An Optimal Sobolev Embedding for L^1

ABSTRACT: 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(a,d)>0 such that 

||I_aF||_{L^d/(d-a),1(R^d;R^d)} <_C||F||_L1(R^d;R^d)

for all fields FEL¹(R^d;R^d) such that curl F=0 in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regine p=1 of the well-established result for p>1.