Let $\phi\in P(M,g)$ be such that $e^{F}={\omega_{\phi}^{n}\over\omega_{0}^{n}}$ with $e^{F}\in L^{2+s}(M,\omega_{0}),\;$ for some $s>0$. Then $||\phi||_{0}\leq C_{5.21}$, with $C_{5.21}$ depending only on the metric $\omega_{0}$, $s>0$ and $||e^{F}||_{L^{2+s}(M,\omega_{0})}$.