Integral operators defined "up to a polynomial''

TL;DR

This paper develops a new framework for integral operators, including the fractional Laplacian, acting on functions with minimal decay at infinity by quotienting out polynomials, and studies their stability, compatibility, and solvability.

Contribution

It introduces a novel approach to define integral operators up to polynomials, addressing divergence issues and establishing stability, compatibility, and Dirichlet problem solvability.

Findings

Framework for integral operators acting on functions with minimal decay.

Stability results under convergence of functions.

Compatibility results between polynomials of different orders.

Abstract

We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent expressions, thus we replace it with an appropriate framework obtained by a cut-off procedure. The notion obtained in this way quotients out the polynomials which produce the divergent pattern once the cut-off is removed. We also present results of stability under the appropriate notion of convergence and compatibility results between polynomials of different orders. Additionally, we address the solvability of the Dirichlet problem. The theory is developed in general in the pointwise sense. A viscosity counterpart is also presented under the additional assumption that the interaction kernel has a sign, in conformity with the maximum principle structure.