2009/03/27 by Emanuel Milman, Milman, Emanuel
Mathematics · #Analytic Number Theory Research #Mathematical Inequalities and Applications #Point processes and geometric inequalities #math.FA #math.MG
paper · pdf · doi:10.48550/arxiv.0903.4822
26 pages, to appear in Springer's International Mathematical Series Vols. 10-13
arxiv created 2009/03/27 · arxiv updated 2009/12/01
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.