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∈L¹(R³;R³) satisfies divF=0 in the sense of distributions, then the function Z=curl(-Δ)^-1F satisfies (Formula presented.) and there exists a constant C>0 such that (Formula presented.) 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.