Regularity of degenerate k-Hessian equations on closed Hermitian manifolds

In this article, we are concerned with the existence of weak C 1, 1 solution of the k k-Hessian equation on a closed Hermitian manifold under the optimal assumption of the function in the right-hand side of the equation. The key points are to show the weak C 1, 1 estimates. We prove a Cherrier-Type inequality to obtain the C 0 estimate, and the complex Hessian estimate is proved by using an auxiliary function, which was motivated by Hou et al. and Tosatti and Weinkove. Our result generalizes the Kähler case proved by Dinew et al.