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April 2020 Radial variation of Bloch functions on the unit ball of $\mathbb{R}^d$
Paul F. X. Müller, Katharina Riegler
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Ark. Mat. 58(1): 161-178 (April 2020). DOI: 10.4310/ARKIV.2020.v58.n1.a10


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.


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.


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Paul F. X. Müller. Katharina Riegler. "Radial variation of Bloch functions on the unit ball of $\mathbb{R}^d$." Ark. Mat. 58 (1) 161 - 178, April 2020.


Received: 14 January 2019; Revised: 21 November 2019; Published: April 2020
First available in Project Euclid: 16 January 2021

Digital Object Identifier: 10.4310/ARKIV.2020.v58.n1.a10

Primary: 30H30 , 31A20 , 31B25

Keywords: Bloch functions , radial variation

Rights: Copyright © 2020 Institut Mittag-Leffler


Vol.58 • No. 1 • April 2020
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