ScalingStacks

Remark 2.11. Thm. 2.7 implies a famous result of Kolodziej stating that if we fix $(X,\omega)$ and $p>1$, then $\phi$ has a $C^{0}$-bound depending only on $X,\omega,\left\lVert\frac{\omega_{\phi}^{n}}{\omega^{n}}\right\rVert_{L^{p}}$. It is enough to check (2), which reduces by Hölder inequality to the standard Skoda inequality (cf. Thm 2.4), with modified constants. The strength of Thm. 2.7 is that it still applies when the complex/Kähler structures are highly degenerate, as it distills the dependence on $(X,\omega)$ to only 3 constants $n,\alpha,A$.