## S. Krantz - M. Peloso - D. Spector

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

created by spector on 06 Jun 2019

[

BibTeX]

*Preprint*

**Inserted:** 6 jun 2019

**Last Updated:** 6 jun 2019

**Pages:** 10 pages

**Year:** 2019

**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 $Q\geq 2$ 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
\[
| | \mathcal{I}_\alpha f | |_{L^{Q/(Q-\alpha),1}(G)} \leq C| | X\mathcal{I}_1 f | |_{L^1(G)}
\]
for distributions $f$ such that $X \mathcal{I}_1 f \in L^1(G)$, where $X$ denotes the horizontal gradient.