On uniform estimates for $(n-1)-$form fully nonlinear partial differential equations on compact Hermitian manifolds

We obtain a priori $L^{\mathrm{\infty }}$ estimate for a general class of $(n-1)-$form fully nonlinear partial differential equations on compact Hermitian manifolds. Our method relies on the local version of comparison with auxiliary Monge-Ampère equations, developed earlier by B. Guo and D. H. Phong. The key is to find the appropriate elliptic operator such that the maximum principle applies.