Obstacle problems for integro-differential operators: Higher regularity of free boundaries

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are $C^{1,\alpha }$, then they are $C^{\mathrm{\infty }}$. This completes the study of regular points, initiated in [5]. In order to achieve this, we need to establish optimal boundary regularity estimates for solutions to linear nonlocal equations in $C^{k,\alpha }$ domains. These new estimates are the core of our paper, and extend previously known results by Grubb (for $k=\mathrm{\infty }$) and by the second author and Serra (for $k=1$).