Estimates of partial derivatives for harmonic functions on the unit disc

TL;DR

This paper extends previous results on the regularity of harmonic functions' derivatives in the unit disk, showing new conditions for their belonging to Hardy spaces based on boundary data and Hilbert transforms.

Contribution

It generalizes prior work by establishing that harmonic derivatives are in Hardy spaces for all p in (1,∞) without extra conditions, and characterizes the cases p=1 and p=∞.

Findings

Harmonic derivatives belong to Hardy spaces for p in (1,∞) without additional assumptions.

For p=1 and p=∞, membership in Hardy spaces is characterized by the Hilbert transform of boundary derivatives.

Provides explicit formulas for harmonic derivatives in terms of boundary data and Hilbert transforms.

Abstract

Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ is an absolute continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L^p(\mathbb{T})$, where $\dot{F}(e^{it}) = \frac{d}{dt} F(e^{it})$ and $p \in [1,\infty]$. Recently, Chen et al. (J. Geom. Anal., 2021) extended Zhu's results (J. Geom. Anal., 2020) and proved that (i) if $f$ is a harmonic mapping and $1 \leq p < \infty$, then $f_z$ and $\overline{f_{\overline{z}}} \in B^p(\mathbb{D})$, the Bergman spaces of $\mathbb{D}$. Moreover, (ii) under additional conditions as $f$ being harmonic quasiregular mapping in \cite{Zhu} or $f$ being harmonic elliptic mapping in \cite{CPW}, they proved that $f_z$ and $\overline{f_{\overline{z}}}\in H^p(\mathbb{D})$, the Hardy space of $\mathbb{D}$, for $1 \leq p \leq \infty$. The aim of this paper is to extend these results by showing that (ii) holds for $p\in(1,\infty)$ without any extra conditions and for $p=1$ or $p=\infty$, $f_z$ and $\overline{f_{\bar{z}}}\in H^p(\mathbb{D})$ if and only if $H(\dot{F})\in L^p(\mathbb{T})$, the Hilbert transform of $\dot{F}$ and in that case, it yields $zf_z=P[\frac{\dot{F}+iH(\dot{F})}{2i}]$.