  We derive a priori second order estimates for fully nonlinear elliptic
equations which depend on the gradients of solutions in critical ways on
Hermitian manifolds. The global estimates we obtained apply to an equation
arising from a conjecture by Gauduchon which extends the Calabi conjecture;
this was one of the original motivations to this work. We were also motivated
by the fact that there had been increasing interests in fully nonlinear pde's
from complex geometry in recent years, and aimed to develop general methods to
cover as wide a class of equations as possible.
