Liouville theorem for singular solutions to nonlocal equations

TL;DR

This paper investigates the behavior of singular solutions to nonlocal equations, establishing Bôcher-type results and Liouville theorems, and constructing fundamental solutions for these operators.

Contribution

It introduces new Liouville theorems and fundamental solutions for nonlocal linear operators with measurable kernels, advancing understanding of singular solutions.

Findings

Characterization of singular solutions near singular points

Liouville theorems for nonlocal equations

Construction of fundamental solutions for nonlocal operators

Abstract

We study singular solutions to the fractional Laplace equation and, more generally, to nonlocal linear equations with measurable kernels. We establish B\^ocher type results that characterize the behavior of singular solutions near the singular point. In addition, we prove Liouville theorems for singular solutions. To this end, we construct fundamental solutions for nonlocal linear operators and establish a localized comparison principle.