Interior Hessian estimates for a class of Hessian type equations

In this paper, we introduce some Hessian operators σ k ( η ) and σ k ( η ) σ l ( η ) by a self-adjoint mapping and the corresponding convex cone Γ k ~ , and derive interior a priori Hessian estimates for the equation σ k ( η ) σ l ( η ) = f ( x ) in Γ k ~ with 0 ≤ l < k < n . As an application we prove Pogorelov type estimates which imply Liouville theorem for such equation.