2**.**

Let $h:\mathbb{R}\to\mathbb{R}$ be a quasisymmetric homeomorphism whose quasiconformal extension $H=H(\mu):\mathbb{H}\to\mathbb{H}$ has a complex dilatation $\mu$ belonging to $M_{p}(\mathbb{H})$ for $p\geq 1$. Then, $h$ is locally absolutely continuous and $\phi=\log h^{\prime}$ belongs to $\widehat{B}_{p}(\mathbb{R})$.

Proof.

We represent $h$ by conformal welding. Let $F_{1}=F^{\mu}:\mathbb{C}\to\mathbb{C}$ be the normalized quasiconformal homeomorphism that is conformal on $\mathbb{H}^{-}$ and has the complex dilatation $\mu$ on $\mathbb{H}^{+}$. Let $F_{2}=F_{\bar{\mu}^{-1}}:\mathbb{C}\to\mathbb{C}$ be the normalized quasiconformal homeomorphism that is conformal on $\mathbb{H}^{+}$ and has the complex dilatation $\bar{\mu}^{-1}$ on $\mathbb{H}^{-}$. Here, $\mu^{-1}$ is the complex dilatation of the quasiconformal homeomorphism $H({\mu^{-1}}):\mathbb{H}^{+}\to\mathbb{H}^{+}$ satisfying $(H(\mu))^{-1}=H(\mu^{-1})$, and $\bar{\mu}^{-1}$ stands for its reflection defined on $\mathbb{H}^{-}$. By choosing a suitable representative $\mu$ in the Teichmüller equivalence class $[\mu]$, we see that $\mu^{-1}$ also belongs to $M_{p}(\mathbb{H})$ (see [42, Lemma 3.1]). We note that the normalization of $F_{1}$ and $F_{2}$ guarantees $F_{1}(\mathbb{R})=F_{2}(\mathbb{R})$.

Let $f_{1}=F_{1}|_{\mathbb{R}}$ and $f_{2}=F_{2}|_{\mathbb{R}}$. Then, we have that $f_{2}\circ h=f_{1}$. Here, Lemma 6.1 asserts that $f_{1}$ and $f_{2}$ are locally absolutely continuous, and $\log f_{1}^{\prime}$ and $\log f_{2}^{\prime}$ belong to $\widehat{B}_{p}(\mathbb{R})$. Moreover, $f_{2}^{\prime}(x)\neq 0\ {\rm(a.e.)}$ implies that $h$ is also locally absolutely continuous. Then, taking the logarithm and the derivatives for the conformal welding $f_{2}\circ h=f_{1}$, we have

[TABLE]

In the case $p=1$, we see that $\log f_{2}^{\prime}\circ h\in\dot{W}_{1}^{1}(\mathbb{R})$ because

[TABLE]

is finite by $\log f_{2}^{\prime}\in\widehat{B}_{1}(\mathbb{R})\subset\dot{W}_{1}^{1}(\mathbb{R})$. Since $\log f_{1}^{\prime}$ also belongs to $\widehat{B}_{1}(\mathbb{R})\subset{\dot{W}}^{1}_{1}(\mathbb{R})$, we have $\log h^{\prime}\in{\dot{W}}^{1}_{1}(\mathbb{R})$ by (7). This is the assumption of Theorem 5.2, so we can apply it to conclude that the composition operator $C_{h}$ is bounded on $\widehat{B}_{1}(\mathbb{R})$. This implies that $C_{h}(\log f_{2}^{\prime})=\log f_{2}^{\prime}\circ h\in\widehat{B}_{1}(\mathbb{R})$. Hence, we obtain $\log h^{\prime}\in\widehat{B}_{1}(\mathbb{R})$ again by (7).

In the case of $p>1$, Proposition 5.1 claims that the composition operator $C_{h}$ is bounded on $\widehat{B}_{p}(\mathbb{R})$. It follows from (7) that $\log h^{\prime}\in\widehat{B}_{p}(\mathbb{R})$ in the same manner as above. ∎

For the case $p=1$, Theorem 6.2 in particular implies that $\log h^{\prime}$ belongs to

[TABLE]

From this, the following result follows.

Corollary 6.3**.**

Every quasisymmetric homeomorphism $h$ in $T_{1}$ is a $C^{1}$-diffeomorphism of $\widehat{\mathbb{R}}\cong\mathbb{S}$ (including $\infty$).

This result was proved in [2]. Its proof utilizes the Teichmüller–Wittich–Belinskiǐ theorem (see [33]), which asserts that for a quasiconformal map $H$ with complex dilatation $\mu$, the local integrability condition at a point $z_{0}$ given by

[TABLE]

for some $r>0$ implies that $H$ has a non-zero complex derivative $H^{\prime}(z_{0})$ at $z_{0}$. If the complex dilatation $\mu$ of a quasiconformal map $H:\mathbb{H}^{+}\to\mathbb{H}^{+}$ belongs to $M_{1}(\mathbb{H}^{+})$, then the extension of $H$ to a quasiconformal homeomorphism of $\mathbb{C}$ by taking the reflection with respect to $\mathbb{R}$ (with complex dilatation $\bar{\mu}$ on $\mathbb{H}^{-}$) satisfies the above condition at any $z_{0}\in\mathbb{R}$.

7. Weil–Petersson Embedding in Bers Coordinates

We regard Weil–Petersson curves as the images of certain embeddings $\gamma:\mathbb{R}\to\mathbb{C}$, and we use Teichmüller space to coordinate these embeddings. We first recall the integrable Teichmüller space $T_{p}$ defined in Section 2.

For $p\geq 1$, the $p$-integrable Teichmüller space $T_{p}$ is the set of all normalized quasisymmetric homeomorphisms $h:\mathbb{R}\to\mathbb{R}$ that can be extended to quasiconformal homeomorphisms $H:\mathbb{H}\to\mathbb{H}$ whose complex dilatations belong to $M_{p}(\mathbb{H})$. By this correspondence from Beltrami coefficients $\mu$ to quasisymmetric homeomorphisms $h(\mu)$ via quasiconformal ones $H(\mu)$, we have the Teichmüller projection $\pi:M_{p}(\mathbb{H})\to T_{p}$. An element $h(\mu)$ of $T_{p}$ can be represented by a Teichmüller equivalence class $[\mu]$ for $\mu\in M_{p}(\mathbb{H})$.

Let $F^{\mu}$ denote the normalized quasiconformal homeomorphism of $\mathbb{C}$ whose complex dilatation is $\mu\in M_{p}(\mathbb{H}^{+})$ on $\mathbb{H}^{+}$ and [math] on $\mathbb{H}^{-}$. The Schwarzian derivative map $S:M_{p}(\mathbb{H}^{+})\to{\mathcal{A}}_{p}(\mathbb{H}^{-})$ is defined by the correspondence $\mu\mapsto S(F^{\mu}|_{\mathbb{H}^{-}})$. This induces the Bers embedding $\alpha:T_{p}\to{\mathcal{A}}_{p}(\mathbb{H}^{-})$ such that $\alpha\circ\pi=S$. By showing that $\alpha$ is a topological embedding, we provide the complex Banach structure for $T_{p}$.

In a similar way, the pre-Schwarzian derivative map $L:M_{p}(\mathbb{H}^{+})\to\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$ is defined by the correspondence $\mu\mapsto\log(F^{\mu}|_{\mathbb{H}^{-}})^{\prime}$. This induces a well-defined injection $\beta:T_{p}\to\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$ such that $\beta\circ\pi=L$. We call $\beta$ the pre-Bers embedding. The following claim is known to be true for all $p\geq 1$ in [42, Theorem 5.1].

Proposition 7.1**.**

The pre-Bers embedding $\beta:T_{p}\to\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$ for $p\geq 1$ is a biholomorphic homeomorphism onto the connected open subset $\beta(T_{p})=L(M_{p}(\mathbb{H}^{+}))$ in $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$.

The integrable Teichmüller space $T_{p}$ possesses the group structure as a subgroup of $T$. For every $[\nu]\in T_{p}$, the right translation $r_{[\nu]}:T_{p}\to T_{p}$ of the group elements of $T_{p}$ is defined by $[\mu]\mapsto[\mu]\ast[\nu]$. The following results are proved in [41, Theorem 6.1] and [40, Section 4].

