关于次椭圆域上$L^1$嵌入的几点注记
Some remarks on $L^1$ embeddings in the subelliptic setting
摘要：
本文在分层群$G$的背景下建立了$L^1$体系中Riesz势的最优Lorentz估计：设$Q\geq 2$是$G$的齐次维数，$\mathcal{I}_α$表示$G$上$α$阶Riesz势.然后，对于每个$α\in (0,Q)$，存在常数$C=C(α,Q)>0$，使得对于分布$f$，使得$X \mathcal{I}_1 f \in L^1(G)$，\begin{align} \| \mathcal{I}_αf \|_{L^{Q/(Q-α),1}(G)} \leq C\| X \mathcal{I}_1 f \|_{L^1(G)} \end{align}，其中$X$表示水平梯度。
英文摘要：
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 $\mathcal{I}_α$ denote the Riesz potential of order $α$ on $G$. Then, for every $α\in (0,Q)$, there exists a constant $C=C(α,Q)>0$ 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.
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