We discuss a class of fully nonlinear PDEs arising from convex geometry, of the form of linear combinations of different Hessians. This type of equations also includes the special Lagrangian equation and Fu-Yau complex Hessian equation. It is desirable to have a general theory for these equations, but not much is known. We will focus on a special case, which extends the class of Hessian equations studied by Krylov. The main results are related to the regularity estimates, by exploring concavity properties associated to these special equations.

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