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

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 $\alpha$-Hölder continuous for $\alpha <1-\beta$, with the Hölder 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.

arXiv comment12 pages, no figures

References in corpus (1)

• Harmonic quasiconformal mappings between ${\mathcal{C}}^1$ smooth Jordan domains