Capacities in metric spaces

Abstract

Abstract. We discuss the potential theory related to the variational capacity and the Sobolev capacity on metric measure spaces. We prove our results in the axiomatic framework of [16].

A NOTE ON THE EFFECT OF PROJECTIONS ON BOTH MEASURES AND THE GENERALIZATION OF q-DIMENSION CAPACITY

“…The generalised capacity C s,q (E) plays a role in the study of potential theory, for example one can compare the generalised capacity to the variational q-capacity, the relative q-capacity and the Riesz capacity in metric spaces (see [37]- [39]). Now, we define the generalized q-dimension Riesz capacity by…”

A perturbation result for the Riesz transform

“…In this section we recall some notions concerning p-hyperbolicity that will be needed in the sequel. General references are [8] and [14]. We will assume that the manifold is connected.…”

“…The last inequality in this definition follows from the fact that the "truncation" of a function u up to height 1 on U decreases the energy R M |ru| p . For a detailed proof, see [14,Corollary 7.5]. With this definition, we have the following characterisation of the p-hyperbolicity: Theorem 2.5.…”

The connectivity at infinity of a manifold andLq,p-Sobolev inequalities

“…By the monotonicity properties of the p-capacity, [17], [11], and recalling that E 1,t |∇k 1,t | p is decreasing in t, we deduce…”