Zygmund theorem for harmonic quasiregular mappings

TL;DR

The paper extends Zygmund-type inequalities to $K$-quasiregular harmonic mappings in the unit disk and unit ball. It establishes that if the real part of such a map lies in the Hardy space with a log-corrected integrability ($ ext{Re} f\in \mathbf{h}\,\log^+\mathbf{h}$) and $\text{Im} f(0)=0$, then the map belongs to $\mathbf{h}^1$ with a quantitative bound $\|f\|_1\le C(K)(1+\|\text{Re} f\log^+|\text{Re} f|\|_1)$. In higher dimensions, for $f=(f_1,...,f_n)$ with $f_1>0$ and $f_1\in \mathbf{h}\,\log^+\mathbf{h}$, the inequality $\|f\|_1-|f(0)|\le (n-1)K^2(\int_{\mathbb{S}}(f_1\log f_1-f_1(0)\log f_1(0))d\sigma)$ is shown to be asymptotically sharp as $K\to 1$. The proofs combine Calderón-type estimates for harmonic components with Green's formula and subharmonicity arguments, and the sharpness is demonstrated via a simple affine test mapping. These results generalize classical Zygmund and M. Riesz-type inequalities to quasiregular harmonic settings and illuminate the dependence on the quasiregularity constant $K$.

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.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• proof : Proof of Theorem \ref{['th1']}
• Lemma 2.1
• proof
• Lemma 2.2
• proof : Proof of Lemma \ref{['larmi']}
• proof : Proof of Theorem \ref{['zyg']}