Hardy-Littlewood property and $\alpha$-quasihyperbolic metric
Commun. Korean Math. Soc. 2020 Vol. 35, No. 1, 243-250
Published online January 31, 2020
Ki Won Kim, Jeong Seog Ryu
Silla University; Hongik University
Abstract : Hardy and Littlewood found a relation between the smoothness of the radial limit of an analytic function on the unit disk $D \subset {\mathbb C}$ and the growth of its derivative. It is reasonable to expect an analytic function to be smooth on the boundary if its derivative grows slowly, and conversely. Gehring and Martio showed this principle for uniform domains in ${\mathbb R}^{2}$. Astala and Gehring proved quasiconformal analogue of this principle for uniform domains in ${\mathbb R}^n$. We consider \textit{$\alpha$-quasihyperbolic metric}, $k_D^{\alpha}$ and we extend it to proper domains in ${\mathbb R}^n$.
Keywords : Hardy-Littlewood property, quasiconformal mapping, quasihyperbolic metric
MSC numbers : Primary 30C65
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