Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

Abstract

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball $\B \sub \C^n$ with its relative logarithmic capacity in $\C^n$ with respect to the same ball $\B$. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of $\C^n$ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at infinity on $\C^n$ as well as the Cegrell class of \psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain $\W \Sub \C^n.$ Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of \psh functions.