Degenerate complex Hessian equations with arbitrary measure in bounded domains

Abstract

Let $\hat{I}\text{c◯}$ be a bounded strictly $m$-pseudoconvex domain of ${\mathbb{C}}^n$. We solve degenerate complex Hessian equations of the form $(ω+ dd^c φ)^m\wedgeβ^{n-m} = μ$ in the generalized Cegrell classes ${\mathcal{K}}_m(\hat{I}\text{c◯},\ddot{I}‰,\ddot{I}•)$, where $\ddot{I}•\in {\mathcal{E}}_m(\hat{I}\text{c◯})$ is a $m$-maximal function, $\ddot{I}‰$ is a smooth real $(1,1)$-form defined in a neighborhood of $\bar{\hat{I}}\text{c◯}$ and $\hat{I}¼$ is a positive Radon measure which is dominated by a Hessian measure of $m$-subharnomic functions in Cegrell class.

32 pages