Boundary behaviour of harmonic functions on hyperbolic manifolds

Camille Petit (1)

1 JYU - University of Jyväskylä

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 Calderón-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.

Mots clés: Harmonic functions; Gromov hyperbolic manifolds; Brownian motion; boundary at infinity; Fatou's theorem; non-tangential convergence

Domaines: Géométrie métrique [math.MG]; Analyse classique [math.CA]; Probabilités [math.PR]

Camille Petit. Boundary behaviour of harmonic functions on hyperbolic manifolds. 2013. ⟨hal-00794012⟩