Suppose $\alpha>-1$ and $1\leq p \leq \infty$. 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.
Categories: math.CV