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

Abstract

Suppose $p\ge 1$, $w=P[F]$ 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., $\Vert w_z{\Vert }_{L^p}$ and $\Vert \overline{w_{\bar{z}}}{\Vert }_{L^p}$, where $1\le p<2$. 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 $1\le p\le \infty$. The corresponding Hardy norm estimates, $\Vert w_z{\Vert }_p$ and $\Vert \overline{w_{\bar{z}}}{\Vert }_p$, are also obtained.

17 pages