Proposition 7.2**.**

For $p\geq 1$, $T_{p}$ is a topological group. Moreover, every right translation $r_{[\nu]}$ is a biholomorphic automorphism of $T_{p}$.

We introduce the Weil–Petersson embeddings as follows.

Definition 7.1**.**

For $\mu^{+}\in M_{p}(\mathbb{H}^{+})$ and $\mu^{-}\in M_{p}(\mathbb{H}^{-})$, we denote by $G(\mu^{+},\mu^{-})$ the normalized quasiconformal self-homeomorphism of $\mathbb{C}$ whose complex dilatation on $\mathbb{H}^{+}$ is $\mu^{+}$ and on $\mathbb{H}^{-}$ is $\mu^{-}$. The restriction of $G(\mu^{+},\mu^{-})$ to $\mathbb{R}$ is called a $p$-Weil–Petersson embedding. Its image is referred to as a $p$-Weil–Petersson curve.

The following claim guarantees that $p$-Weil–Petersson embeddings are parametrized by the product of Teichmüller spaces. The proof is the same as that in [39, Proposition 4.1].

Proposition 7.3**.**

$G(\mu^{+},\mu^{-})|_{\mathbb{R}}=G(\nu^{+},\nu^{-})|_{\mathbb{R}}$* if and only if $[\mu^{+}]=[\nu^{+}]$ and $[\mu^{-}]=[\nu^{-}]$.*

Hence, a $p$-Weil–Petersson embedding $G(\mu^{+},\mu^{-})|_{\mathbb{R}}$ is determined by the pair of the Teichmüller equivalence classes, which is denoted by $\gamma([\mu^{+}],[\mu^{-}])$. Let $T_{p}^{+}=\pi(M_{p}(\mathbb{H}^{+}))$ and $T_{p}^{-}=\pi(M_{p}(\mathbb{H}^{-}))$. Then, $T_{p}^{+}\times T_{p}^{-}$ becomes the coordinates of the space of $p$-Weil–Petersson embeddings. For the space of quasi-Fuchsian groups, this method is called simultaneous uniformization, which is originally due to Bers. The entire space of quasi-Fuchsian groups can be coordinated as a product of Teichmüller spaces. Similarly, the entire space of $p$-Weil–Petersson embeddings can be coordinated as $T_{p}^{+}\times T_{p}^{-}$. We refer to this as the Bers coordinates.

Let $\bar{\mu}\in M_{p}(\mathbb{H}^{-})$ denote the reflection $\overline{\mu(\bar{z})}$ of a Beltrami coefficient $\mu(z)$ for $z\in\mathbb{H}^{+}$. Then, $G(\mu,\bar{\mu})$ is nothing but the normalized quasiconformal homeomorphism $H(\mu):\mathbb{C}\to\mathbb{C}$ preserving $\mathbb{R}$, and $h=G(\mu,\bar{\mu})|_{\mathbb{R}}=\gamma([\mu],[\bar{\mu}])$ is the corresponding quasisymmetric homeomorphism of $\mathbb{R}$, which can be regarded as an element of the Teichmüller space $T_{p}$. The axis of symmetry of the product space $T_{p}^{+}\times T_{p}^{-}$ is defined as

[TABLE]

The canonical map $\iota:T_{p}\to{\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})\subset T_{p}^{+}\times T_{p}^{-}$ defined by $[\mu]\mapsto([\mu],[\bar{\mu}])$ is a real-analytic embedding, and hence ${\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})$ is a real-analytic submanifold of $T_{p}^{+}\times T_{p}^{-}$.

Regarding the group structure of $T_{p}$, the right transformation $r_{[\nu]}$ defined by $[\nu]\in T_{p}$ is extended to $T_{p}^{+}\times T_{p}^{-}$, defining the parallel translation

[TABLE]

This is a biholomorphic automorphism of $T_{p}^{+}\times T_{p}^{-}$ that preserves ${\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})$.

Theorem 6.2 shows that any element $h=\gamma([\mu],[\bar{\mu}])\in T_{p}$ with $\mu\in M_{p}(\mathbb{H}^{+})$ for $p\geq 1$ is locally absolutely continuous and $\log h^{\prime}$ belongs to ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$. This claim can be generalized as follows using the method of conformal welding.

Corollary 7.4**.**

Let $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ be a $p$-Weil–Petersson embedding for $p\geq 1$. Then, $\gamma:\mathbb{R}\to\mathbb{C}$ is locally absolutely continuous and $\log\gamma^{\prime}$ belongs to $\widehat{B}_{p}(\mathbb{R})$.

Proof.

We represent $\gamma$ as the following composition:

[TABLE]

Here, $\gamma_{1}=\gamma([0],[\mu^{-}]\ast[\overline{\mu^{+}}]^{-1})=F_{\nu}|_{\mathbb{R}}$ for the quasiconformal self-homeomorphism $F_{\nu}$ of $\mathbb{C}$ that is conformal on $\mathbb{H}^{+}$ and has complex dilatation $\nu$ on $\mathbb{H}^{-}$ with $[\nu]=[\mu^{-}]\ast[\overline{\mu^{+}}]^{-1}$, and $\gamma_{2}=\gamma([\mu^{+}],[\overline{\mu^{+}}])=H(\mu^{+})|_{\mathbb{R}}$ for the quasiconformal self-homeomorphism $H(\mu^{+})$ of $\mathbb{C}$ with the indicated complex dilatation. Then, $\gamma_{1}$ is locally absolutely continuous and $\log\gamma_{1}^{\prime}\in\widehat{B}_{p}(\mathbb{R})$ by Lemma 6.1. In addition, $\gamma_{2}$ is locally absolutely continuous and $\log\gamma_{2}^{\prime}\in\widehat{B}_{p}(\mathbb{R})$ by Theorem 6.2. Therefore, $\gamma=\gamma_{1}\circ\gamma_{2}$ is locally absolutely continuous and $\log\gamma^{\prime}=\log\gamma_{1}^{\prime}\circ\gamma_{2}+\log\gamma_{2}^{\prime}$ belongs to $\widehat{B}_{p}(\mathbb{R})$ by Proposition 5.1 and Theorem 5.2. ∎

8. Holomorphy to the Besov space and the characterization of $T_{p}$

For a Weil–Petersson embedding $\gamma:\mathbb{R}\to\mathbb{C}$, $\log\gamma^{\prime}$ belongs to the Besov space by Corollary 7.4. In this section, we first prove that this correspondence is holomorphic in the Bers coordinates. Then, using this fact, we extend Theorem 6.2 to the complete characterization of $T_{p}$ in terms of the real Besov space ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ (Theorem 1.1).

Lemma 8.1**.**

The map $\Lambda:T_{p}^{+}\times T_{p}^{-}\to\widehat{B}_{p}(\mathbb{R})$ defined by $([\mu^{+}],[\mu^{-}])\mapsto\log\gamma^{\prime}$ for $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ is a holomorphic injection for $p\geq 1$.

Proof.

By the Hartogs theorem for Banach spaces (see [8, §14.27]), to see that $\Lambda$ is holomorphic it suffices to show that $\Lambda$ is separately holomorphic. Namely, we fix, say $[\mu^{+}_{0}]\in T_{p}^{+}$, and prove that $\Lambda([\mu^{+}_{0}],[\mu^{-}])$ is holomorphic in $[\mu^{-}]\in T_{p}^{-}$. The other case is treated in the same way.

Let $h_{0}=\pi(\mu^{+}_{0})\in T_{p}$ be the quasisymmetric homeomorphism of $\mathbb{R}$, and $C_{h_{0}}$ the composition operator on $\widehat{B}_{p}(\mathbb{R})$ induced by $h_{0}$. We define the affine translation $Q_{h_{0}}$ on $\widehat{B}_{p}(\mathbb{R})$ by $Q_{h_{0}}(\phi)=C_{h_{0}}(\phi)+\log h_{0}^{\prime}$. Then,

[TABLE]

holds (see [41, Proposition 4.1]), and this relation yields a useful representation

[TABLE]

Here, $\Lambda([0],\,\cdot\,)$ is regarded as the trace operator in Proposition 4.3 if we compose the pre-Bers embedding $\beta^{-}:T_{p}^{-}\to\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$, that is,

[TABLE]

Since $E^{+}$ is a bounded linear operator and $r_{[\,\bar{\mu}^{+}_{0}\,]}^{-1}$ is holomorphic, we conclude that $\Lambda([\mu^{+}_{0}],\,\cdot\,)$ is holomorphic.

