Some remarks on [formula omitted] embeddings in the subelliptic setting

Abstract

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 Iα denote the Riesz potential of order α on G. Then, for every α∈(0,Q), there exists a constant C=C(α,Q)>0 such that (0.1)‖Iαf‖LQ∕(Q−α),1(G)≤C‖XI1f‖L1(G)for all f∈Cc∞(G) such that XI1f∈L1(G), where X denotes the horizontal gradient.