Besov and Triebel–Lizorkin regularity for the Hodge decomposition and applications to magnetic potentials

Abstract

The Hodge decomposition is a fundamental result in the theory of differential forms. It has been extensively studied from different perspectives and it is the source of several applications in physics and mathematics. For example, it is important in the analysis of Navier–Stokes' and Maxwell's equations, it can be used to prove results on inverse scattering problems in quantum mechanics and it is useful for the study of magnetic potentials associated to a magnetic field. The regularity properties of the Hodge decomposition have been studied in several works in the last decades, most prominently it has been proved that Sobolev regularity holds true. In this paper we prove regularity for the Hodge decomposition in a much larger scale of function spaces: Besov and Triebel–Lizorkin spaces on compact Riemannian manifolds with boundary. We prove regularity in such spaces for the Hodge–Morrey decomposition and the Hodge–Morrey–Friedrichs decomposition. We, furthermore, state and prove a refinement for the Hodge–Morrey–Friedrichs decomposition (with regularity) that allows expressing every differential form as the sum of the differential of a form plus the co-differential of another form plus a harmonic field belonging to a finite dimensional space. This holds true even if the original form is barely differentiable (with sth order of differentiability, for every real number s>0). We finally provide an application to the study of magnetic potentials associated to magnetic fields. More precisely, we prove the existence of a magnetic potential associated to a magnetic field. The former being one time more differentiable than the latter, in terms of Besov and Triebel–Lizorkin regularity. This is an important motivation for us, since we have the intention to apply our results to further investigations in relativistic inverse (time dependent) scattering problems in quantum mechanics, where Besov and Triebel–Lizorkin regularity for magnetic potentials is relevant.