Radial variation of Bloch functions on the unit ball of $\mathbb{R}^d$

In [9] Anderson’s conjecture was proven by comparing values of Bloch functions with the variation of the function. We extend that result on Bloch functions from two to arbitrary dimension and prove that\[\int \limits_{[0, x]} \lvert \nabla b(\zeta) \rvert e^{b(\zeta)} \: d \lvert \zeta \rvert \lt \infty \; \textrm{.}\]In the second part of the paper, we show that the area or volume integral\[\int \limits_{B^d} \lvert \nabla u(w) \rvert p(w,\theta) \: dA(w)\]for positive harmonic functions $u$ is bounded by the value $cu(0)$ for at least one $\theta$. The integral is also transferred to simply connected domains and interpreted from the point of view of stochastics. Several emerging open problems are presented.

Keywords

radial variation, Bloch functions

2010 Mathematics Subject Classification

30H30, 31A20, 31B25

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This paper is part of the second named author’s PhD thesis written at the Department of Analysis, Johannes Kepler University Linz. The research has been supported by the Austrian Science foundation (FWF) Pr.Nr P28352-N32.

Received 14 January 2019

Received revised 21 November 2019

Accepted 17 January 2020

Published 21 July 2022