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Radial variation of Bloch functions on the unit ball of ℝ^d

Paul F. X. Müller Institute of Analysis, Johannes Kepler University, Linz, Austria Katharina Riegler Institute of Analysis, Johannes Kepler University, Linz, Austria

Analysis of PDEs mathscidoc:2203.03005

Arkiv for Matematik, 58, (1), 161-178, 2020.4
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 ∫_{[0,x]}|∇b(ζ)|e^{b(ζ)}d|ζ|<∞.. In the second part of the paper, we show that the area or volume integral ∫_{B^d}|∇u(w)|p(w,θ)dA(w) for positive harmonic functions u is bounded by the value cu(0) for at least one θ. The integral is also transferred to simply connected domains and interpreted from the point of view of stochastics. Several emerging open problems are presented.
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@inproceedings{paul2020radial,
  title={Radial variation of Bloch functions on the unit ball of ℝ^d},
  author={Paul F. X. Müller, and Katharina Riegler},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310171354972935938},
  booktitle={Arkiv for Matematik},
  volume={58},
  number={1},
  pages={161-178},
  year={2020},
}
Paul F. X. Müller, and Katharina Riegler. Radial variation of Bloch functions on the unit ball of ℝ^d. 2020. Vol. 58. In Arkiv for Matematik. pp.161-178. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310171354972935938.
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