Nonlocal operators of small order

In this work we study nonlocal operators and corresponding spaces with a focus on operators of order near zero. We investigate the interior regularity of eigenfunctions and of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side. Our method exploits the variational structure of the problem and we prove that eigenfunctions are of class C^∞ if the kernel satisfies this property away from its singularity. Similarly in this case, if in the Poisson problem the right-hand is of class C^∞ , then also any weak solution is of class C^∞ .