Fractional Integration and Optimal Estimates for Elliptic Systems

Felipe Hernandez, Daniel Spector

Published 2020-08-13Version 1

In this paper we prove the following optimal Lorentz embedding for the Riesz potentials: Let $\alpha \in (0,d)$. There exists a constant $C=C(\alpha,d)>0$ such that \[ \|I_\alpha F \|_{L^{d/(d-\alpha),1}(\mathbb{R}^d;\mathbb{R}^d)} \leq C \|F\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)} \] for all fields $F \in L^1(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{div} F=0$ in the sense of distributions. We then show how this result implies optimal Lorentz regularity for a Div-Curl system (which can be applied, for example to obtain new estimates for the magnetic field in Maxwell's equations), as well as for a vector-valued Poisson equation in the divergence free case.