Endpoint $$L^1$$ estimates for Hodge systems

Abstract

AbstractLet $$d\ge 2$$ d ≥ 2 . In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$\begin{aligned} \Vert I_\alpha F\Vert _{{\dot{B}}^{0,1}_{d/(d-\alpha ),1}(\mathbb {R}^d;\mathbb {R}^k)} \le C \Vert F \Vert _{L^1(\mathbb {R}^d;\mathbb {R}^k)} \end{aligned}$$ ‖ I α F ‖ B ˙ d / ( d - α ) , 1 0 , 1 ( R d ; R k ) ≤ C ‖ F ‖ L 1 ( R d ; R k ) for all $$F \in L^1(\mathbb {R}^d;\mathbb {R}^k)$$ F ∈ L 1 ( R d ; R k ) which satisfy a first order cocancelling differential constraint, where $$\alpha \in (0,d)$$ α ∈ ( 0 , d ) and $$I_\alpha $$ I α is a Riesz potential. We show how this implies endpoint Besov–Lorentz estimates for Hodge systems with $$L^1$$ L 1 data via fractional integration for exterior derivatives.