Sharp capacity estimates for annuli in weighted R^n and in metric spaces

Abstract

We obtain estimates for the nonlinear variational capacity of annuli in weighted R^n and in metric spaces. We introduce four different (pointwise) exponent sets, show that they all play fundamental roles for capacity estimates, and also demonstrate that whether an end point of an exponent set is attained or not is important. As a consequence of our estimates we obtain, for instance, criteria for points to have zero (resp. positive) capacity. Our discussion holds in rather general metric spaces, including Carnot groups and many manifolds, but it is just as relevant on weighted R^n. Indeed, to illustrate the sharpness of our estimates, we give several examples of radially weighted R^n, which are based on quasiconformality of radial stretchings in R^n.

42 pages

References in corpus (1)

• The annular decay property and capacity estimates for thin annuli

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• The Liouville theorem for $p$-harmonic functions and quasiminimizers with finite energy
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• Volume growth, capacity estimates, $p$-parabolicity and sharp integrability properties of $p$-harmonic Green functions
• Sharp Besov capacity estimates for annuli in metric spaces with doubling measures
• Pointwise Assouad dimension for measures
• Poincaré inequalities and $A_p$ weights on bow-ties
• Uniqueness and nonuniqueness of $p$-harmonic Green functions on weighted ${\mathbf{R}}^n$ and metric spaces