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 $L^1$ regime in the setting of a stratified group $G$: Let $Q\geq 2$ 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}_\alpha f \|_{L^{Q/(Q-\alpha),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.