Conformally natural extensions of vector fields and applications

In this paper, we study an integral operator $L_0$ extending tangent vector fields $V$ along the unit circle $\mathbb{S}^1$ to tangent vector fields $L_0(V)$ defined on the closure of the open unit disk $\mathbb{D}$. We first show that $L_0$ is conformally natural. Then we show: (1) the cross-ratio distortion norm ${\lVert V \lVert}_{cr}$ of $V$ on $\mathbb{S}^1$ is equivalent to ${\lVert \overline{\partial}L_0 (V) \rVert}_{\infty}$; (2) $\overline{\partial}L_0 (V)$ is uniformly vanishing near the boundary of $\mathbb{D}$ if and only if $V$ satisfies the little Zygmund bounded condition; (3) for each $0 \lt \alpha \lt 1$, $\overline{\partial} L_0(V)(z) = O((1- {\lvert z \rvert})^{\alpha)}$ if and only if $V$ is $C^{1+\alpha}$-smooth. As applications, the collection of $V$ with ${\lVert V \rVert}_{cr} \lt \infty$ (resp. being uniformly vanishing near the boundary) recapitulates a known model of the tangent space of the universal Teichmüller space $T (\mathbb{D})$ (resp. the little Teichmüller space $T_0 (\mathbb{D})$); the collection of $V$ with ${\lVert V \rVert}_{cr} \lt \infty$ and satisfying a group compatible condition characterizes the tangent space of the Teichmüller space $T(\mathcal{R})$ of a hyperbolic Riemann surface $\mathcal{R}$; the collection of the $C^{1+\alpha}$-smooth vector fields $V$ provides a model for the tangent space of the Teichmüller space $T^\alpha_0 (\mathbb{D})$ of the $C^{1+\alpha}$ diffeomorphisms of $\mathbb{S}^1$.

Keywords

conformally natural extension, Zygmund norm, Teichmüller space, little Teichmüller space, Hölder continuity

2010 Mathematics Subject Classification

Primary 30C62, 30E25, 30F60. Secondary 30C40.

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The first-named author’s work is supported by NNSF of China (No. 11871215).

The second-named author’s work is partially supported by PSC-CUNY grants, and by a fellowship leave of CUNY in Spring 2018 and Spring 2019.

Received 29 November 2021

Received revised 4 May 2022

Accepted 9 May 2022

Published 24 July 2022