Random harmonic functions in growth spaces and Bloch-type spaces

TL;DR

This paper characterizes when random harmonic functions in growth and Bloch-type spaces belong almost surely, improving previous estimates and generalizing known results in harmonic analysis.

Contribution

It provides new conditions on coefficients for random harmonic functions to be in growth spaces, with sharper estimates and broader applicability to Bloch-type spaces.

Findings

Conditions on coefficients for almost sure inclusion in growth spaces.

Improved and sharp estimates compared to previous work.

Generalization of results to Bloch-type spaces.

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 $u (re^{i\ta},\xi) = \sum_{j=0}^\infty (a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta)$ where $\xi =\{\xi_{ji}\}%_{k=0}^\infty $ 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.