Fractional integration and optimal estimates for elliptic systems

Abstract

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 ∈ L 1 ( R 3 ; R 3 ) satisfies $$\text {div}F=0$$ div F = 0 in the sense of distributions, then the function $$Z=\text {curl} (-\Delta )^{-1} F$$ Z = curl ( - Δ ) - 1 F satisfies $$\begin{aligned} \text {curl } Z&= F \\ \text {div } Z&= 0 \end{aligned}$$ curl Z = F div Z = 0 and there exists a constant $$C>0$$ C > 0 such that $$\begin{aligned} \Vert Z\Vert _{L^{3/2,1}(\mathbb {R}^3;\mathbb {R}^3)} \le C\Vert F\Vert _{L^{1}(\mathbb {R}^3;\mathbb {R}^3)}. \end{aligned}$$ ‖ Z ‖ L 3 / 2 , 1 ( R 3 ; R 3 ) ≤ C ‖ F ‖ L 1 ( R 3 ; R 3 ) . 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.