Sharp regularity estimates for second order fully nonlinear parabolic equations

We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form u(t) - F (D(2)u, Du, X, t) = f (X, t) in Q(1), ( Eq) where F is elliptic with respect to the Hessian argument and f is an element of L-p,L-q (Q(1)). The quantity Xi(n, p, q) := n/p + 2/q determines to which regularity regime a solution of (Eq) belongs. We prove that when 1 < Xi(n, p, q) < 2 - epsilon F, solutions are parabolically alpha-Holder continuous for a sharp, quantitative exponent 0 < alpha(n, p, q) < 1. Precisely at the critical borderline case, Xi(n, p, q) = 1, we obtain sharp parabolic Log-Lipschitz regularity estimates. When 0 < Xi(n, p, q) < 1, solutions are locally of class C-1+sigma,C- 1+sigma/2 and in the limiting case Xi(n, p, q) = 0, we show parabolic C-1,C-Log-Lip regularity estimates provided F has "better" a priori estimates.