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

We obtain a priori local pointwise second derivative estimates for solutions u to a class of augmented Hessian equations on Riemannian manifolds, in terms of the C1 norm and certain W2,p norms of u. We consider the case that no structural assumptions are imposed on either the augmenting term or the right hand side of the equation, and the case where these terms are convex in the gradient variable. In the latter case, under an additional ellipticity condition we prove that the dependence on any W2,p norm can be dropped. Our results are derived using integral estimates.