Volume growth, capacity estimates, p-parabolicity and sharp integrability properties of p-harmonic Green functions

In a complete metric space equipped with a doubling measure supporting a p -Poincaré inequality, we prove sharp growth and integrability results for p -harmonic Green functions and their minimal p -weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general p -harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted R n and on manifolds. The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for p -harmonic Green functions. The capacity estimate is valid under considerably milder assumptions than above. We also use it, under these milder assumptions, to characterize singletons of zero capacity and the p -parabolicity of the space. This generalizes and improves earlier results that have been important especially in the context of Riemannian manifolds.