The nonlocal Harnack inequality for antisymmetric functions: an approach via Bochner's relation and harmonic analysis

TL;DR

This paper extends Harnack inequalities for antisymmetric functions to general nonlocal elliptic operators using a novel approach based on Bochner's relation and harmonic analysis, connecting boundary and interior inequalities.

Contribution

The paper introduces a new method leveraging Bochner's relation to extend Harnack inequalities to broader classes of nonlocal operators for antisymmetric functions.

Findings

Extended Harnack inequality to general nonlocal elliptic operators

Reduced boundary Harnack inequalities to interior inequalities

Established a harmonic analysis framework for antisymmetric functions

Abstract

We revisit a Harnack inequality for antisymmetric functions that has been recently established for the fractional Laplacian and we extend it to more general nonlocal elliptic operators. The new approach to deal with these problems that we propose in this paper leverages Bochner's relation, allowing one to relate a one-dimensional Fourier transform of an odd function with a three-dimensional Fourier transform of a radial function. In this way, Harnack inequalities for odd functions, which are essentially Harnack inequalities of boundary type, are reduced to interior Harnack inequalities.