Fully nonlinear parabolic fixed transmission problems

We consider transmission problems for parabolic equations governed by distinct fully nonlinear operators on each side of a time-dependent interface. We prove that if the interface is $C^{1,α}$, in the parabolic sense, then viscosity solutions are piecewise $C^{1,α}$ up to the interface. As byproducts, we obtain a new ABP-Krylov-Tso estimate, and establish existence, uniqueness, a comparison principle, and regularity results for the flat interface problem.