Interpolated time-H\"older regularity of solutions of fully nonlinear parabolic equations

We show by the maximum principle that a quantitative estimate of the H\"older continuity of a solution of a fully nonlinear parabolic equation with respect to the spatial variable xⁿ, with exponent α∈(0,1], implies an estimate of its H\"older continuity in the time variable with the (natural) exponent α/2. This holds without assuming any H\"older continuity of the data of the equation. We apply this nonvariational approach to prove interior Schauder estimates for a special class of fully nonlinear parabolic Isaacs equations, providing an Evans-Krylov result for the model equation mininfβL_β u,_γ L_γ u\-∂ₜ u=0, where L_β,L_γ are linear operators with possibly variable H\"older coefficients. Along the way, we provide a short survey on the regularity theory for fully nonlinear parabolic equations of second order.