Suppose that $\Lambda([\mu_{1}^{+}],[\mu_{1}^{-}])=\Lambda([\mu_{2}^{+}],[\mu_{2}^{-}])$. Then, $\gamma([\mu_{1}^{+}],[\mu_{1}^{-}])=\gamma([\mu_{2}^{+}],[\mu_{2}^{-}])$ by the normalization. This implies that $[\mu_{1}^{+}]=[\mu_{2}^{+}]$ and $[\mu_{1}^{-}]=[\mu_{2}^{-}]$, which can be verified by the same proof as that of [39, Proposition 4.1]. Hence, $\Lambda$ is injective. ∎

We utilize a certain smaller subspace of $T_{p}$ to prove the converse of Theorem 6.2. Let $M_{0}(\mathbb{H})$ denote the set of all Beltrami coefficients on $\mathbb{H}$ with compact support, and define $T_{0}=\pi(M_{0}(\mathbb{H}))$. Then, $T_{0}$ is a dense subset of $T_{p}$ for every $p\geq 1$. This is because for every $[\mu]\in T_{p}$ there exists a representative $\nu\in M_{p}(\mathbb{H})$ of $[\mu]$ such that the cut-off sequence $\nu 1_{W_{n}}\in M_{0}(\mathbb{H})$ for an exhaustion of $\mathbb{H}$ by compact subsets $W_{n}$ converges to $\nu$ in $M_{p}(\mathbb{H})$ in the norm $\|\cdot\|_{p}+\|\cdot\|_{\infty}$. The appropriate representative $\nu$ can be constructed by a finite composition of quasiconformal self-homeomorphisms of $\mathbb{H}$ given by the Ahlfors–Weill section as in [40, Lemma 3.4].

A quasisymmetric homeomorphism $h:\mathbb{R}\to\mathbb{R}$ in the universal Teichmüller space $T$ belongs to $T_{0}$ if and only if $h$ is a real-analytic self-diffeomorphism of $\widehat{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$, where the regularity at $\infty$ is given by that of $h(1/x)$ at $x=0$. Moreover, we see that a holomorphic function $\Phi\in\beta(T)$ on $\mathbb{H}$ belongs to $\beta(T_{0})$ if and only if $\Phi$ extends to $\widehat{\mathbb{R}}$ analytically. We can easily generalize this property as follows.

Proposition 8.2**.**

A pair $([\mu^{+}],[\mu^{-}])\in T^{+}\times T^{-}$ belongs to $T_{0}^{+}\times T_{0}^{-}$ if and only if $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ is a real-analytic diffeomorphism of $\widehat{\mathbb{R}}$ into the Riemann sphere $\widehat{\mathbb{C}}$. In this case, $\phi=\log\gamma^{\prime}$ is a real-analytic function on $\widehat{\mathbb{R}}$.

Let $B_{0}(\mathbb{R})$ be the complex linear subspace of ${\rm BMO}(\mathbb{R})$ consisting of all complex-valued real-analytic functions on $\widehat{\mathbb{R}}$. Then, $B_{0}(\mathbb{R})\subset\widehat{B}_{p}(\mathbb{R})$ for every $p\geq 1$. Since $T_{0}$ is dense in $T_{p}$, $T_{0}^{+}\times T_{0}^{-}$ is also dense in $T_{p}^{+}\times T_{p}^{-}$. Then, by Lemma 8.1 and Proposition 8.2, if we have that $\Lambda:T_{p}^{+}\times T_{p}^{-}\to\widehat{B}_{p}(\mathbb{R})$ is surjective near the origin, we see that $B_{0}(\mathbb{R})$ is a dense subspace of $\widehat{B}_{p}(\mathbb{R})$.

We denote the tangent space of $T_{p}$ at $[\mu]$ by ${\mathscr{T}}_{[\mu]}T_{p}$. The tangent space of $T_{p}^{+}\times T_{p}^{-}$ at $([\mu^{+}],[\mu^{-}])$ is represented by the direct sum

[TABLE]

By the identification $T_{p}^{+}\cong\beta(T_{p}^{+})\subset\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$ and $T_{p}^{-}\cong\beta(T_{p}^{-})\subset\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$ under the pre-Bers embedding by Proposition 7.1, we may assume that ${\mathscr{T}}_{[\mu^{+}]}T_{p}^{+}\cong\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})$ and ${\mathscr{T}}_{[\mu^{-}]}T_{p}^{-}\cong\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$. Then, the derivative $d_{([\mu^{+}],[\mu^{-}])}\Lambda$ of $\Lambda$ at $([\mu^{+}],[\mu^{-}])$ is regarded as the linear mapping

[TABLE]

taking the direct product decomposition as in Proposition 4.5 into account.

The derivative $d_{([0],[0])}\,\Lambda$ at the origin can be easily understood. By checking that the restriction of $\Lambda$ to $T_{p}^{+}$ and $T_{p}^{-}$ coincides with the pre-Bers embeddings

[TABLE]

we see that the linearization $d_{([0],[0])}\,\Lambda$ is the identity map of $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})\oplus\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$. This implies the following claim by the inverse mapping theorem (see [8, §7.18]).

Proposition 8.3**.**

The derivative $d_{([0],[0])}\,\Lambda$ is surjective, and hence $\Lambda^{-1}$ is holomorphic in some neighborhood $U$ of [math] in $\widehat{B}_{p}(\mathbb{R})$.

Since $U$ is in the image of $\Lambda$, we also obtain that $B_{0}(\mathbb{R})$ is dense in $\widehat{B}_{p}(\mathbb{R})$ as mentioned after Proposition 8.2.

Theorem 6.2 implies that $\Lambda([\mu],[\bar{\mu}])\in{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ for every $([\mu],[\bar{\mu}])\in{\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})$. The converse of this claim also holds.

Lemma 8.4**.**

For every $\phi\in{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$, there exists $([\mu],[\bar{\mu}])\in{\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})$ such that $\Lambda([\mu],[\bar{\mu}])=\phi$.

Proof.

We take any $\phi_{0}$ from ${\rm Re}\,B_{0}(\mathbb{R})$, which is dense in ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$. Then, $\gamma_{0}(x)=\int_{0}^{x}\exp{\phi_{0}(t)}dt$ is a real-analytic self-diffeomorphism of $\widehat{\mathbb{R}}$, and by Proposition 8.2, we can find $([\mu_{0}],[\bar{\mu}_{0}])\in{\rm Sym}\,(T_{0}^{+}\times T_{0}^{-})$ such that $\Lambda([\mu_{0}],[\bar{\mu}_{0}])=\phi_{0}$. By proposition 8.3, there exists a neighborhood $U$ of $0\in\widehat{B}_{p}(\mathbb{R})$ that is contained in the image of $\Lambda$. Then, by relation (8), $R_{[\mu_{0}]}\circ\Lambda^{-1}|_{U}\circ Q_{h_{0}}^{-1}$ for $h_{0}=\pi(\mu_{0})$ gives the local holomorphic inverse of $\Lambda$ on the neighborhood $Q_{h_{0}}(U)$ of $\phi_{0}$. In particular, we see that $\Lambda^{-1}(\phi)$ belongs to ${\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})$ for every $\phi\in{\rm Re}\,\widehat{B}_{p}(\mathbb{R})\cap Q_{h_{0}}(U)$. ∎

Remark 8.1**.**

For $p>1$, the proof can also be demonstrated by applying the variant of the Beurling–Ahlfors quasiconformal extension, which replaces the convolution kernel in the original definition with a Gaussian function (heat kernel) (see [13], [37]), as shown in [38]. In fact, this yields a holomorphic right inverse of $\Lambda\circ(\pi^{+}\times\pi^{-})$ from a neighborhood of ${\rm Re}\,{B}_{p}(\mathbb{R})$ in ${B}_{p}(\mathbb{R})$ to $M_{p}(\mathbb{H}^{+})\times M_{p}(\mathbb{H}^{-})$.

