  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.
