arXiv subjects

Hadhami Elaini

Publications and source records attributed to Hadhami Elaini.

2 recordsLinked to original sources

arXiv2023

Weighted Green functions for complex Hessian operators

Let $1\leq m\leq n$ be two fixed integers. Let $Ω\Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset Ω\times ]0,+ \infty[$ a finite set of weighted poles. We define and study properties of the $m$-subharmonic Green function of $Ω$ with prescribed behaviour near the weighted set $A$. In particular we prove uniform continuity of the exponential Green function in both variables $(z,\mathcal A)$ in the metric space $\bar Ω\times \mathcal F$, where $\mathcal F$ is a suitable family of sets of weighted poles in $Ω\times ]0,+ \infty[$ endowed with the Hausdorff distance. Moreover we give a precise estimates on its modulus of continuity. Our results generalize and improve previous results concerning the pluricomplex Green function du to P. Lelong.

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↗