Weighted Green functions for complex Hessian operators

TL;DR

This paper introduces weighted Green functions for complex Hessian operators in hyperconvex domains, establishing their continuity properties and providing estimates, thus extending classical pluricomplex Green function results.

Contribution

It defines and analyzes the properties of weighted Green functions for complex Hessian operators, generalizing and improving upon previous pluricomplex Green function results.

Findings

Proved uniform continuity of the exponential Green function in both variables.

Provided precise estimates on the modulus of continuity.

Extended classical results to weighted Green functions for complex Hessian operators.

Abstract

Let $1\leq m\leq n$ be two fixed integers. Let $\Omega \Subset \mathbb C^n$ 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.