Nonlocal operators of small order

Abstract

In this work we study nonlocal operators and corresponding spaces of order strictly below one and investigate interior regularity properties 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, in particular, we prove that if the right-hand is of class $C^{\infty}$ and the kernel satisfies similar regularity properties away from its singularity, then any weak solution is of class $C^{\infty}$.