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