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A p = 2 dichotomy for uniform Riesz transform bounds on stratified Lie groups

  • Sheng-Chen Mao
  • Yaojun Wang
  • Ye Zhang
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Abstract

Let 𝔾 be a stratified Lie group and β„’ be its sub-Laplacian. We prove that the full horizontal Riesz transform βˆ‡H β„’βˆ’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 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 β„’ instead of the Fourier transform. As a consequence, by interpolation we obtain uniform Lp bounds for the full horizontal Riesz transform βˆ‡H β„’βˆ’1/2 for p ∈ (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 Lp norms of the horizontal Riesz transforms diverge.

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Cite

BibTeXRIS
BibTeX
@misc{mao2026dichotomy,
  title = {{A $p = 2$ dichotomy for uniform Riesz transform bounds on stratified Lie groups}},
  author = {Sheng-Chen Mao and Yaojun Wang and Ye Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.20267},
  eprint = {2608.20267},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.20267},
}
RIS
TY  - UNPB
TI  - A p = 2 dichotomy for uniform Riesz transform bounds on stratified Lie groups
AU  - Sheng-Chen Mao
AU  - Yaojun Wang
AU  - Ye Zhang
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2608.20267
UR  - https://arxiv.org/abs/2608.20267
AB  - Let 𝔾 be a stratified Lie group and β„’ be its sub-Laplacian. We prove that the full horizontal Riesz transform βˆ‡_H β„’^βˆ’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 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 β„’ instead of the Fourier transform. As a consequence, by interpolation we obtain uniform L^p bounds for the full horizontal Riesz transform βˆ‡_H β„’^βˆ’1/2 for p ∈ (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.
ER  -