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 C n , when m+p ≥ n. In the special case of m−subharmonic function ϕ, we give a relationship between the Lelong numbers of dd c ϕ and the mean values of ϕ on spheres or balls. As an application we study the integrability exponent of ϕ. We express the integrability exponent of ϕ in terms of volume of sub-level sets of ϕ and we give a link between this exponent and its Lelong number.2010 Mathematics Subject Classification. 32U25; 32U40; 32U05.

Poincaré- and Sobolev- Type Inequalities for Complex m-Hessian Equations

“…Later, Dinew and Ko lodziej partially confirmed this conjecture under the extra assumption that the m-subharmonic functions [24]). For the relation of this conjecture with the so called integrability exponent, and the Lelong number, of m-subharmonic functions see [14]. As an immediate consequence of our Theorem 5.4 is that we get that B locki's conjecture is true for functions in the Cegrell class E m (Ω) (Corollary 5.8).…”

Lelong-Jensen formula, Demailly-Lelong numbers and weighted degree of positive supercurrents

“…In the complex setting, this corollary is a variant of the well-known result for positive plurisubharmonic currents (see Demailly [4] and Skoda [12]). Next, we give a version of a result recently obtained by Benali-Ghiloufi [1] in the complex hessian theory, which can be viewed as a generalization of corollary 2.…”

“…2 and log |x| otherwise in our setting. Next, by following almost verbatim the proof of proposition 2 in [1] and by using lemma 1, we can formulate a variant of the Lelong-Jensen formula similar to that given in proposition 2 in [1]. Finally, it is not hard to see that such formula leads to the following conclusion :…”

“…Proof. Thanks to the weak continuity of dd # , it is clear that ( 2) is a direct consequence of (1), then it suffices to prove (1). We proceed by induction on q.…”

“…For this aim, we let ourselves be inspired by the complex setting. Indeed, we follow the method of Lelong in the closed case, which has generalized by Demailly [4] and Skoda [12] for the plurisubharmonic case and by Benali-Ghiloufi [1] in the complex hessian theory.…”