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In this paper we give an affirmative answer to the Euclidean analogue of aquestion of Bourgain and Brezis concerning the optimal Lorentz estimate for aDiv-Curl system: The function $Z=\operatorname*{curl} (-\Delta)^{-1} F$satisfies \begin{align*} \operatorname*{curl} Z = F \newline  \operatorname*{div} Z = 0 \end{align*} and there exists a constant $C>0$ suchthat \begin{align*} \| Z\|_{L^{3/2,1}(\mathbb{R}^3;\mathbb{R}^3)} \leq C\|F\|_{L^{1}(\mathbb{R}^3;\mathbb{R}^3)}. \end{align*} Our proof relies on a newendpoint Hardy-Littlewood-Sobolev inequality for divergence free measures whichwe obtain via a result of independent interest, an atomic decomposition of suchobjects.

Fractional Integration and Optimal Estimates for Elliptic Systems