$L^p$-parabolicity of Riemannian manifolds

Abstract

We describe the $L^p$-parabolicity of a Riemannian manifold in terms of a unified theory of nonlinear $L^p$-capacities, for the whole range $1\le p\le \infty$. For example, the \(L^{1}\)-parabolicity is equivalent to the usual notion of parabolicity/recurrence, while \(L^{2}\)-parabolicity is equivalent to biparabolicity. For any $1\le p\le \infty ,$ the $L^p$-parabolicity turns out to be equivalent to the $L^q$-Liouville property for positive superharmonic functions, where $p$ and $q$ are Hölder conjugate exponents. We also provide a new capacitary characterization of the $L^1$-Liouville property. Finally we obtain an almost optimal volume growth condition for the $L^p$-parabolicity for $1<p\le 2$ as well as a sharp volume condition for all $1<p<\infty$ for manifolds of non-negative Ricci curvature and for model manifolds.

arXiv commentThis is a revised and expanded version of a paper originally posted on A. Grigor'yan's webpage and also available on ResearchGate. The main new result is the proof, in the Riemannian setting, that the (2,1)-Laplacian capacity is twice the usual variational capacity, thereby extending a theorem of Brezis, Marcus, and Ponce. This answers a question raised in the first version of the paper