  This paper studies a priori and regularity estimates of Evans-Krylov type in
H\"older spaces for fully nonlinear uniformly elliptic and parabolic equations
of second order when the operator fails to be concave or convex in the space of
symmetric matrices. In particular, it is assumed that either the level sets are
convex or the operator is concave, convex or close to a linear function near
infinity. As a byproduct, these results imply polynomial Liouville theorems for
entire solutions of elliptic equations and for ancient solutions to parabolic
problems.
