Lelong numbers of $m-$subharmonic functions

TL;DR

This paper investigates the existence and properties of Lelong numbers for m-subharmonic functions and currents, establishing relationships with mean value properties and integrability exponents, thus advancing understanding in pluripotential theory.

Contribution

It introduces new results on the existence of Lelong numbers for m-subharmonic currents and relates these to mean values and integrability exponents of m-subharmonic functions.

Findings

Lelong numbers exist for m-subharmonic currents when m+p≥n

Lelong numbers of functions relate to mean values on spheres or balls

Integrability exponents are expressed via volume of sub-level sets and linked to Lelong numbers

Abstract

In this paper we study the existence of Lelong numbers of $m-$subharmonic currents of bidimension $(p,p)$ on an open subset of $\Bbb C^n$, when $m+p\geq n$. In the special case of $m-$subharmonic function $\varphi$, we give a relationship between the Lelong numbers of $dd^c\varphi$ and the mean values of $\varphi$ on spheres or balls. As an application we study the integrability exponent of $\varphi$. We express the integrability exponent of $\varphi$ in terms of volume of sub-level sets of $\varphi$ and we give a link between this exponent and its Lelong number.