A priori estimates for Donaldson's equation over compact Hermitian manifolds

In this paper we prove a priori estimates for Donaldson’s equation $\omega \wedge (\chi +\sqrt{-1}\mathrm{\partial }\bar{\mathrm{\partial }}\phi {)}^{n-1}=e^F(\chi +\sqrt{-1}\mathrm{\partial }\bar{\mathrm{\partial }}\phi {)}^n$ over a compact Hermitian manifold X of complex dimension n, where $\omega$ and $\chi$ are arbitrary Hermitian metrics. Our estimates answer a question of Tosatti-Weinkove.