An optimal Sobolev embedding for L¹

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(α,d)>0 such that ‖I_αF‖_{L^d/(d−α),1(R^d;R^d)}≤C‖F‖_L¹(R^d;R^d) for all fields F∈L¹(R^d;R^d) such that curlF=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.