Endpoint $L^1$ estimates for Hodge systems

TL;DR

This paper presents a straightforward proof of endpoint Besov-Lorentz estimates for certain differential systems with $L^1$ data, extending classical results to the critical case.

Contribution

It provides a simple proof of endpoint estimates for Hodge systems with $L^1$ data under cocancelling constraints, using fractional integration techniques.

Findings

Proved endpoint Besov-Lorentz estimates for $L^1$ data.

Extended estimates to Hodge systems with fractional integration.

Demonstrated applicability to differential constraints.

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.