Weighted Green functions for complex Hessian operators

Let $1\leq m\leq n$ be fixed integers. Let $\Omega \Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset \Omega \times \mathopen {]}0,+ \infty \mathclose {[}$ a finite set of weighted poles. We define and study properties of the $m$-subharmonic Green function of $\Omega $ with prescribed behavior near the weighted set $\mathcal A$. 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 $\Omega \times \mathopen {]}0,+ \infty \mathclose {[}$ endowed with the Hausdorff distance. Moreover, we give a precise estimate on its modulus of continuity. Our results generalize and improve previous results concerning the pluricomplex Green function due to P. Lelong.