The first partial derivatives of generalized harmonic functions

Suppose $\alpha ,\beta \in \mathbb{R}\mathrm{\backslash }{\mathbb{Z}}^{-}$ such that $\alpha +\beta >-1$ and $1\le p\le \mathrm{\infty }$. 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 ${\mathrm{\partial }}_{z}u$ and ${\mathrm{\partial }}_{\bar{z}}u$ in the space $H_G^p(\mathbb{D})$, the generalized Hardy space. We prove, if $\alpha +\beta >0$, then both ${\mathrm{\partial }}_{z}u$ and ${\mathrm{\partial }}_{\bar{z}}u$ are in $H_G^p(\mathbb{D})$. For $\alpha +\beta <0$, we show if ${\mathrm{\partial }}_{z}u$ or ${\mathrm{\partial }}_{\bar{z}}u\in H_G^1(\mathbb{D})$ then $u=0$ or $u$ is a polyharmonic function.