Estimates on Monge-Amp\`ere operators derived from a local algebra inequality

Abstract

The goal of this short note is to relate the integrability property of the exponential $e^{-2\phi}$ of a plurisubharmonic function $\phi$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Amp\`ere mass of $(dd^c\phi)^n$. The inequality is valid locally or globally on an arbitrary open subset $\Omega$ in $\bC^n$. We show that $\int_\Omega(dd\phi)^n<n^n$ implies $\int_Ke^{-2\phi}<+\infty$ for every compact subset $K$ in $\Omega$, while functions of the form $\phi(z)=n\log|z-z_0|$, $z_0\in\Omega$, appear as limit cases. The result is derived from an inequality of pure local algebra, which turns out a posteriori to be equivalent to it, proved by A.Corti in dimension $n=2$, and later extended by L.Ein, T.De Fernex and M.Musta\c{t}\v{a} to arbitrary dimensions.