Therefore, combined with Lemma 8.4, Theorem 6.2 is improved to the complete characterization of $T_{p}$ in terms of the real Besov space ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ for $p\geq 1$. Thus, we obtain Theorem 1.1.

9. Biholomorphic correspondence

We have seen that $p$-Weil–Petersson embeddings $\gamma$ are represented in the Bers coordinates and the map $\Lambda$ to the Besov space via $\log\gamma^{\prime}$ is a holomorphic injection. In this section, we prove that this is in fact a biholomorphic homeomorphism onto its image.

Theorem 9.1**.**

Let $\Lambda:T_{p}^{+}\times T_{p}^{-}\to\widehat{B}_{p}(\mathbb{R})$ be the holomorphic injection given by

[TABLE]

for $p\geq 1$. Then, its image ${\rm Ran}\,\Lambda$ is a connected open subset of $\widehat{B}_{p}(\mathbb{R})$ containing ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$, and $\Lambda$ is a biholomorphic homeomorphism onto ${\rm Ran}\,\Lambda$.

In virtue of the fact that $\Lambda$ is a holomorphic injection, to see that the inverse $\Lambda^{-1}$ is holomorphic, we have only to show that its derivative $d\Lambda$ is surjective by the inverse mapping theorem. The proof for the surjectivity of the derivative

[TABLE]

at $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$ involves the following steps:

• (a)

The image of the tangent space ${\mathscr{T}}_{([\mu^{+}],[\mu^{-}])}(T_{p}^{+}\times T_{p}^{-})\cong\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-})\oplus\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$ under the derivative $d\Lambda$ is given as the algebraic sum

[TABLE]

where $h^{+}=\pi(\mu^{+})$ and $h^{-}=\pi(\mu^{-})$.

• (b)

If $\Lambda([\mu^{+}],[\mu^{-}])$ is on the subspace $Z_{p}=i\,{\rm Re}\,\widehat{B}_{p}(\mathbb{R})\cap{\rm Ran}\,\Lambda$, then $d\Lambda$ is surjective on $\Lambda^{-1}(Z_{p})$.

• (c)

For $\Lambda([\mu^{+}],[\mu^{-}])=\log\gamma^{\prime}\in Z_{p}$, we translate $([\mu^{+}],[\mu^{-}])$ in parallel by $R_{[\nu]}$ and move $\log\gamma^{\prime}$ through the affine translation $Q_{h}$ with $\pi(\nu)=h$. Then, by $\Lambda\circ R_{[\nu]}=Q_{h}\circ\Lambda$ in (8), the surjectivity of $d\Lambda$ at any point of $T_{p}^{+}\times T_{p}^{-}$ is demonstrated.

We show these steps by partially relying on the previous result in [39] regarding the biholomorphy of $\Lambda$ on a certain larger space. Let $T_{B}$ be the BMO Teichmüller space consisting of all normalized strongly quasisymmetric homeomorphisms $h:\mathbb{R}\to\mathbb{R}$ such that $\log h^{\prime}\in{\rm BMO}(\mathbb{R})$. Likewise to the case of $T_{p}$, we can define the corresponding map $\widetilde{\Lambda}:T_{B}^{+}\times T_{B}^{-}\to{\rm BMO}(\mathbb{R})$. The embedding $\gamma:\mathbb{R}\to\mathbb{C}$ similarly defined by $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ for $([\mu^{+}],[\mu^{-}])\in T_{B}^{+}\times T_{B}^{-}$ is called a BMO embedding. Since $T_{p}\subset T_{B}$, we have $\Lambda=\widetilde{\Lambda}|_{T_{p}^{+}\times T_{p}^{-}}$.

A simple arc $\Gamma$ in $\mathbb{C}$ that extends to $\infty$ in both directions is called a chord-arc curve if it is the image of $\mathbb{R}$ under a bi-Lipschitz homeomorphism of $\mathbb{C}$. Chord-arc curves are locally rectifiable. Let $\widetilde{T}_{C}$ be the open subset of $T_{B}^{+}\times T_{B}^{-}$ consisting of BMO embeddings $\gamma:\mathbb{R}\to\mathbb{C}$ whose images are chord-arc curves. These embeddings are characterized by the property that $\gamma$ is locally absolutely continuous and $|\gamma^{\prime}|$ is an $A_{\infty}$-weight. We note that $T_{p}^{+}\times T_{p}^{-}\subset\widetilde{T}_{C}$. It has been shown in [39, Theorem 6.1] that $\widetilde{\Lambda}$ maps $\widetilde{T}_{C}$ biholomorphically onto its image.

Proposition 9.2**.**

$\widetilde{\Lambda}:\widetilde{T}_{C}\to{\rm BMO}(\mathbb{R})$* is a biholomorphic homeomorphism onto its image. In particular, the derivative $d\widetilde{\Lambda}$ is a Banach isomorphism onto ${\rm BMO}(\mathbb{R})$ at every point of $\widetilde{T}_{C}$.*

Concerning item (a) in the above steps, we have the following.

Proposition 9.3**.**

For the holomorphic injection $\Lambda:T_{p}^{+}\times T_{p}^{-}\to\widehat{B}_{p}(\mathbb{R})$ with $p\geq 1$, the image ${\rm Ran}\,d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$ of the derivative of $\Lambda$ at $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$ is given as the algebraic direct sum

[TABLE]

of the two closed subspaces, where $h^{+}=\pi(\mu^{+})$ and $h^{-}=\pi(\mu^{-})$ are the corresponding quasisymmetric homeomorphisms of $\mathbb{R}$.

Proof.

Under the direct sum decomposition of the tangent space ${\mathscr{T}}_{[\mu^{+}]}T_{p}^{+}\oplus{\mathscr{T}}_{[\mu^{-}]}T_{p}^{-}$ as in (10), we will show that

[TABLE]

To this end, we use relation (8) to obtain

[TABLE]

By fixing $[\mu^{+}]$, we take the partial derivative of this formula along the direction of $T_{p}^{-}$. Then,

[TABLE]

where $d_{([0],[\mu^{-}]\ast[\bar{\mu}^{+}]^{-1})}\Lambda$ can be regarded as the identity map on the tangent subspace ${\mathscr{T}}_{[\mu^{-}]\ast[\bar{\mu}^{+}]^{-1}}T_{p}^{-}\cong\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$. This implies that $d_{([\mu^{+}],[\mu^{-}])}\,\Lambda({\mathscr{T}}_{[\mu^{-}]}T_{p}^{-})=C_{h^{+}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))$. The other equation in (11) is similarly proved.

Finally, we show that $C_{h^{-}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-}))\cap C_{h^{+}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))=\{0\}$. Suppose that $\phi\in\widehat{B}_{p}(\mathbb{R})$ belongs to both $C_{h^{-}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-}))$ and $C_{h^{+}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))$. The derivative $d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$ is injective because

[TABLE]

and $d_{([\mu^{+}],[\mu^{-}])}\,\widetilde{\Lambda}$ is injective by Proposition 9.2 for $\widetilde{\Lambda}:\widetilde{T}_{C}\to{\rm BMO}(\mathbb{R})$. Hence, there is a unique tangent vector $u\in{\mathscr{T}}_{([\mu^{+}],[\mu^{-}])}(T_{p}^{+}\times T_{p}^{-})$ such that $d_{([\mu^{+}],[\mu^{-}])}\,\Lambda(u)=\phi$. However, by belonging of $\phi$, $u$ has to lie in both ${\mathscr{T}}_{[\mu^{-}]}T_{p}^{-}$ and ${\mathscr{T}}_{[\mu^{+}]}T_{p}^{+}$, and thus $u=o$. This implies that $\phi=0$. ∎

Remark 9.1**.**

After proving Theorem 9.1, we see that the above algebraic direct sum in fact coincides with $\widehat{B}_{p}(\mathbb{R})$ and thus obtain the topological direct sum decomposition

[TABLE]

Item (b) is proved by using the following lemma. The assumption of this statement will be verified by applying the corresponding result for $\widetilde{\Lambda}:\widetilde{T}_{C}\to{\rm BMO}(\mathbb{R})$.

