Note on real and imaginary parts of harmonic quasiregular mappings

TL;DR

The paper investigates whether the real and imaginary parts of planar harmonic $K$-quasiregular mappings share the same growth behavior and boundary regularity. It provides a self-contained argument showing that $M_p(r,u)=O((1-r)^{-eta})$ implies $M_p(r,v)=O((1-r)^{-eta})$ for all $p>0$ and $eta>0$, and it gives a refined bound when $eta=0$ and $0<p<1$; it also proves that boundary Hölder continuity of $u$ transfers to $v$ for any $ alpha ightarrow(0,1)$, independent of $K$. The methods hinge on decomposing $f$ as $h+ar{g}$, introducing $F=h+g$, and using elementary derivative estimates to control $h'$ and $g'$, while a Poisson-integral/Hardy–Littlewood-type approach underpins the boundary regularity result. Collectively, these results extend Riesz-type symmetry from analytic to harmonic quasiregular mappings and illuminate boundary smoothness for the real and imaginary parts within Hardy-space theory.

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.

Theorems & Definitions (14)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem 1
• proof
• Lemma A
• Theorem 2
• proof
• Theorem E