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

Abstract

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.