The annular decay property and capacity estimates for thin annuli

Abstract

We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted ${\mathbf{R}}^n$ and in metric spaces, primarily under the assumptions of an annular decay property and a Poincaré inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincaré inequality at $x_0$, then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at $x_0$, which generalizes the known estimate for the usual variational capacity in unweighted ${\mathbf{R}}^n$. Most of our estimates are sharp, which we show by supplying several key counterexamples. We also characterize the $1$-annular decay property.

20 pages

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