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We obtain upper and lower bounds for the nonlinear variational capacity ofthin annuli in weighted $\mathbf{R}^n$ and in metric spaces, primarily underthe assumptions of an annular decay property and a Poincar\'e inequality. Inparticular, if the measure has the $1$-annular decay property at $x_0$ and themetric space supports a pointwise $1$-Poincar\'e inequality at $x_0$, then theupper and lower bounds are comparable and we get a two-sided estimate for thinannuli centred at $x_0$, which generalizes the known estimate for the usualvariational capacity in unweighted $\mathbf{R}^n$. Most of our estimates aresharp, which we show by supplying several key counterexamples. We alsocharacterize the $1$-annular decay property.

The annular decay property and capacity estimates for thin annuli