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If $U$ is a $C^{\infty}$ function with compact support in the plane, we let $u$ be its restriction to the unit circle $\mathbb{S}$, and denote by $U_i,\,U_e$ the harmonic extensions of $u$ respectively in the interior and the exterior of $\mathbb S$ on the Riemann sphere. About a hundred years ago, Douglas has shown that \begin{align*} \iint_{\mathbb{D}}|\nabla U_i|^2(z)dxdy&= \iint_{\bar{\mathbb{C}}\backslash\bar{\mathbb{D}}}|\nabla U_e|^2(z)dxdy &= \frac{1}{2\pi}\iint_{\mathbb S\times\mathbb S}\left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|, \end{align*} thus giving three ways to express the Dirichlet norm of $u$. On a rectifiable Jordan curve $\Gamma$ we have obvious analogues of these three expressions, which will of course not be equal in general. The main goal of this paper is to show that these $3$ (semi-)norms are equivalent if and only if $\Gamma$ is a chord-arc curve., Comment: 19 pages, 1 figure, Accepted by Math. Ann

We consider the embedding of integrable Teichm\"uller spaces $T_p$ into analytic Besov spaces by using pre-Schwarzian derivatives. Unlike the case of the Bers embedding by Schwarzian derivatives, there is a big difference between the cases $p>1$ and $p=1$. In this paper, we focus on the case $p=1$ and generalize the previous results obtained for $p>1$. This gives a unified approach to the complex analytic theory of integrable Teichm\"uller spaces $T_p$ for all $p \geq 1$.

We verify that the $p$-integrable Teichm\"uller space $T_p$ admits the canonical complex Banach manifold structure for any $p \geq 1$. Moreover, we characterize a quasisymmetric homeomorphism corresponding to an element of $T_p$ in terms of the $p$-Besov space for any $p>1$.

We investigate strongly symmetric homeomorphisms of the real line which appear in harmonic analysis aspects of quasiconformal Teichm\"uller theory. An element in this class can be characterized by a property that it can be extended quasiconformally to the upper half-plane so that its complex dilatation induces a vanishing Carleson measure. However, differently from the case on the unit circle, strongly symmetric homeomorphisms on the real line are not preserved under either the composition or the inversion. In this paper, we present the difference and the relation between these two cases. In particular, we show that if uniform continuity is assumed for strongly symmetric homeomorphisms of the real line, then they are preserved by those operations. We also show that the barycentric extension of uniformly continuous one induces a vanishing Carleson measure and so do the composition and the inverse of those quasiconformal homeomorphisms of the upper half-plane.

We consider a univalent analytic function $f$ on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on $f$, we show that there exists a chordal Loewner chain initiated from $f$ until some finite time, and this Loewner chain defines a quasiconformal extension of $f$ over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result developed on the disk by Becker and also the generalization of the Ahlfors-Weill formula. As an application of this quasiconformal extension, we complete the characterization of an element of the VMO-Teichm\"uller space on the half-plane using the vanishing Carleson measure condition induced by the (pre-)Schwarzian derivative.

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

We consider the space of chord-arc curves on the plane passing through the infinity with their parametrization $\gamma$ on the real line, and embed this space into the product of the BMO Teichm\"uller spaces. The fundamental theorem we prove about this representation is that $\log \gamma'$ also gives a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Riemann mapping and its dependence on the arc-length parametrization. As a consequence, we can solve a conjecture of Katznelson, Nag, and Sullivan in 1990 by showing that this dependence is not continuous.

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

We consider the correspondence between the space of $p$-Weil-Petersson curves $\gamma$ on the plane and the $p$-Besov space of $u=\log \gamma'$ on the real line for $p >1$. We prove that the variant of the Beurling-Ahlfors extension defined by using the heat kernel yields a holomorphic map for $u$ on a domain of the $p$-Besov space to the space of $p$-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichm\"uller projection from the space of $p$-integrable Beltrami coefficients to the $p$-Weil-Petersson Teichm\"uller space., Comment: To appear in Journal of Geometric Analysis

