Some remarks on $L^1$ embeddings in the subelliptic setting

Abstract

In this paper we establish an optimal Lorentz estimate for the Riesz potential in the $L^1$ regime in the setting of a stratified group $G$: Let $Q\ge 2$ be the homogeneous dimension of $G$ and ${\mathcal{I}}_{\alpha }$ denote the Riesz potential of order $\alpha$ on $G$. Then, for every $\alpha \in (0,Q)$, there exists a constant $C=C(\alpha ,Q)>0$ such that \begin{align} \| \mathcal{I}_αf \|_{L^{Q/(Q-α),1}(G)} \leq C\| X \mathcal{I}_1 f \|_{L^1(G)} \end{align} for distributions $f$ such that $X{\mathcal{I}}_{1}f\in L^1(G)$, where $X$ denotes the horizontal gradient.

10 pages