Endpoint $L^1$ estimates for Hodge systems

Abstract: In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$ |I_\alpha F|{\dot{B}^{0,1}{d/(d-\alpha),1}(\mathbb{R}^d;\mathbb{R}^k)} \leq C |F |_{L^1(\mathbb{R}^d;\mathbb{R}^k)} $$ for all $F \in L^1(\mathbb{R}^d;\mathbb{R}^k)$ which satisfy a first order cocancelling differential constraint. We show how this implies endpoint Besov-Lorentz estimates for Hodge systems with $L^1$ data via fractional integration for exterior derivatives.