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

TL;DR

This paper extends previous results on the regularity of harmonic mappings' derivatives, showing that certain integrability conditions imply membership in Bergman or Hardy spaces, and clarifies the limits of these results.

Contribution

It generalizes earlier theorems to broader p ranges and introduces harmonic elliptic mappings, expanding the understanding of derivative estimates in harmonic analysis.

Findings

For 1 ≤ p < ∞, the derivative estimates hold for harmonic mappings.

The results do not extend to p = ∞ for harmonic mappings.

Theorems remain valid when replacing quasiregular with elliptic harmonic mappings.

Abstract

Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ being absolutely continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L_p(0, 2\pi)$, where $\dot{F}(e^{it})=\frac{d}{dt} F(e^{it})$ and $p\geq 1$. Recently, the author in \cite{Zhu} proved that $(1)$ if $f$ is a harmonic mapping and $1\leq p< 2$, then $f_{z}$ and $\overline{f_{\overline{z}}}\in \mathcal{B}^{p}(\mathbb{D}),$ the classical Bergman spaces of $\mathbb{D}$ \cite[Theorem 1.2]{Zhu}; $(2)$ if $f$ is a harmonic quasiregular mapping and $1\leq p\leq \infty$, then $f_{z},$ $\overline{f_{\overline{z}}}\in \mathcal{H}^{p}(\mathbb{D}),$ the classical Hardy spaces of $\mathbb{D}$ \cite[Theorem 1.3]{Zhu}. These are the main results in \cite{Zhu}. The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, \cite[Theorem 1.2]{Zhu} is true when $1\leq p< \infty$. Also, we show that \cite[Theorem 1.2]{Zhu} is not true when $p=\infty$. Second, we demonstrate that \cite[Theorem 1.3]{Zhu} still holds true when the assumption $f$ being a harmonic quasiregular mapping is replaced by the weaker one $f$ being a harmonic elliptic mapping.