ScalingStacks

Lemma 2.9 . [00PS]

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Lemma 2.9.

(Volume-capacity estimate) In the setting of Thm. 2.7, for any compact set K⊂XK\subset X,

∫KωϕnVol​(X)≤A​eα​exp⁡(−α​(Vol​(X)Capω​(K))1/n).\frac{\int_{K}\omega_{\phi}^{n}}{\text{Vol}(X)}\leq Ae^{\alpha}\exp\left(-\alpha(\frac{\text{Vol}(X)}{\text{Cap}_{\omega}(K)})^{1/n}\right). (3)

In particular there is a constant B=B⁡(n,α,A)B=B(n,\alpha,A) verifying the power law bound

∫KωϕnVol​(X)≤B2​n​(Capω​(K)Vol​(X))2.\frac{\int_{K}\omega_{\phi}^{n}}{\text{Vol}(X)}\leq B^{2n}\left(\frac{\text{Cap}_{\omega}(K)}{\text{Vol}(X)}\right)^{2}.

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