Quasiregular families bounded in $L^p$ and elliptic estimates

TL;DR

The paper proves that families of quasiregular mappings bounded in L^p are normal and shows how elliptic estimates imply higher regularity and quasiregularity of derivatives, enhancing understanding of their regularity properties.

Contribution

It establishes normality of L^p-bounded quasiregular families and links elliptic estimates to higher regularity and quasiregularity of derivatives.

Findings

L^p-bounded quasiregular families are normal.

Elliptic estimates on functional differences imply higher regularity.

Complex gradients of such functions are quasiregular.

Abstract

We prove that a family of quasiregular mappings of a domain $\Omega$ which are uniformly bounded in $L^p$ for some $p>0$ form a normal family. From this we show how an elliptic estimate on a functional differences implies all directional derivatives, and thus the complex gradient to be quasiregular. Consequently the function enjoys much higher regularity than apriori assumptions suggest.