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Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings (2007.12827v1)

Published 25 Jul 2020 in math.CV

Abstract: Suppose $p\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{D}$ satisfying $F$ is absolutely continuous and $\dot{F}\in Lp(0, 2\pi)$, 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., $|w_z|{Lp}$ and $|\overline{w{\bar{z}}}|{Lp}$, 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 $Hp$, 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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