The first partial derivatives of generalized harmonic functions

Abstract: Suppose $\alpha,\beta \in \mathbb{R}\backslash \mathbb{Z}^-$ such that $\alpha+\beta>-1$ and $1\leq p \leq \infty$. Let $u=P_{\alpha,\beta}[f]$ be an $(\alpha,\beta)$-harmonic mapping on $\mathbb{D}$, the unit disc of $\mathbb{C}$, with the boundary $f$ being absolutely continuous and $\dot{f}\in L^p(0,2\pi)$, where $\dot{f}(e^{i\theta}):=\frac{d}{d\theta}f(e^{i\theta})$. In this paper, we investigate the membership of the partial derivatives $\partial_z u$ and $\partial_{\overline{z}}u$ in the space $H_G^{p}(\mathbb{D})$, the generalized Hardy space. We prove, if $\alpha+\beta>0$, then both $\partial_z u$ and $\partial_{\overline{z}}u$ are in $H_G^{p}(\mathbb{D})$. For $\alpha+\beta<0$, we show if $\partial_z u$ or $\partial_{\overline{z}}u \in H_G^1(\mathbb{D})$ then $u=0$ or $u$ is a polyharmonic function.