Schauder and Calderón-Zygmund type estimates for fully nonlinear parabolic equations under "small ellipticity aperture" and applications

In this manuscript, we derive Schauder estimates for viscosity solutions to non-convex fully nonlinear second-order parabolic equations $${\mathrm{\partial }}_{t}u-F(x,t,D^{2}u)=f(x,t)\text{in}{\mathrm{Q}}_1=B_1\times (-1,0],$$ provided that the source $f$ and the coefficients of $F$ are Hölder continuous functions and $F$ enjoys a small ellipticity aperture. Furthermore, for problems with merely bounded data, we prove that such solutions are $C^{1,\text{Log-Lip}}$ smooth in the parabolic metric. We also address Calderón-Zygmund estimates for such a class of non-convex operators. Finally, we connect our findings with recent estimates for fully nonlinear models in certain solution classes.