Lemma 9.4**.**

If the real subspace $i{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ is contained in the image ${\rm Ran}\,d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$ of the derivative at $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$, then $d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$ is surjective, that is to say, ${\rm Ran}\,d_{([\mu^{+}],[\mu^{-}])}\,\Lambda=\widehat{B}_{p}(\mathbb{R})$.

Proof.

It suffices to show that ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})\subset{\rm Ran}\,d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$. We note that Proposition 9.3 implies that $C_{h^{+}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))\subset{\rm Ran}\,d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$.

We take any $\phi\in{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$. Since $C_{h^{+}}$ maps ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ isomorphically onto itself, $C_{h^{+}}^{-1}(\phi)$ also belongs to ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$. We consider its Szegö projection

[TABLE]

which is in $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})$. The application of $2C_{h^{+}}$ to this implies

[TABLE]

Here, $iC_{h^{+}}\circ\mathcal{H}\circ C_{h^{+}}^{-1}(\phi)\in i{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ also belongs to ${\rm Ran}\,d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$ by the assumption. Therefore, we have $\phi\in{\rm Ran}\,d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$. ∎

Remark 9.2**.**

Likewise to the above lemma, we can prove that if ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ is contained in ${\rm Ran}\,d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$, then $d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$ is surjective.

Having these preparatory arguments, we can provide the required proof as follows. Conducting item (c) is included in it.

Proof of Theorem 9.1.

Proposition 8.1 asserts that $\Lambda$ is a holomorphic injection. To show that $\Lambda$ is biholomorphic, we prove that the derivative $d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$ at every $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$ is surjective onto $\widehat{B}_{p}(\mathbb{R})$. Then, by the inverse mapping theorem (see [8, §7.18]), we will obtain the required claim.

Let $\phi=\Lambda([\mu^{+}],[\mu^{-}])\in\widehat{B}_{p}(\mathbb{R})$. We can find $\phi_{0}\in i{\rm Re}\,\widehat{B}_{p}(\mathbb{R})\cap{\rm Ran}\,\Lambda$ and $[\nu]\in T_{p}$ such that $Q_{h}(\phi_{0})=\phi$ with $\pi(\nu)=h$. Indeed, we take $[\nu]\in T_{p}$ such that $\Lambda([\nu],[\bar{\nu}])=\log h^{\prime}={\rm Re}\,\phi$ by Lemma 8.4, and set $\phi_{0}=iC_{h}^{-1}({\rm Im}\,\phi)$. Then,

[TABLE]

Since $Q_{h}$ preserves ${\rm Ran}\,\Lambda$ by relation (8), we see that $\phi_{0}\in{\rm Ran}\,\Lambda$.

We consider the derivative of $\Lambda$ at $([\mu^{+}_{0}],[\mu^{-}_{0}])=R_{[\nu]}^{-1}([\mu^{+}],[\mu^{-}])$, where $\Lambda([\mu^{+}_{0}],[\mu^{-}_{0}])=Q_{h}^{-1}\circ\Lambda([\mu^{+}],[\mu^{-}])=\phi_{0}$ by (8). For an arbitrary tangent vector $v\in i{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ at $\phi_{0}$, we take a sufficiently short segment $c(t)=\phi_{0}+tv$ in $i{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ defined for $t\in(-\varepsilon,\varepsilon)$. Then, under the biholomorphic homeomorphism $\widetilde{\Lambda}:\widetilde{T}_{C}\to{\rm BMO}(\mathbb{R})$ onto its image which contains $c(t)$, each point in the inverse image $(\widetilde{\Lambda}^{-1}\circ c)(t)$ corresponds to a chord-arc curve of arc-length parametrization generated by an element of $i{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$. In this case, we can apply [41, Proposition 5.5] to show that each $(\widetilde{\Lambda}^{-1}\circ c)(t)$ is a $p$-Weil–Petersson embedding in $T_{p}^{+}\times T_{p}^{-}$. Hence, the tangent vector $u=\frac{d}{dt}(\widetilde{\Lambda}^{-1}\circ c)(t)|_{t=0}$ at $([\mu_{0}^{+}],[\mu_{0}^{-}])$ belongs to ${\mathscr{T}}_{([\mu_{0}^{+}],[\mu_{0}^{-}])}(T_{p}^{+}\times T_{p}^{-})$ and satisfies $d_{([\mu_{0}^{+}],[\mu_{0}^{-}])}\widetilde{\Lambda}(u)=v$. Then, we use the fact that $d\widetilde{\Lambda}|_{{\mathscr{T}}_{([\mu_{0}^{+}],[\mu_{0}^{-}])}(T_{p}^{+}\times T_{p}^{-})}=d\Lambda$ (see [41, Claim 2]) to conclude that $v\in{\rm Ran}\,d_{([\mu_{0}^{+}],[\mu_{0}^{-}])}\,\Lambda$. Thus, we have $i{\rm Re}\,\widehat{B}_{p}(\mathbb{R})\subset{\rm Ran}\,d_{([\mu_{0}^{+}],[\mu_{0}^{-}])}\,\Lambda$.

Under this condition, Lemma 9.4 yields that $d_{([\mu_{0}^{+}],[\mu_{0}^{-}])}\,\Lambda$ is surjective. Hence, by the inverse mapping theorem, there exists some neighborhood $U_{0}$ of $\phi_{0}$ in $\widehat{B}_{p}(\mathbb{R})$ such that $U_{0}\subset{\rm Ran}\,\Lambda$, and $\Lambda^{-1}$ is holomorphic on $U_{0}$. Since

[TABLE]

this is also surjective (and hence isomorphic). Alternatively, we can construct the local holomorphic inverse $R_{[\nu]}\circ\Lambda^{-1}|_{U_{0}}\circ Q_{h}^{-1}$ on the neighborhood $Q_{h}(U_{0})$ of $\phi$. ∎

Finally, as an application of Theorem 9.1, we obtain a result about the real-analytic structure of the integrable Teichmüller space $T_{p}$ for $p\geq 1$. We restrict the biholomorphic map $\Lambda$ to the real-analytic submanifold ${\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})$, as in the setting of Theorem 1.1. By composing $\Lambda$ with the canonical real-analytical embedding $\iota:T_{p}\to{\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})$ given by $\iota([\mu])=([\mu],[\bar{\mu}])$, we have the following result. With some restrictions on $p$, it has appeared in [31, Theorem 2.3] and [41, Corollary 5.2].

Corollary 9.5**.**

For $p\geq 1$, the map $\Lambda\circ\iota:T_{p}\to{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ given by $h\mapsto\log h^{\prime}$ is a real-analytic diffeomorphism.

Thus, the integrable Teichmüller space $T_{p}$ is real-analytically equivalent to the entire real Banach space ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$. Hence, the complex Banach structure of $T_{p}$ is subordinate to the real Banach structure of ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$.

10. Analysis on curves via biholomorphic correspondence

The biholomorphic correspondence from Weil–Petersson embeddings in the Bers coordinates to the Besov space not only describes the structure of the integrable Teichmüller spaces but also provides a method of analysis on Weil–Petersson curves for classical real analysis problems. In this section, we first introduce the Cauchy transform and the Cauchy projection by considering the Cauchy integral on a Weil–Petersson curve $\Gamma=\gamma(\mathbb{R})$, where $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ is a $p$-Weil–Petersson embedding for $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$ with $p\geq 1$.

We define the Banach space of Besov functions on the $p$-Weil–Petersson curve $\Gamma=\gamma(\mathbb{R})$ by the push-forward of $\widehat{B}_{p}(\mathbb{R})$ by $\gamma$ and identify this pair. Namely,

[TABLE]

with norm $\|\gamma_{*}\phi\|_{\widehat{B}_{p}(\gamma)}=\|\phi\|_{\widehat{B}_{p}}$.

Remark 10.1**.**

Usually, the function spaces on the locally rectifiable curve $\Gamma$ is defined by using its arc-length parametrization. In our case, for a $p$-Weil–Petersson embedding $\gamma_{0}:\mathbb{R}\to\mathbb{C}$ with $\gamma_{0}(\mathbb{R})=\Gamma$ such that $\gamma_{0}$ is the arc-length parametrization of $\Gamma$, we may consider $\widehat{B}_{p}(\gamma_{0}(\mathbb{R}))$ in the above notation. However, the difference between $\gamma$ and $\gamma_{0}$ is given by the composition operator $C_{h}$ for a quasisymmetric homeomorphism $h\in T_{p}$ (as explained in the next section), and it can be controlled well. Hence, we adopt the representation $\widehat{B}_{p}(\gamma(\mathbb{R}))$ because it helps the arguments to be more natural when we consider dependence of operators on $\gamma$.

