Carleson Measure of Harmonic Schwarzian Derivatives Associated with a Finitely Generated Fuchsian Group of the Second Kind

Guangming Hu,Yutong Liu,Yu Sun,Xinjie Qian,Dumitru Motreanu

doi:10.1155/2021/5523454

Abstract

Let S H f be the Schwarzian derivative of a univalent harmonic function f in the unit disk D , compatible with a finitely generated Fuchsian group G of the second kind. We show that if S H f 2 1 − z 2 3 d x d y satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G , then S H f 2 1 − z 2 3 d x d y is a Carleson measure in D .

Highlights

Throughout this paper, we adopt the conventional symbols, D = fz : jzj < 1g and Bðz, rÞ, to denote the unit disk in the extended complex plane Cb and the disk with center z and radius r, respectively
Let ω = g′/h′ be the dilatation of f = h + g. It follows from [1] that, if a locally univalent harmonic function f = h + g is sense-preserving, its analytic part h is locally univalent and ω = g′/h′ is analytic with jωj < 1
Huo [7] considered a Beltrami coefficient μ in D compatible with a finitely generated Fuchsian group G of the second kind and showed that if ðjμj/ð1 − jzj2Þ1 − jzj2Þdx dy satisfies the Carleson condition on Fð∞Þ, ðjμj/ð1 − jzj2Þ1 − jzj2Þdxdy is a Carleson measure in D

Summary

Introduction

Let ω = g′/h′ be the (second complex) dilatation of f = h + g It follows from [1] that, if a locally univalent harmonic function f = h + g is sense-preserving, its analytic part h is locally univalent and ω = g′/h′ is analytic with jωj < 1. On the basis of the above definitions, we call a locally univalent harmonic function f compatible with a Fuchsian group G if and only if f ∘ A = f , for any A ∈ G. Huo [7] considered a Beltrami coefficient μ in D compatible with a finitely generated Fuchsian group G of the second kind and showed that if ðjμj/ð1 − jzj2Þ1 − jzj2Þdx dy satisfies the Carleson condition on Fð∞Þ, ðjμj/ð1 − jzj2Þ1 − jzj2Þdxdy is a Carleson measure in D.

Some Lemmas

Proof of Theorem 1