Weak Solutions to the Complex m-Hessian Equation on Open Subsets of ${{\mathbb {C}}}^{n}$

Abstract

In this paper, we prove the existence of weak solutions to the complex m-Hessian equations in the class ${\mathcal {D}}_{m}(\Omega )$ on an open subset $\Omega$ of ${\mathbb {C}}^n$. In the end of the paper we give an example shows that in the unit ball ${\mathbb {B}}^{2}(0,1)\subset {\mathbb {C}}^{2}$ the complex Monge-Ampère equation $(dd^{c} .)^{2}=\mu$ is solvable but the complex Hessian equation $H_{1}(.)=\mu$ has not any weak solutions where $\mu$ is a nonnegative Radon measure on ${\mathbb {B}}^{2}(0,1)$.