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

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}.