Potential Theory on Gromov Hyperbolic Spaces

TL;DR

This paper extends potential theory to a broad class of Schrödinger operators on Gromov hyperbolic spaces, unifying geometric and analytic frameworks for applications in singular spaces.

Contribution

It generalizes Ancona's potential theory to Gromov hyperbolic metric measure spaces, including singular spaces like RCD spaces and minimal hypersurfaces.

Findings

Boundary Harnack inequalities established

Complete classification of positive harmonic functions

Martin boundary identified with Gromov boundary

Abstract

Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Here we extend Ancona's potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schr\"odinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geometric Gromov boundary.