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Transactions of the American Mathematical Society
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
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, $ \vert f\vert< 1 $, in $ \vert z\vert< 1 $, the condition that is necessary and sufficient for $ f$ to be continuous on $ \vert z\vert\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 $ \vert z\vert=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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0662051-5
PII: S 0002-9947(1982)0662051-5
Keywords: Lipschitz functions, mean Lipschitz functions, Hardy class, non-Euclidean hyperbolic distance
Article copyright: © Copyright 1982 American Mathematical Society