$H^p$-Norm estimates of the partial derivatives and Schwarz lemma for $\alpha$-harmonic functions

Abstract

Suppose $\alpha >-1$ and $1\le p\le \infty$. Let $f=P_{\alpha }[F]$ be an $\alpha$-harmonic mapping on $\mathbb{D}$ with the boundary $F$ being absolute continuous and $\dot{F}\in L^p(0,2\pi )$, where $\dot{F}(e^{i\theta }):=\frac{dF(e^{i\theta })}{d\theta }$. In this paper, we investigate the membership of $f_z$ and $f_{\overline{z}}$ in the space ${\mathcal{H}}_{\mathcal{G}}^p(\mathbb{D})$, the generalized Hardy space. We prove, if $\alpha >0$, then both $f_z$ and $f_{\overline{z}}$ are in ${\mathcal{H}}_{\mathcal{G}}^p(\mathbb{D})$. If $\alpha <0$, then $f_z$ and $f_{\overline{z}}\in {\mathcal{H}}_{\mathcal{G}}^p(\mathbb{D})$ if and only if $f$ is analytic. Finally, we investigate a Schwartz Lemma for $\alpha$-harmonic functions.