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

Abstract

Abstract. First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean bail B C C ~ with its relative logarithmic capacity in C ~ with respect to the same ball B. An analogous 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 plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity on C ~ as well as the Cegrell class of plurisubharmonic functions of bounded Monge Ampere mass on a hypereonvex domain f~ 9 ~ .Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonie functions.

A minimum principle for plurisubharmonic functions

“…2) When n ≥ 2 our estimate shows how the relative Hausdorff content of the exceptionnal set with respect to the ball B R is asymptotically small when R → +∞. 3) Observe that for α = 2 similar estimates in terms of the relative logarithmic capacity was obtained in [8].…”

“…Uniform estimates on the size of sublevel sets of some classes of plurisubharmonic functions, called plurisubharmonic lemniscates, have been obtained in our earlier papers (see [39], [40], [8]). …”

Weighted Pluricomplex Energy II

“…[21] for more details). G will be endowed with the induced Euclidean structure and the corresponding Lebesgue measure which will be denoted by G .…”

A Note on the approximation of plurisubharmonic functions

“…Let ϕ ∈ PSH − (Ω), from [4] there exist a sequence of functions ϕ j ∈ E 0 (Ω) such that ϕ j ϕ in Ω, and then we have (cf. [2,1,6]), for every j s n cap {ϕ j −s}, Ω Ω dd c ϕ j n , ∀s > 0.…”