LAUSR

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-Ampere 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 \({\mathbb {B}}^{2}(0,1)\).