A converse to Maz'ya's inequality for capacities under curvature lower bound

TL;DR

This paper explores inequalities relating capacities in measure-metric spaces, extending Maz'ya's bounds and establishing conditions under which these inequalities can be reversed, with implications for geometric analysis under curvature constraints.

Contribution

It extends Maz'ya's capacity inequalities to arbitrary q > 1 and weak semi-convexity, providing new semi-group estimates and a converse inequality under curvature bounds.

Findings

Maz'ya's lower bound for q-capacity extended to arbitrary q > 1

Reversal of Maz'ya's inequality under convexity assumptions

New semi-group estimates for semi-convex spaces

Abstract

We survey some classical inequalities due to Maz'ya relating isocapacitary inequalities with their functional and isoperimetric counterparts in a measure-metric space setting, and extend Maz'ya's lower bound for the $q$-capacity ($q>1$) in terms of the 1-capacity (or isoperimetric) profile. We then proceed to describe results by Buser, Bakry, Ledoux and most recently by the author, which show that under suitable convexity assumptions on the measure-metric space, Maz'ya's inequality for capacities may be reversed, up to dimension independent numerical constants: a matching lower bound on 1-capacity may be derived in terms of the $q$-capacity profile. We extend these results to handle arbitrary $q > 1$ and weak semi-convexity assumptions, by obtaining some new delicate semi-group estimates.