Estimates for the complex Monge-Amp\`ere equation on Hermitian and balanced manifolds

We generalize Yau’s estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\mathrm{\infty }}$ a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced (in higher dimensions), giving an alternative proof of a theorem of Cherrier. We relate this to recent results of Guan-Li.