Weighted Green functions for complex Hessian operators

Abstract

Let $1\le m\le n$ be two fixed integers. Let $\Omega ⋐{\mathbb{C}}^n$ be a bounded $m$-hyperconvex domain and $\mathcal{A}\subset \Omega \times ]0,+\infty [$ a finite set of weighted poles. We define and study properties of the $m$-subharmonic Green function of $\Omega$ 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 $\bar{\Omega }\times \mathcal{F}$, where $\mathcal{F}$ is a suitable family of sets of weighted poles in $\Omega \times ]0,+\infty [$ 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.

31 pages

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• Subextension and approximation of $m$-subharmonic functions with given boundary values