First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball B⊂Cn with its relative logarithmic capacity in Cn with respect to the same ball B. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of Cn is also proved.
Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity on Cn as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(Cn.
Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions.
This work was partially supported by the programmes PARS MI 07 and AI.MA 180.
"Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates." Ark. Mat. 43 (1) 85 - 112, April 2005. https://doi.org/10.1007/BF02383612