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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Smoothness of the boundary values of functions bounded and holomorphic in the disk


Author: Shinji Yamashita
Journal: Trans. Amer. Math. Soc. 272 (1982), 539-544
MSC: Primary 30D50
DOI: https://doi.org/10.1090/S0002-9947-1982-0662051-5
MathSciNet review: 662051
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Abstract: The non-Euclidean counterparts of Hardy-Littlewood’s theorems on Lipschitz and mean Lipschitz functions are considered. Let $1\le p < \infty$ and $0 < \alpha \le 1$. For $f$ holomorphic and bounded, $|f|< 1$, in $|z|< 1$, the condition that is necessary and sufficient for $f$ to be continuous on $|z|\le 1$ with the boundary function $f({e^{it}}) \in \sigma {\Lambda _\alpha }$, the hyperbolic Lipschitz class. Furthermore, the condition that the $p$th mean of $f^{\ast }$ on the circle $|z|=r < 1$ is $O({(1 - r)^{\alpha - 1}})$ is necessary and sufficient for $f$ to be of the hyperbolic Hardy class $H_\sigma ^{p}$ and for the radial limits to be of the hyperbolic mean Lipschitz class $\sigma \Lambda _\alpha ^{p}$.


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Keywords: Lipschitz functions, mean Lipschitz functions, Hardy class, non-Euclidean hyperbolic distance
Article copyright: © Copyright 1982 American Mathematical Society