ScalingStacks

Lemma 3.2. There is a uniform constant $C$ so that for any $0<t\leq 1$, for any $y\in Y\backslash f(S)$ and for any $u\in C^{\infty}(X_{y})$ we have

(3.3)

$$\left(\int_{X_{y}}|u|^{\frac{2(n-m)}{n-m-1}}\omega_{y}^{n-m}\right)^{\frac{n-m-1}{n-m}}\leq C\int_{X_{y}}(|\nabla u|^{2}_{\omega_{y}}+|u|^{2})\omega_{y}^{n-m}.$$