An Optimal Sobolev Embedding for $L^1$

In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant $C=C(\alpha ,d)>0$ such that $$\mathrm{\parallel }{I}_{\alpha }F{\mathrm{\parallel }}_{L^{d\mathrm{/}(d-\alpha ),1}({\mathbb{R}}^d;{\mathbb{R}}^d)}\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{L^1({\mathbb{R}}^d;{\mathbb{R}}^d)}$$ for all fields $F \in L^1($\mathbb{R}$^d;$\mathbb{R}$^d)$suchthat$\operatorname*{curl} F=0$ in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime $p=1$ of the well-established results for $p>1$.