Fractional integration and optimal estimates for elliptic systems

In this paper we give an affirmative answer to the Euclidean analogue of a question of Bourgain and Brezis concerning the optimal Lorentz estimate for a Div–Curl system: If \(F \in L^1(\mathbb {R}^3;\mathbb {R}^3)\) $F\in L^1({\mathbb{R}}^3;{\mathbb{R}}^3)$ satisfies \(\text {div}F=0\) $\text{div}F=0$ in the sense of distributions, then the function \(Z=\text {curl} (-\Delta )^{-1} F\) $Z=\text{curl}{(-\mathrm{\Delta })}^{-1}F$ satisfies \(\begin{aligned} \text {curl } Z&= F \\ \text {div } Z&= 0 \end{aligned}\) $$\begin{array}{cc}\hfill \text{curl}Z& =F\hfill \\ \hfill \text{div}Z& =0\hfill \end{array}$$ and there exists a constant \(C>0\) $C>0$ such that \(\begin{aligned} \Vert Z\Vert _{L^{3/2,1}(\mathbb {R}^3;\mathbb {R}^3)} \le C\Vert F\Vert _{L^{1}(\mathbb {R}^3;\mathbb {R}^3)}. \end{aligned}\) $$\begin{array}{c}\hfill {\Vert Z\Vert }_{L^{3/2,1}({\mathbb{R}}^3;{\mathbb{R}}^3)}\le C{\Vert F\Vert }_{L^1({\mathbb{R}}^3;{\mathbb{R}}^3)}.\end{array}$$ Our proof relies on a new endpoint Hardy–Littlewood–Sobolev inequality for divergence free measures which we obtain via a result of independent interest, an atomic decomposition of such objects.