Suppose $p\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{ D}$ satisfying $F$ is absolutely continuous and $\dot{ F}\in L^p(0, 2\pi)$, where $\dot{ F}(e^{ it})=\frac{ \mathrm{ d}}{ \mathrm{ d}t}F(e^{ it})$. In this paper, we obtain Bergman norm estimates of the partial derivatives for $w$, i.e., $\|w_z\|_{ L^p}$ and $\|\overline{ w_{ \bar{ z}}}\|_{ L^p}$, where $1\leq p<2$. Furthermore, if $w$ is a harmonic quasiregular mapping of $\mathbb{ D}$, then we show that $w_z$ and $\overline{ w_{ \bar{ z}}}$ are in the Hardy space $H^p$, where $1\leq p\leq\infty$. The corresponding Hardy norm estimates, $\|w_z\|_{ p}$ and $\|\overline{ w_{ \bar{ z}}}\|_{ p}$, are also obtained.
Categories: math.CV