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

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