Calculus of Variations and Geometric Measure Theory
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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

ArXiv: 1906.01896 PDF

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.

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