On $C^{1+\alpha }$ regularity of solutions of Isaacs parabolic equations with VMO coefficients

We prove that boundary value problems for fully nonlinear second-order parabolic equations admit $L_p$-viscosity solutions, which are in $C^{1+\alpha }$ for an $\alpha \in (0,1)$. The equations have a special structure that the “main” part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.