Estimates of partial derivatives for harmonic functions on the unit disc

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 $1\le p<\infty$, 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 $1\le p\le \infty$. 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 $z{f}_z=P[\frac{\dot{F}+iH(\dot{F})}{2i}]$.