Let $\Omega^{+}$ and $\Omega^{-}$ be the left and the right domains bounded by $\Gamma$, respectively. Let $F^{\pm}:\mathbb{H}^{\pm}\to\Omega^{\pm}$ be the normalized Riemann mappings. Then, we also define the Banach space of analytic Besov functions on $\Omega^{\pm}$ by the push-forward of $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})$ by $F^{\pm}$ and identify these pairs. Namely,

[TABLE]

with norm $\|(F^{\pm})_{*}\Phi^{\pm}\|_{\widehat{\mathcal{B}}_{p}(\Omega^{\pm})}=\|\Phi^{\pm}\|_{\widehat{\mathcal{B}}_{p}}$.

For any functions $(F^{\pm})_{*}\Phi^{\pm}\in\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$, their boundary extensions to $\Gamma$ is defined by

[TABLE]

where the Riemann mappings $F^{\pm}$ are assumed to extend to the homeomorphisms of $\mathbb{R}$ onto $\Gamma$.

Proposition 10.1**.**

It holds that $E_{\Gamma}^{\pm}(\widehat{\mathcal{B}}_{p}(\Omega^{\pm}))\subset\widehat{B}_{p}(\gamma(\mathbb{R}))$ for any $\gamma=\gamma([\mu^{+}],[\mu^{-}])$. Moreover, these trace operators $E_{\Gamma}^{\pm}$ are Banach isomorphisms onto their images, where their operator norms are estimated in terms of the norms of the composition operators $C_{h^{\pm}}$ for $h^{\pm}=\pi(\mu^{\pm})$.

Proof.

The norm on $\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$ is induced from $\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm})$ by $F^{\pm}$ whereas the norm of $\widehat{B}_{p}(\gamma(\mathbb{R}))$ is induced from $\widehat{B}_{p}(\mathbb{R})$ by $\gamma$. Because $\gamma=F^{\pm}\circ h^{\pm}$, the differences between $\|E^{\pm}\|$ and $\|E_{\Gamma}^{\pm}\|$ are caused by the composition operators $C_{h^{\pm}}$. ∎

By this proposition, we see that $E_{\Gamma}^{\pm}(\widehat{\mathcal{B}}_{p}(\Omega^{\pm}))$ are closed subspaces of $\widehat{B}_{p}(\gamma(\mathbb{R}))$. Moreover, under the Banach isomorphism $E_{\Gamma}^{\pm}$, we may identify $E_{\Gamma}^{\pm}(\widehat{\mathcal{B}}_{p}(\Omega^{\pm}))$ with $\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$. Hence, we regard $\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$ as closed subspaces of $\widehat{B}_{p}(\gamma(\mathbb{R}))$ without noticing $E_{\Gamma}^{\pm}$ hereafter. However, the Banach isomorphisms $E_{\Gamma}^{\pm}$ are not uniform but they depend on $\gamma$ as shown in Proposition 10.1.

Definition 10.1**.**

The Cauchy transform of $\psi\in\widehat{B}_{p}(\gamma(\mathbb{R}))$ on the $p$-Weil–Petersson curve $\Gamma=\gamma(\mathbb{R})$ (oriented by $\mathbb{R}$ consistently with $\pm$) is defined by the singular integral

[TABLE]

where $dz=\gamma^{\prime}(t)dt$. The Cauchy integrals of $\psi$ on $\Gamma$ are defined by

[TABLE]

which are holomorphic functions on $\Omega^{\pm}$.

The point-wise (a.e.) convergence of the Cauchy transform and the Cauchy integrals for $\widehat{B}_{p}(\gamma(\mathbb{R}))$ are guaranteed by using the regularized kernel. By the Privalov theorem (see [15, p.431]), if the Cauchy transform $({\mathcal{H}}_{\Gamma}\psi)(\xi)$ exists a.e. on $\Gamma$, then the Cauchy integrals $({P}^{\pm}_{\Gamma}\psi)(\zeta)$ have non-tangential limits a.e. on $\Gamma$, and vice versa. The boundary extensions $E_{\Gamma}^{\pm}({P}^{\pm}_{\Gamma}\psi)$ of ${P}^{\pm}_{\Gamma}\psi$ to $\Gamma$ are also denoted by the same symbol ${P}^{\pm}_{\Gamma}\psi$ and called the Cauchy projections of $\psi$. Later, we see that ${P}^{\pm}_{\Gamma}\psi\in\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$, and the Cauchy transform ${\mathcal{H}}_{\Gamma}\psi$ is in $\widehat{B}_{p}(\gamma(\mathbb{R}))$.

The Plemelj formula (also named after Sokhotski and Privalov) for the Riemann–Hilbert problem asserts the following relation between the Cauchy transform and the Cauchy projections. This is a generalization of the relation between the Hilbert transform and the Szegö projections.

Proposition 10.2**.**

For a function $\psi\in\widehat{B}_{p}(\gamma(\mathbb{R}))$ on the $p$-Weil–Petersson curve $\Gamma=\gamma(\mathbb{R})$, the Cauchy transform ${\mathcal{H}}_{\Gamma}$ and Cauchy projections $P^{\pm}_{\Gamma}$ satisfy

[TABLE]

In other words,

[TABLE]

holds.

For $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$, we have obtained the images of the tangent spaces ${\mathscr{T}}_{[\mu^{+}]}T_{p}^{+}$ and ${\mathscr{T}}_{[\mu^{-}]}T_{p}^{-}$ under the derivative $d_{([\mu^{+}],[\mu^{-}])}\Lambda$ as in (11), and since $d_{([\mu^{+}],[\mu^{-}])}\Lambda$ is surjective, which is the main ingredient of the proof of Theorem 9.1, we have the topological direct sum decomposition (13) of $\widehat{B}_{p}(\mathbb{R})$ as mentioned in Remark 9.1. Then, every $\phi\in\widehat{B}_{p}(\mathbb{R})$ is uniquely represented by $\phi=\phi^{+}+\phi^{-}$ for $\phi^{+}\in C_{h^{+}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+}))$ and $\phi^{-}\in C_{h^{-}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{-}))$, and this defines the bounded linear projections

[TABLE]

Theorem 10.3**.**

In the Besov space $\widehat{B}_{p}(\gamma(\mathbb{R}))$ on the $p$-Weil–Petersson curve $\Gamma=\gamma(\mathbb{R})$ of $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ for $p\geq 1$, the Cauchy projections $P^{\pm}_{\Gamma}$ satisfy

[TABLE]

In particular, $P^{\pm}_{\Gamma}$ maps $\widehat{B}_{p}(\gamma(\mathbb{R}))$ onto $\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$, they are bounded linear operators, and their operator norms are estimated in terms of $\|C_{h^{\pm}}\|$.

Proof.

Let $\phi=\gamma_{*}^{-1}(\psi)=\psi\circ\gamma$ for $\psi\in\widehat{B}_{p}(\gamma(\mathbb{R}))=\gamma_{*}(\widehat{B}_{p}(\mathbb{R}))$. Then, $\phi\in\widehat{B}_{p}(\mathbb{R})$ and $P^{\pm}_{([\mu^{+}],[\mu^{-}])}\circ\gamma_{*}^{-1}(\psi)=\phi^{\pm}\in C_{h^{\pm}}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm}))$. Therefore,

[TABLE]

since $\gamma=F^{\pm}\circ h^{\pm}$ and $\widehat{\mathcal{B}}_{p}(\Omega^{\pm})=F^{\pm}_{*}(\widehat{\mathcal{B}}_{p}(\mathbb{H}^{\pm}))$ for the Riemann mappings $F^{\pm}:\mathbb{H}^{\pm}\to\Omega^{\pm}$. By the definition of the projections $P^{\pm}_{([\mu^{+}],[\mu^{-}])}$, we have

[TABLE]

Here, if we consider a measurable function $\Psi_{1}$ on $\mathbb{C}$ defined by holomorphic functions $\gamma_{*}\circ P^{+}_{([\mu^{+}],[\mu^{-}])}\circ\gamma_{*}^{-1}(\psi)$ on $\Omega^{+}$ and $\gamma_{*}\circ P^{-}_{([\mu^{+}],[\mu^{-}])}\circ\gamma_{*}^{-1}(\psi)$ on $\Omega^{-}$, it is locally integrable and is of growth order $\Psi_{1}(z)=o(|z|)$ as $z\to\infty$.

