Radial variation of Bloch functions on the unit ball of \(\mathbb{R}^d\) (Q2180948)

The authors prove that if \(b\) is a Bloch function on the unit ball \(B\) in \(\mathbb R^d\) (i.e., \(b\) is harmonic and \(\sup_{z\in B}|\nabla b(z)|(1-|z|)<\infty\) so that \(b\) is Lipschitz with respect to the hyperbolic metric) then there is a point \(x\) on the unit sphere such that \[ \int_{[0,x]} |\nabla b(\zeta)|e^{b(\zeta)}\, d|\zeta| <\infty. \] This result has been established in [\textit{P. W. Jones} and \textit{P. F. X. Müller}, Math. Res. Lett. 4, No. 2--3, 395--400 (1997; Zbl 0882.30020)] in the case \(d=2\). In addition they show that if \(u\) is a positive harmonic function on the unit ball \(B\) in \(\mathbb R^d\) then there is a point \(\theta\) on the unit sphere such that \[ \int_B |\nabla u(w)| p(w,\theta) dA(w) < cu(0), \] where \(p\) is the Poisson kernel and \(c\) is a constant depending on \(d\) only. Using conformal invariance the authors transfer this result to arbitrary simply connected domains and provide a stochastic interpretation using Brownian motion. The work is complemented with several connected open problems.