The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(H)$

TL;DR

This paper extends the understanding of the Dirichlet problem for the complex Hessian operator by characterizing measures dominated by Hessian measures as those associated with functions in the class \(\\mathcal{N}_m(H)\").

Contribution

It generalizes previous results to m-hyperconvex domains, establishing a measure characterization for solutions in the class \(\mathcal{N}_m(H)\).

Findings

Characterization of measures dominated by complex Hessian measures.

Extension of previous results to m-hyperconvex domains.

Solutions exist in the class \(\mathcal{N}_m(H)\).

Abstract

We prove that, in a $m$-hyperconvex domain in $\mathbb{C}^{n},$ if a non-negative Borel measure is dominated by a complex Hessian measure, then it is a complex Hessian measure of a function in the class $\mathcal{N}_m(H)$. This is an extension of P. {\AA}hg, U. Cegrell, R. Czy\.z and P.H. Hiep's result in \cite{ACCH}.