Assume $\omega_{\varphi}^{n}=f\omega^{n}$, $\omega_{\psi}=g\omega^{n}$, where $\varphi,\psi\in PSH(X,\omega)$ are continuous and $f,g\in L^{p}(\omega^{n})$, $p>1$. Then for all $0<\gamma<2/(2+nq)$,

$$||\varphi-\psi||_{L^{\infty}(X)}\leq C||\varphi-\psi||_{L^{2}(\omega^{n})}^{\gamma},$$

where $q=p/(p-1)$ denotes the conjugate exponent to $p$.