$C^{2,\alpha }$ estimates and existence results for certain nonconcave PDE

We establish $C^{2,\alpha }$ estimates for PDE of the form convex $+$ a sum of weakly concave functions of the Hessian, thus generalising a recent result of Collins which is in turn inspired by a theorem of Caffarelli and Yuan. Independently, we also prove an existence result for a certain generalised Monge-Amp`ere PDE.