ScalingStacks

Lemma 3.4. There are uniform constants $\lambda,B,C$ so that for any $0<t\leq 1$, for any $y\in Y\backslash f(S)$ and for any $u\in C^{\infty}(X_{y})$ with $\int_{X_{y}}u\omega_{y}^{n-m}=0$ we have

(3.6)

$$\int_{X_{y}}|u|^{2}\omega_{y}^{n-m}\leq Ce^{B\sigma^{-\lambda}}\int_{X_{y}}|\nabla u|^{2}_{\omega_{y}}\omega_{y}^{n-m}.$$