ScalingStacks

Let $(Y,\omega)$ be a compact Kähler manifold, and $d\mu$ be a measure on $Y$. We say $(Y,\omega,d\mu)$ satisfies the Skoda type inequality, if for any Kähler potential $u\in PSH(Y,\omega)$ normalised to $\sup u=0$,

$$\int_{Y}e^{-\alpha u}d\mu\leq A,$$

(1)

where $\alpha,A$ are independent of $u$. A prototype theorem is