Muckenhoupt weights and Lindel\"of theorem for harmonic mappings

TL;DR

This paper generalizes classical harmonic measure results to quasiconformal harmonic mappings, establishing conditions for boundary smoothness based on the argument function's extension, using Muckenhoupt weights and Lindel"of theorem extensions.

Contribution

It extends Lavrentiev's result on harmonic measure equivalence to quasiconformal harmonic mappings and characterizes boundary smoothness via argument function extension.

Findings

Harmonic measure and arc-length measure are $A_ abla$-equivalent in chord-arc Jordan domains.

The argument function of quasiconformal harmonic mappings extends continuously iff the image domain has $C^1$ boundary.

The paper generalizes Lindel"of's theorem to quasiconformal harmonic mappings.

Abstract

We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are $A_\infty$ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindel\"of to the class of quasiconformal (q.c.) harmonic mappings by proving the following assertion. Assume that $f$ is a quasiconformal harmonic mapping of the unit disk $\mathbf{U}$ onto a Jordan domain. Then the function $A(z)=\arg(\partial_\varphi(f(z))/z)$ where $z=re^{i\varphi}$, is well-defined and smooth in $\mathbf{U}^*=\{z: 0<|z|<1\}$ and has a continuous extension to the boundary of the unit disk if and only if the image domain has $C^1$ boundary.