We present a new characterization of Muckenhoupt $A_{\infty}$-weights whose logarithm is in $\operatorname{VMO}(\mathbb{R})$ in terms of vanishing Carleson measures on $\mathbb{R}_+^2$ and vanishing doubling weights on $\mathbb{R}$. This also gives a novel description of strongly symmetric homeomorphisms on the real line by using a geometric quantity., Comment: In the new version, we adjusted some proofs of the main theorem and deleted the appendix. This version is accepted by PAMS

With the use of real-variable techniques, we construct a weight function $\omega$ on the interval $[0, 2\pi)$ that is doubling and satisfies $\log \omega$ is a BMO function, but which is not a Muckenhoupt weight ($A_\infty$). Applications to the BMO-Teichm\"uller space and the space of chord-arc curves are considered., Comment: 12 pages

We investigate a variant of the Beurling-Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (i.e. its derivative is an ${\rm A}_\infty$-weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.

We interpolate a new family of Teichm\"uller spaces $T_{\sharp}^X$ between the universal Teichm\"uller space $T$ and its little subspace $T_0$, which we call the Teichm\"uller space of piecewise symmetric homeomorphisms. This is defined by prescribing a subset $X$ of the unit circle. The inclusion relation of $X$ induces a natural inclusion of $T_{\sharp}^X$, and an approximation of $T$ is given by an increasing sequence of $T_{\sharp}^X$. In this paper, we discuss the fundamental properties of $T_{\sharp}^X$ from the viewpoint of the quasiconformal theory of Teichm\"uller spaces. We also consider the quotient space of $T$ by $T_{\sharp}^X$ as an analog of the asymptotic Teichm\"uller space., Comment: arXiv admin note: text overlap with arXiv:1908.06618

We introduce the concept of a new kind of symmetric homeomorphisms on the unit circle, which is derived from the generalization of symmetric homeomorphisms on the real line. By the investigation of the barycentric extension for this class of circle homeomorphisms and the biholomorphic automorphisms induced by trivial Beltrami coefficients, we endow a complex Banach manifold structure on the space of those generalized symmetric homeomorphisms.

We show an example of a symmetric homeomorphism $h$ of the real line $\mathbb{R}$ onto itself such that $h^{-1}$ is not symmetric. This implies that the set of all symmetric self-homeomorphisms of $\mathbb{R}$ does not constitute a group under the composition. We also deal with strongly symmetric self-homeomorphisms of $\mathbb{R}$ along the same line. These results reveal the difference of the sets of such self-homeomorphisms of the real line from those of the unit circle., Comment: Title is changed and Section 5 is added (v2)

The paper presents some recent results on the BMO Teichm\"uller space, its subspaces and quotient spaces. We first consider the chord-arc curve subspace and prove that every element of the BMO Teichm\"uller space is represented by its finite composition. Moreover, we show that these BMO Teichm\"uller spaces have affine foliated structures induced by the VMO Teichm\"uller space. By which, their quotient spaces have natural complex structures modeled on the quotient Banach space. Then, a complete translation-invariant metric is introduced on the BMO Teichm\"uller space and is shown to be a continuous Finsler metric in a special case.

We study the relation between the boundary of a simply connected domain being Ahlfors-regular and the invariance of Carleson measures under the push-forward operator induced by a conformal mapping from the unit disk onto the domain. As an application, we characterize the chord-arc curve with small norm and the asymptotically smooth curve in terms of the complex dilatation of some quasiconformal reflection with respect to the curve., Comment: 11 pages

In a paper of Cui and Zinsmeister the equivalence among three definitions of BMO-Teichm\"uller spaces associated with a Fuchsian group was proven using the Douady-Earle extension operator. In this paper, we show that these equivalences are actually biholomorphisms. It was further shown in the above quoted paper that the Douady-Earle extension operator is continuous at the origin. We improve this result by showing G\^ateaux-differentiability at this point., Comment: 19 pages

Following Semmes and Zinsmeister, we continue the study of Carleson measures and their invariance under pull-back and push-forward operators. We also study the analogous statements for vanishing Carleson measures. As an application, we show that some quotient space of the space of chord-arc curves has a natural complex structure., Comment: 21 pages