Nonlocal operators of small order

TL;DR

This paper investigates nonlocal operators of order less than one, establishing regularity results for solutions based on the smoothness of the data and kernel, using variational methods.

Contribution

It provides new regularity results for weak solutions of nonlocal Poisson problems with small order operators, under smoothness assumptions on data and kernels.

Findings

Weak solutions are infinitely differentiable if the right-hand side and kernel are smooth.

The variational approach effectively proves interior regularity.

Results extend regularity theory to 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 $C^{\infty}$.