arXiv subjects

Fredj Elkhadhra

Publications and source records attributed to Fredj Elkhadhra.

5 recordsLinked to original sources

arXiv2021

$m$-potential theory and $m$-generalized Lelong numbers associated with $m$-positive supercurrents

In this study, we first define the local potential associated to a weakly positive closed supercurrent in analogy to the one investigated by Ben Messaoud and El Mir in the complex setting. Next, we study the definition and the continuity of the $m$-superHessian operator for unbounded $m$-convex functions. As an application, we generalize our previous work on Demailly-Lelong numbers and several related results in the superformalism setting. Furthermore, strongly inspired by the complex Hessian theory, we introduce the Cegrell-type classes as well as a generalization of some $m$-potential results in the class of $m$-convex functions.

math.CV↗

arXiv2021

Complex Hessian Operator associated to an $m$-positive closed current and weighted $m$-capacity

In this paper, we first study the definition and the continuity of the complex Hessian operator associated to an $m$-positive closed current $T$, for some classes of unbounded $m$-subharmonic functions as well as when we consider a regularization sequence of $T$. Next, we introduce the notion of weighted $(m,T)$-capacity in the complex Hessian setting and we investigate the link with the weighted $m$-extremal function. As an application we give a characterization of the Cegrell classes ${\mathscr F}^m$ and ${\mathscr E}^m$ by means of the weighted $(m,1)$-capacity. Furthermore, we prove a subsolution theorem for a general complex Hessian equation relatively to $T$.

math.CV↗

arXiv2019

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

The goal of this work is to extend the concepts of generalized Lelong number of positive currents investigated by Skoda, Demailly and Ghiloufi in complex analysis, to weakly positive supercurrents on the real superspaces. We generalize then a result of Lagerberg when the supercurrent is closed as well as a very recent result of Berndtsson for minimal supercurrents associated to submanifolds of $\mathbb{R}^n$. The main tool is a variant of the well-known Lelong-Jensen formula in the superformalism case. Moreover, we extend to our setting various interesting theorems in complex analysis such as Demailly and Rashkovskii comparison theorems. We also complete the work begun by Lagerberg on the degree of positive closed supercurrents and we prove a removable singularities result for positive supercurrents.

math.CV↗

arXiv2015

Complex Hessian Operator, $m$-capacity, Cegrell's classes and $m$-Potential associated to a Positive Closed Current

In this paper we firstly introduce the concepts of capacity and Cegrell's classes associated to any $m$-positive closed current $T$. Next, after investigating the most imporant related properties, we study the definition and the continuity of the complex hessian operator in several cases, generalizing then the work of Demailly and Xing in this direction. We also prove a Xing-type comparison principle for the analogous Cegrell class $\mathcal{F}^{m,T}$ of negative $m$-subharmonic functions. Finally, we generalize the work of Ben Messaoud-El Mir on the complex Monge-Ampère operator and the Lelong-Skoda potential associated to a positive closed current.

math.CV↗

arXiv2012

Poincaré-Lelong formula, $J$-analytic subsets and Lelong numbers of currents on almost complex manifolds

In this paper, we first establish a Poincaré-Lelong type formula in the almost complex setting. Then, after introducing the notion of $J$-analytic subsets, we study the restriction of a closed positive current defined in an almost complex manifold $(M,J)$ on a $J$-analytic subset. Finally, we prove that the Lelong numbers of a plurisubharmonic current defined on an almost complex manifold are independent of the coordinate systems.

math.CV↗