Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings

TL;DR

This paper establishes norm estimates for the partial derivatives of harmonic and harmonic quasiregular mappings in the unit disk, providing bounds in Bergman and Hardy spaces for derivatives under certain regularity conditions.

Contribution

It introduces new Bergman and Hardy space norm estimates for derivatives of harmonic and harmonic quasiregular mappings, extending previous results to broader classes of functions.

Findings

Bergman norm estimates for $w_z$ and $ar{w}_{ar{z}}$ when $F$ is absolutely continuous with $ ext{dot}F extin L^p$

Harmonic quasiregular mappings have derivatives in Hardy spaces with norm bounds

Explicit Hardy space norm estimates for derivatives of harmonic quasiregular mappings

Abstract

Suppose $p\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{D}$ satisfying $F$ is absolutely continuous and $\dot{F}\in L^p(0, 2\pi)$, where $\dot{F}(e^{it})=\frac{\mathrm{d}}{\mathrm{d}t}F(e^{it})$. In this paper, we obtain Bergman norm estimates of the partial derivatives for $w$, i.e., $\|w_z\|_{L^p}$ and $\|\overline{w_{\bar{z}}}\|_{L^p}$, where $1\leq p<2$. Furthermore, if $w$ is a harmonic quasiregular mapping of $\mathbb{D}$, then we show that $w_z$ and $\overline{w_{\bar{z}}}$ are in the Hardy space $H^p$, where $1\leq p\leq\infty$. The corresponding Hardy norm estimates, $\|w_z\|_{p}$ and $\|\overline{w_{\bar{z}}}\|_{p}$, are also obtained.