Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings

Suppose $p\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{D}$ satisfying $F$ is absolutely continuous and $\dot{F}\in L^p(0, 2π)$, 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.

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