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# Random Harmonic Functions in Growth Spaces and Bloch-type Spaces

Published:2013-08-26
Printed: Apr 2014
• Kjersti Solberg Eikrem,
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
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## Abstract

Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v$ that fulfill a doubling condition. In the two-dimensional case let $u (re^{i\theta},\xi) = \sum_{j=0}^\infty (a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta)$ where $\xi =\{\xi_{ji}\}$ is a sequence of random subnormal variables and $a_{ji}$ are real; in higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients $a_{ji}$ which imply that $u$ is in $h^\infty_v(\mathbf B)$ almost surely. Our estimate improves previous results by Bennett, Stegenga and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces, which generalizes results by Anderson, Clunie and Pommerenke and by Guo and Liu.
 Keywords: harmonic functions, random series, growth space, Bloch-type space
 MSC Classifications: 30B20 - Random power series 31B05 - Harmonic, subharmonic, superharmonic functions 30H30 - Bloch spaces 42B05 - Fourier series and coefficients