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

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 article 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.