Abstract
We survey some classical inequalities due to Maz'ya relating iso- capacitary 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, the Maz'ya 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 semiconvexity assumptions, by obtaining some new delicate semigroup estimates.
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