$$C^{2,\alpha }$$ regularity of free boundaries in parabolic non-local obstacle problems

We study the regularity of free boundary in parabolic obstacle problem for fractional Laplacian $$(-\Delta )^s$$ (and more general integro-differential operators) regime $$s>\frac{1}{2}$$ . prove that once is $$C^1$$ it actually $$C^{2,\alpha }$$ To do so, we establish a Harnack inequality and $$C^{1,\alpha (moving) domains, providing quotient two solutions linear equation, vanish on boundary, as smooth boundary. As consequence our results also first time optimal such to nonlocal equations moving domains.