Hölder Continuity and Differentiability Almost Everywhere of $(K_1,K_2)$-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 Hölder condition with exponent $\hat{I}\pm$ on compact subsets of its domain, where \begin{align} α=\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.

This is an English version of our published Chinese article, available online at ( http://www.actamath.com/Jwk_sxxb_cn/CN/abstract/abstract21741.shtml ) and ( http://123.57.41.99/Jwk_sxxb_cn//CN/article/downloadArticleFile.do?attachType=PDF&id=21741 )