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Radial limits of inner functions and Bloch spaces

Author: Evgueni Doubtsov
Journal: Proc. Amer. Math. Soc. 136 (2008), 2177-2182
MSC (2000): Primary 32A40; Secondary 30D40
Published electronically: February 20, 2008
MathSciNet review: 2383523
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Abstract: Let $ f$ be an inner function in the unit ball $ B_n \subset\mathbb{C}^n$, $ n\ge 1$. Assume that

$\displaystyle \sup_{z\in B_n} \frac{\vert\mathcal{R} f(z)\vert(1-\vert z\vert^2)^{1+\beta}}{\left(1-\vert f(z)\vert^2 \right)^2} < \infty, $

where $ \beta\in (0,1)$ and $ \mathcal{R} f$ is the radial derivative. Then, for all $ \alpha\in\partial B_1$, the set $ \{\zeta\in \partial B_n:\, f^*(\zeta) =\alpha\}$ has a non-zero real Hausdorff $ t^{2n-1-\beta}$-content, and it has a non-zero complex Hausdorff $ t^{n-\beta}$-content.

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Additional Information

Evgueni Doubtsov
Affiliation: St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia

Keywords: Boundary behavior, inner functions
Received by editor(s): January 26, 2007
Received by editor(s) in revised form: April 28, 2007
Published electronically: February 20, 2008
Additional Notes: The author is partially supported by RFFI grant no. 08-01-00358-a and by the Russian Science Support Foundation.
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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