  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 $C^{\infty}$.
