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

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