Estimates of the first partial derivatives of $(\alpha ,\beta )$-harmonic functions on the unit disc

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 function 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.