On estimates for the Fu-Yau generalization of a Strominger system

We study an equation proposed by Fu and Yau as a natural $n$-dimensional generalization of a Strominger system that they solved in dimension $2$. It is a complex Hessian equation with right hand side depending on gradients. Building on the methods of Fu and Yau, we obtain $C^0$, $C^2$ and $C^{2,\alpha}$ a priori estimates. We also identify difficulties in extending the Fu-Yau arguments for non-degeneracy from dimension $2$ to higher dimensions.