Weighted Green functions for complex Hessian operators

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