The Cauchy projections $P^{\pm}_{\Gamma}(\psi)$ satisfy the same properties as $\gamma_{*}\circ P^{\pm}_{([\mu^{+}],[\mu^{-}])}\circ\gamma_{*}^{-1}(\psi)$. They are holomorphic functions on $\Omega^{\pm}$ such that the boundary extension satisfies $P^{+}_{\Gamma}(\psi)+P^{-}_{\Gamma}(\psi)=\psi$ by Proposition 10.2. Moreover, the measurable function $\Psi_{2}$ on $\mathbb{C}$ defined by $P^{\pm}_{\Gamma}(\psi)$ on $\Omega^{\pm}$ is locally integrable and is of growth order $\Psi_{2}(z)=o(|z|)$ as $z\to\infty$. These follow from the arguments in [29, Section 3]. Then, $\bar{\partial}$-derivative of $\Psi_{1}-\Psi_{2}$ is [math] on $\mathbb{C}$ in the distribution sense, and hence it is holomorphic on $\mathbb{C}$. The growth order $\Psi_{1}(z)-\Psi_{2}(z)=o(|z|)$ as $z\to\infty$ forces it to be a constant function. Thus, we have $P^{\pm}_{\Gamma}(\psi)=\gamma_{*}\circ P^{\pm}_{([\mu^{+}],[\mu^{-}])}\circ\gamma_{*}^{-1}(\psi)$. ∎

Remark 10.2**.**

If we assume the fact that $P^{\pm}_{\Gamma}(\psi)$ belongs to ${\mathcal{B}}_{p}(\Omega^{\pm})$ for $\psi\in B_{p}(\gamma(\mathbb{R}))$ taking a larger $p>1$ (see [22, Corollary 5.4]), then the proof of Theorem 10.3 becomes easier. Indeed, we have only to apply the direct sum decomposition (13) to

[TABLE]

for $\phi=\psi\circ\gamma\in B_{p}(\mathbb{R})$.

Remark 10.3**.**

$P^{\pm}_{\Gamma}\circ\gamma_{*}=\gamma_{*}\circ P^{\pm}_{([\mu^{+}],[\mu^{-}])}$ maps $\widehat{B}_{p}(\mathbb{R})$ onto $\widehat{\mathcal{B}}_{p}(\Omega^{\pm})$ and ${\rm BMO}(\mathbb{R})$ onto ${\rm BMOA}(\Omega^{\pm})$. When $[\mu^{+}]=[0]$ or $[\mu^{-}]=[0]$, this coincides with what is called the Faber operator though the curve $\Gamma$ has assumed certain smoothness according to the function spaces in our case. See [21, 22, 43] for related arguments to ours. In our setting, we see not only the boundedness of the operator but also its holomorphic dependence when the embeddings vary in the Teichmüller space as is discussed in the next section.

This leads to the Calderón theorem (see [10]) for Weil–Petersson curves. For chord-arc curves, the argument on the Bers coordinates for BMO embeddings in [39] can be also developed to the methods described here, which will appear in [23]. The boundedness of the Cauchy transform and projections in ${\rm BMO}(\gamma(\mathbb{R}))$ is verified in [21]. Results in a more general setting of the Cauchy transform and projections in $B_{p}(\gamma(\mathbb{R}))$ for $p>1$ can be found in [22].

Corollary 10.4**.**

The Cauchy projections $P_{\Gamma}^{\pm}$ on the $p$-Weil–Petersson curve $\Gamma=\gamma(\mathbb{R})$ for $p\geq 1$ are associated with the topological direct sum decomposition

[TABLE]

Remark 10.4**.**

Both the conjugate $C_{h^{\pm}}\circ P^{\pm}\circ C_{h^{\pm}}^{-1}$ of the Szegö projections and the standardized Cauchy projections $P^{\pm}_{([\mu^{+}],[\mu^{-}])}$ map ${\widehat{B}}_{p}(\mathbb{R})$ onto $C_{h^{\pm}}({\mathcal{B}}_{p}(\mathbb{H}^{\pm}))\subset{\widehat{B}}_{p}(\mathbb{R})$, but they are different. In the special case $p=2$, ${B}_{2}(\mathbb{R})$ is a Hilbert space and we can think of the conjugate operator for any bounded linear operator. Then, the Szegö projection is an orthogonal projection and self-adjoint, but the Cauchy projection is not in general. They are related as follows:

[TABLE]

This is the same as the Kerzman–Stein formula (see [18]) for $L_{2}(\mathbb{R})$ and the Hardy space ${H}_{2}(\mathbb{H}^{\pm})$.

11. Holomorphic dependence of the Cauchy transform

In this section, we consider how the Cauchy transforms ${\mathcal{H}}_{\Gamma}$ vary when $\Gamma=\gamma(\mathbb{R})$ move according to $\gamma=\gamma([\mu^{+}],[\mu^{-}])$. To formulate this problem, we take the conjugate of ${\mathcal{H}}_{\Gamma}$ so that it acts on ${\widehat{B}}_{p}(\mathbb{R})$.

Let $\gamma=\gamma([\mu^{+}],[\mu^{-}])$ be a $p$-Weil–Petersson embedding for $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$ with $p\geq 1$. By Proposition 10.2, the relationship between the Cauchy transform ${\mathcal{H}}_{\Gamma}$ on the $p$-Weil–Petersson curve $\Gamma=\gamma(\mathbb{R})$ and the Cauchy projections $P_{\Gamma}^{\pm}$ is given as

[TABLE]

Moreover, Theorem 10.3 shows $P^{\pm}_{\Gamma}=\gamma_{*}\circ P^{\pm}_{([\mu^{+}],[\mu^{-}])}\circ\gamma_{*}^{-1}$. Then, pulling back ${\mathcal{H}}_{\Gamma}$ to $\mathbb{R}$ via $\gamma$, we define

[TABLE]

which is a Banach automorphism of $\widehat{B}_{p}(\mathbb{R})$. We refer to ${\mathcal{H}}_{([\mu^{+}],[\mu^{-}])}$ as the standardization of the Cauchy transform ${\mathcal{H}}_{\Gamma}$. More explicitly,

[TABLE]

for $\phi\in{\widehat{B}}_{p}(\mathbb{R})$.

For $([0],[0])\in T_{p}^{+}\times T_{p}^{-}$, ${\mathcal{H}}_{([0],[0])}$ coincides with the Hilbert transform $\mathcal{H}$. For $([\mu],[\bar{\mu}])\in{\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})$, ${\mathcal{H}}_{([\mu],[\bar{\mu}])}$ is the conjugate of $\mathcal{H}$ by the composition operator $C_{h}$ for $h=\pi(\mu)\in T_{p}$. Indeed, (20) applied to the case of $\Gamma=h(\mathbb{R})=\mathbb{R}$ with $\gamma([\mu],[\bar{\mu}])=H(\mu)|_{\mathbb{R}}=h$ yields ${\mathcal{H}}_{([\mu],[\bar{\mu}])}=C_{h}\circ{\mathcal{H}}\circ C_{h}^{-1}$.

Let ${\mathcal{L}}(\widehat{B}_{p}(\mathbb{R}))$ be the Banach space of all bounded linear operators $\widehat{B}_{p}(\mathbb{R})\to\widehat{B}_{p}(\mathbb{R})$ equipped with the operator norm. We consider the map

[TABLE]

defined by $([\mu^{+}],[\mu^{-}])\mapsto{\mathcal{H}}_{([\mu^{+}],[\mu^{-}])}$.

Theorem 11.1**.**

$\eta:T_{p}^{+}\times T_{p}^{-}\to{\mathcal{L}}(\widehat{B}_{p}(\mathbb{R}))$* is holomorphic for $p\geq 1$.*

Proof.

