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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Smoothness of the boundary values of functions bounded and holomorphic in the disk
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by Shinji Yamashita PDF
Trans. Amer. Math. Soc. 272 (1982), 539-544 Request permission

Abstract:

The non-Euclidean counterparts of Hardy-Littlewoodā€™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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • 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