H$\mathbf{\"o}$lder continuity of QCH mappings from the unit ball to a domain with $C^1$ boundary

TL;DR

This paper proves that quasiconformal harmonic mappings from the harmonic β-Bloch space to domains with C^1 boundary are globally Hölder continuous with a coefficient independent of the mapping, extending classical results from the complex plane.

Contribution

It establishes Hölder continuity for quasiconformal harmonic mappings from harmonic β-Bloch spaces to C^1 boundary domains, generalizing previous planar results.

Findings

Mappings are globally α-Hölder continuous for α<1-β

Hölder coefficient is independent of the mapping and β

Results extend classical complex plane theorems

Abstract

We prove that every quasiconformal mapping from the harmonic $\beta$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $\alpha$-H\"older continuous for $\alpha<1-\beta$, with the H\"older coefficient that does not depend neither on the mapping nor on $\beta$. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for $\alpha <1$. This extends some results from the complex plane obtained by Warschawski in \cite{Warschawski} for conformal mappings and Kalaj in \cite{Kalaj6} for quasiconformal harmonic mappings.