A note on the subsolution theorem in weighted energy classes of $m$-subharmonic functions with given boundary values

Abstract

In this note, we deal with the existence of solutions of the weighted complex $m$-Hessian equation $-\chi (u)H_m(u)=\mu$ in the class ${\mathcal{E}}_{m,\chi }(f,\Omega )$ if there exists a subsolution in this class, where the given boundary value $f\in {\mathcal{E}}_m(\Omega )\cap MS{H}_m(\Omega ).$ This is a generalization of the result in the paper \cite{PDtaiwan} where we proved that the subsolution theorem is true in the class ${\mathcal{E}}_{m,\chi }(f,\Omega )$ in the case when the given boundary value $f\in {\mathcal{N}}_m(\Omega )\cap MS{H}_m(\Omega ).$

arXiv comment14 pages, accepted in Comptes Rendus mathematique