We present a new, recent doubling method for establishing a priori estimates, then classical solvability and regularity for solutions to fully nonlinear PDEs.  The method produces the missing estimates for the quadratic Hessian and prescribed hypersurface scalar curvature PDEs in dimension four.  It also gives new proofs of the estimates and regularity for Monge–Ampère and special Lagrangian equations and provides prospects for classical solvability of alternative Dirichlet problems.