Boundary Behavior of Harmonic Functions on Gromov Hyperbolic Manifolds

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 nontangential convergence coming from the density of energy are shown: at almost all points of the boundary, the harmonic function converges nontangentially 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 nontangential properties of convergence, boundedness, and finiteness of energy are equivalent at almost every point of the boundary.