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We obtain a priori local pointwise second derivative estimates for solutions$u$ to a class of augmented Hessian equations on Riemannian manifolds, in termsof the $C^1$ norm and certain $W^{2,p}$ norms of $u$. We consider the case thatno structural assumptions are imposed on either the augmenting term or theright hand side of the equation, and the case where these terms are convex inthe gradient variable. In the latter case, under an additional ellipticitycondition we prove that the dependence on any $W^{2,p}$ norm can be dropped.Our results are derived using integral estimates.

Augmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimates