Invertibility of operators on atomic subspaces of $L^1$ and an application to the Neumann problem

Abstract

We prove a criterion for invertibility of operators on adequate adaptations to the boundary of a smooth domain of atomic subspaces of $L^1$, originally defined on ${\mathbb{R}^n} $ by Sweezy. As an application, we establish solvability of the Neumann problem for harmonic functions on smooth domains, assuming that the normal derivative belongs to said atomic subspaces of $L^1$.