ScalingStacks

(‘Potential estimate’) If the volume density of $\omega_{\phi}^{n}$ has some integrability control such as an $L^{p}$-bound

$$\int_{Y}|\frac{\omega_{\phi}^{n}}{\omega^{n}}|^{p}\leq C,\quad p>1,$$

(10)

then $\phi$ has an $L^{\infty}$ bound depending only on $(X,\omega),n,p,C$ [45][20][24]. In fact this can be improved to a $C^{\alpha}$ Hölder bound on $\phi$ [47]. (Notice $p=1$ would not suffice, as the $L^{1}$-bound $\int_{Y}\omega_{\phi}^{n}\leq\int_{Y}\omega^{n}$ is automatic).