Lelong numbers of $m-$subharmonic functions

Abstract

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