AI 中文总结
针对分层李群，证明全水平里斯变换的弱型(1,1)界，常数不超过2且与群结构参数无关，是无维度欧氏定理的非交换推广，采用适配分层结构的分数阶障碍问题方法完成证明。
AI 中文摘要
设$G$为分层李群，$\boldsymbol{\textit{L}}$为其次拉普拉斯算子。我们证明全水平里斯变换$\nabla_H \boldsymbol{\textit{L}}^{-1/2}$在实值函数上为弱型$(1,1)$，且常数不超过2。特别地，该常数与$G$的水平维数、齐次维数、步长及底层群结构无关。本结果是Ouyang、Spector与Stockdale（arXiv:2608.18068 [math.CA]）的无维度欧氏定理的非交换推广，采用相同的普适常数。我们的证明通过热半群及相关狄利克雷型构造分数阶障碍问题，适配分层结构，在不使用傅里叶分析要素的前提下扩展了上述学者的方法。
英文摘要
Let $\mathbb{G}$ be a stratified Lie group and $\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homogeneous dimension, the step, and the underlying group structure of $\mathbb{G}$. Our result provides a noncommutative generalization of the dimension-free Euclidean theorem of Ouyang, Spector, and Stockdale arXiv:2608.18068, with the same universal constant. Our proof relies upon a fractional obstacle problem adapted to stratified Lie groups by using the functional calculus of $\mathcal L$ instead of the Fourier transform. As a consequence, by interpolation we obtain uniform $L^p$ bounds for the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ for $p \in (1,2]$. By contrast, for every $p > 2$, we construct a sequence of stratified Lie groups with fixed horizontal dimension $5$ and steps tending to infinity for which the $L^p$ norms of the horizontal Riesz transforms diverge.
Comments25 pages, we added the results for general $p$ and thus changed the title
