论文标题
谐波映射和谐波准映射的部分衍生物的规范估计值
Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings
论文作者
论文摘要
假设$ p \ geq1 $，$ w = p [f] $是单位磁盘$ \ mathbb {d} $满足$ f $的谐波映射，绝对是连续的，$ \ dot {f} \ in l^p（0，2π）$，在哪里$ \ dot {f}（e^{it}）= \ frac {\ mathrm {d}}} {\ mathrm {d} t} t} f（e^{it}）$。在本文中，我们获得了伯格曼的规范估算$ W $的部分衍生物，即$ \ | | | w_z \ | _ {l^p} $和$ \ | \ | \ overline {此外，如果$ w $是$ \ mathbb {d} $的谐波Quasiregular映射，那么我们表明$ w_z $和$ \ overline {w _ {w _ {\ bar {z}}}} $在Hardy Space $ H^p $中，其中$ 1 \ leq P \ leq P \ leq p \ leq leq leq \ efty $。还获得了相应的Hardy Norm估计，$ \ | w_z \ | _ {p} $和$ \ | \ | \ overline {w _ {\ bar {z}}}} \ | _ {p} $。
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.
