H\"older Continuity and Differentiability Almost Everywhere of $(K_1, K_2)$-Quasiregular Mappings

TL;DR

This paper proves that $(K_1, K_2)$-quasiregular mappings are Hölder continuous with specific exponents and are differentiable almost everywhere, extending classical regularity results for these mappings.

Contribution

It establishes Hölder continuity and almost everywhere differentiability for $(K_1, K_2)$-quasiregular mappings using Morrey's Lemma and isoperimetric inequality, generalizing known properties.

Findings

Hölder continuity with explicit exponents depending on $K_1$ and $K_2$

Differentiability almost everywhere of these mappings

Extension of classical regularity results to $(K_1, K_2)$-quasiregular mappings

Abstract

This paper deals with $(K_1, K_2)$-quasiregular mappings. It is shown, by Morrey's Lemma and isoperimetric inequality, that every $(K_1, K_2)$-quasiregular mapping satisfies a H\"older condition with exponent $\alpha$ on compact subsets of its domain, where \begin{align} \alpha=\begin{cases} 1/K_1, & \text{for } K_1>1, \\ \text{any positive number less than } 1, & \text{for } K_1=1 \text{ and } K_2>0, \\ 1, & \text{for } K_1=1 \text{ and } K_2=0, \\ 1, & \text{for } K_1<1,\\ \end{cases} \end{align} Differentiability almost everywhere of $(K_1, K_2)$-quasiregular mappings is also derived.