Zygmund theorem for harmonic quasiregular mappings

TL;DR

This paper extends Zygmund's theorem to harmonic quasiregular mappings in the unit disk and ball, establishing bounds involving the real part and providing asymptotically sharp inequalities as the quasiregularity constant approaches 1.

Contribution

It introduces a Zygmund-type inequality for harmonic quasiregular mappings and derives sharp bounds in higher dimensions for such mappings.

Findings

Established a Zygmund inequality with a constant C(K) for harmonic quasiregular mappings.

Derived an asymptotically sharp inequality in higher dimensions as K approaches 1.

Provided bounds involving the real part and logarithmic terms of the mappings.

Abstract

Let $K\ge 1$. We prove Zygmund theorem for $K-$quasiregular harmonic mappings in the unit disk $\mathbb{D}$ in the complex plane by providing a constant $C(K)$ in the inequality $$\|f\|_{1}\le C(K)(1+\|\mathrm{Re}\,(f)\log^+ |\mathrm{Re}\, f|\|_1),$$ provided that $\mathrm{Im}\,f(0)=0$. Moreover for a quasiregular harmonic mapping $f=(f_1,\dots, f_n)$ defined in the unit ball $\mathbb{B}\subset \mathbb{R}^n$, we prove the asymptotically sharp inequality $$\|f\|_{1}-|f(0)|\le (n-1)K^2(\|f_1\log f_1\|_1- f_1(0)\log f_1(0)),$$ when $K\to 1$, provided that $f_1$ is positive.