Potential Theory on Gromov Hyperbolic Spaces

Abstract

Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger 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.

Keywords

• gromov hyperbolic space
• dirichlet form
• schrödinger operator
• boundary harnack inequality
• gromov boundary
• martin boundary
• 31c25
• 31c35
• 35j10
• 51m10
• 53a10