Estimates on Monge-Ampère operators derived from a local algebra inequality

Abstract

The goal of this short note is to relate the integrability property of the exponential $e^{-2\varphi }$ of a plurisubharmonic function $\varphi$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Ampère mass of $(d{d}^c\varphi {)}^n$. The inequality is valid locally or globally on an arbitrary open subset $\Omega$ in $\bC^n$. We show that ${\int }_{\Omega }(dd\varphi {)}^n<n^n$ implies ${\int }_K{e}^{-2\varphi }<+\infty$ for every compact subset $K$ in $\Omega$, while functions of the form $\varphi (z)=n\log |z-z_0|$, $z_0\in \Omega$, 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.Mustaţǎ to arbitrary dimensions.

14 pages, dedicated to Christer Kiselman on the occasion of his retirement; the second version adds an Appendix by Ahmed Zeriahi (Toulouse 3)

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