The size of plurisubharmonic lemniscates in terms of Hausdorff–Riesz measures and capacities

Abstract

The main goal of this paper is to establish new uniform estimates on the size of sublevel sets of plurisubharmonic functions (called plurisubharmonic lemniscates) in terms of Hausdorff–Riesz measures and capacities of certain orders.

We first prove a new uniform version of Skoda's integrability theorem for a given class of plurisubharmonic functions in terms of Borel measures of Hausdorff–Riesz type of certain orders with a precise estimate of the integrability exponent in terms of Lelong numbers of the class and the order of the measures.

Then we present several applications of this result. We first deduce uniform estimates on the size of plurisubharmonic lemniscates associated to functions from some important classes of plurisubharmonic functions in terms of Hausdorff–Riesz measures.

We also derive a new comparison inequality between certain Hausdorff–Riesz capacities and the pluricomplex logarithmic capacity for borelean sets of a fixed bounded domain in $\mathbb{C}^n$ or more generally in an affine algebraic manifold.

Furthermore, using results from classical potential theory, we finally deduce from this comparison inequality new estimates of the size of polynomial lemniscates in terms of Hausdorff contents in the spirit of the famous lemma of H. Cartan.