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

Abstract

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 ${\mathbf{R}}^n$ 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.

References in corpus (1)

• Resolutivity and invariance for the Perron method for degenerate equations of divergence type

Cited by in corpus (4)

• Sharp Besov capacity estimates for annuli in metric spaces with doubling measures
• Poincaré inequalities and $A_p$ weights on bow-ties
• Uniqueness and nonuniqueness of $p$-harmonic Green functions on weighted ${\mathbf{R}}^n$ and metric spaces
• Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces