Boundary behaviour of harmonic functions on hyperbolic manifolds

TL;DR

This paper investigates the boundary behavior of harmonic functions on Gromov hyperbolic manifolds, establishing criteria for non-tangential convergence and proving a Calderón-Stein type theorem relating convergence, boundedness, and energy finiteness.

Contribution

It introduces new boundary convergence criteria for harmonic functions on hyperbolic manifolds and proves a Calderón-Stein theorem linking various boundary properties.

Findings

Almost everywhere non-tangential convergence characterized by energy density

Equivalence of convergence, boundedness, and energy finiteness at boundary points

Extension of Fatou's theorem to hyperbolic manifolds

Abstract

Let $M$ be a complete simply connected manifold which is in addition Gromov hyperbolic, coercive and roughly starlike. For a given harmonic function on $M$, a local Fatou Theorem and a pointwise criteria of non-tangential convergence coming from the density of energy are shown: at almost all points of the boundary, the harmonic function converges non-tangentially if and only if the supremum of the density of energy is finite. As an application of these results, a Calder\'on-Stein Theorem is proved, that is, the non-tangential properties of convergence, boundedness and finiteness of energy are equivalent at almost every point of the boundary.