## 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-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 $\sigma \Lambda _\alpha ^{p}$.## References

- Peter L. Duren,
*Theory of $H^{p}$ spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655** - G. H. Hardy and J. E. Littlewood,
*A convergence criterion for Fourier series*, Math. Z.**28**(1928), no. 1, 612–634. MR**1544980**, DOI 10.1007/BF01181186 - J. E. Littlewood,
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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