ScalingStacks

Lemma 3.2 . [00VZ]

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Lemma 3.2.

There is a uniform constant CC so that for any 0<t≤10<t\leq 1, for any y∈Y\f⁡(S)y\in Y\backslash f(S) and for any u∈C∞​(Xy)u\in C^{\infty}(X_{y}) we have

(3.3) (∫Xy|u|2​(n−m)n−m−1​ωyn−m)n−m−1n−m≤C​∫Xy(|∇u|ωy2+|u|2)​ωyn−m.\left(\int_{X_{y}}|u|^{\frac{2(n-m)}{n-m-1}}\omega_{y}^{n-m}\right)^{\frac{n-m-1}{n-m}}\leq C\int_{X_{y}}(|\nabla u|^{2}_{\omega_{y}}+|u|^{2})\omega_{y}^{n-m}.

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