Note on real and imaginary parts of harmonic quasiregular mappings

TL;DR

This paper investigates the growth and smoothness properties of the real and imaginary parts of harmonic quasiregular mappings in the unit disk, extending classical results and establishing new growth equivalence and boundary regularity results.

Contribution

It proves that harmonic quasiregular mappings have real and imaginary parts with comparable growth of integral means and boundary smoothness, extending Riesz theorems to this class.

Findings

Real and imaginary parts have the same growth order of integral means.

The real and imaginary parts share the same boundary smoothness degree.

Extension of Riesz type theorems to harmonic quasiregular mappings.

Abstract

If $f=u+iv$ is analytic in the unit disk $\mathbb{D}$, it is known that the integral means $M_p(r,u)$ and $M_p(r,v)$ have the same order of growth. This is false if $f$ is a (complex-valued) harmonic function. However, we prove that the same principle holds if we assume, in addition, that $f$ is $K$-quasiregular in $\mathbb{D}$. The case $0<p<1$ is particularly interesting, and is an extension of the recent Riesz type theorems for harmonic quasiregular mappings by several authors. Further, we proceed to show that the real and imaginary parts of a harmonic quasiregular mapping have the same degree of smoothness on the boundary.