Some remarks on L-1 embeddings in the subelliptic setting

In this paper we establish an optimal Lorentz estimate for the Riesz potential in the L1 regime in the setting of a stratified group G: Let Q >= 2 be the homogeneous dimension of G and Ia denote the Riesz potential of order a on G. Then, for every alpha is an element of (0, Q), there exists a constant C = C(alpha, Q) > 0 such that parallel to I(alpha)f parallel to L-Q/(Q-alpha),L-1(G) <= C parallel to XI(1)f parallel to(L1(G)) (0.1) for all f is an element of C-c(infinity) (G) such that XI(1)f is an element of L-1(G), where X denotes the horizontal gradient.