Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 3

Conformally natural extensions of vector fields and applications

Pages: 1147 – 1186

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n3.a10


Jinhua Fan (School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, China)

Jun Hu (Department of Mathematics and Ph.D. Program in Mathematics, Brooklyn College of CUNY, Brooklyn, New York, U.S.A)


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 $\mathbb{S}^1$.


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

2010 Mathematics Subject Classification

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

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