Due to formula (20), it suffices to show that $P^{\pm}_{([\mu^{+}],[\mu^{-}])}$ depend holomorphically on $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$. We consider the derivative $d_{([\mu^{+}],[\mu^{-}])}\,\Lambda$ of the biholomorphic map $\Lambda$ and its inverse

[TABLE]

According to this direct sum decomposition of the tangent space, we denote the canonical projections by

[TABLE]

Then, by the proof of formula (11) for Proposition 9.3, we see that the standardizations of the Cauchy projections turn out to be

[TABLE]

Since $\Lambda$ is biholomorphic, these operators depend holomorphically on $([\mu^{+}],[\mu^{-}])\in T_{p}^{+}\times T_{p}^{-}$. ∎

Corresponding results for BMO embeddings with chord-arc image are found in [10], though without involving the biholomorphic map $\Lambda$, only demonstrating holomorphic dependence near the origin. A complete extension is also possible, which is in [23].

As an application of the holomorphic dependence of ${\mathcal{H}}_{([\mu^{+}],[\mu^{-}])}$, we consider the modification of the Coifman–Meyer theorem which was originally proved for chord-arc curves.

We define $Z_{p}=i\widehat{B}_{p}(\mathbb{R})\cap{\rm Ran}\,\Lambda$ as the real-analytic submanifold of ${\rm Ran}\,\Lambda$ consisting of purely imaginary-valued Besov functions (which is an open subset of the real Banach subspace $i\widehat{B}_{p}(\mathbb{R})$) as used in the proof of Theorem 9.1. For $\psi\in Z_{p}$,

[TABLE]

serves as the arc-length parameterization of the $p$-Weil–Petersson curve $\gamma_{0}(\mathbb{R})$. In general, for any $p$-Weil–Petersson embedding

[TABLE]

determined by $\phi\in{\rm Ran}\,\Lambda$, we take the quasisymmetric homeomorphism

[TABLE]

and set $\psi=i\,{\rm Im}\,\phi\circ h^{-1}\in Z_{p}$. Then, the arc-length parameter $\gamma_{0}$ determined by $\psi$ allows $\gamma$ to be expressed as the reparameterization of $\gamma_{0}$ by $h$, that is, $\gamma=\gamma_{0}\circ h$. See [41, Lemma 4.2].

We define $Y_{p}=\widehat{\mathcal{B}}_{p}(\mathbb{H}^{+})\cap{\rm Ran}\,\Lambda$ as the complex submanifold of ${\rm Ran}\,\Lambda$ consisting of analytic Besov functions on $\mathbb{H}^{+}$. For $\varphi\in Y_{p}$, the corresponding $p$-Weil–Petersson embedding

[TABLE]

when applied as above, can be expressed as the reparameterization of the arc-length parameter $\gamma_{0}$ by a quasisymmetric map $h$ so that $f=\gamma_{0}\circ h$. This correspondence between $f$ and $\gamma_{0}$ is bijective, thus defining a mapping from $Z_{p}$ to $Y_{p}$, which will be recognized later as a homeomorphism. Similarly, the correspondence from the pair $(f,\gamma_{0})$ to their reparameterization $h$ allows the definition of a mapping from $Z_{p}$ to ${\rm Re}\,\widehat{B}_{p}(\mathbb{R})$.

To investigate these mappings on $T_{p}^{+}\times T_{p}^{-}$ through the biholomorphic homeomorphism $\Lambda$, we define

[TABLE]

Here, $\rho$ is continuous due to the topological group structure of $T_{p}$ given in Proposition 7.2, and $\delta$ is the projection to the symmetric axis, which is real-analytic. The unique decomposition

[TABLE]

corresponds to the decomposition of the $p$-Weil–Petersson embedding $g=\gamma([\mu^{+}],[\mu^{-}])$ into $g=f\circ h$, where $f=\gamma(\rho([\mu^{+}],[\mu^{-}]))$ is the boundary extension of the conformal homeomorphism of $\mathbb{H}^{+}$ to $\mathbb{R}$, and $h=\gamma(\delta([\mu^{+}],[\mu^{-}]))$ is a quasisymmetric homeomorphism of $\mathbb{R}$.

We transform the two maps defined on the submanifolds in $\widehat{B}_{p}(\mathbb{R})$ into those in $T_{p}^{+}\times T_{p}^{-}$ by $\Lambda$. The map $Z_{p}\to Y_{p}$ corresponds to

[TABLE]

and the map $Z_{p}\to{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ corresponds to

[TABLE]

For the map $Z_{p}\to Y_{p}$ and $\rho_{0}$, we know the following. This has been proved in [41, Proposition 6.5] for $p>1$, and the proof is essentially the same in this case.

Proposition 11.2**.**

$\rho_{0}:\Lambda^{-1}(Z_{p})\to\{[0]\}\times T_{p}^{-}=\Lambda^{-1}(Y_{p})$* is a homeomorphism for $p\geq 1$.*

Proof.

Since the map $Z_{p}\to Y_{p}$ is bijective, so is $\rho_{0}$. Since $\rho$ is continuous, so is $\rho_{0}$. Hence, it suffices to show that $\rho_{0}^{-1}$ is continuous. To this end, we consider the conjugation of $\rho_{0}^{-1}$ by $\Lambda$, that is, $\Lambda\circ\rho^{-1}_{0}\circ\Lambda^{-1}:Y_{p}\to Z_{p}$. For any $\varphi\in Y_{p}$, we have

[TABLE]

with $h(x)=\int_{0}^{x}\exp({\rm Re}\,\varphi(t))dt$. Since $h\mapsto h^{-1}$ is continuous in $T_{p}$ by Proposition 7.2, we see that $\widehat{B}_{p}(\mathbb{R})\times T_{p}\to\widehat{B}_{p}(\mathbb{R})$ defined by $(\phi,h)\mapsto C_{h}^{-1}(\phi)$ is continuous by Lemma 11.3 below. Thus, $\Lambda\circ\rho^{-1}_{0}\circ\Lambda^{-1}$ is continuous. ∎

Lemma 11.3**.**

The map $\widehat{B}_{p}(\mathbb{R})\times T_{p}\to\widehat{B}_{p}(\mathbb{R})$ defined by $(\phi,h)\mapsto C_{h}(\phi)$ is continuous. In particular, a sequence of bounded linear operators $C_{h_{n}}$ on $\widehat{B}_{p}(\mathbb{R})$ converges to $C_{h}$ strongly as $h_{n}\to h$ in $T_{p}$.

Proof.

This can be seen in [41, Theorem 6.3] and the remark after that in the case $p>1$. The same argument can be applied to $p=1$. ∎

Remark 11.1**.**

In the circumstances of the above lemma, we do not know whether or not $C_{h_{n}}$ converges to $C_{h}$ in the operator norm.

Coifman and Meyer [11] investigated the map $\delta_{0}$ for the BMO embeddings with chord-arc image. We translate their results to the case of Weil–Petersson embeddings by formalizing the following. The proof follows immediately from what has been demonstrated earlier. For $p>1$, this has been obtained in [41, Theorem 5.4].

Theorem 11.4**.**

$\delta_{0}:\Lambda^{-1}(Z_{p})\to{\rm Sym}\,(T_{p}^{+}\times T_{p}^{-})=\Lambda^{-1}({\rm Re}\,\widehat{B}_{p}(\mathbb{R}))$* is a real-analytic diffeomorphism onto its image for $p\geq 1$.*

Proof.

Since $\delta_{0}$ is a real-analytic injection as is shown in the case $p>1$, it suffices to show that its inverse $\delta_{0}^{-1}$ is also real-analytic. As before, we consider the conjugate

[TABLE]

Then, for any $\phi\in{\rm Re}\,\widehat{B}_{p}(\mathbb{R})\cap{\rm Ran}\,(\Lambda\circ\delta_{0})$, we have

[TABLE]

by formula (*) in the proof of [41, Theorem 5.4]. Because the standardized Cauchy transform ${\mathcal{H}}_{\Lambda^{-1}(\phi)}$ depends real-analytically on $\phi\in{\rm Re}\,\widehat{B}_{p}(\mathbb{R})$ by Theorem 11.1, we see that $\Lambda\circ\delta^{-1}_{0}\circ\Lambda^{-1}$ is real-analytic. ∎