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Radial variation of Bloch functions on the unit ball of \(\mathbb{R}^d\). (English) Zbl 1442.30062

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 [P. W. Jones and 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.

MSC:

30H30 Bloch spaces
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions

Citations:

Zbl 0882.30020
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