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An endpoint estimate for product singular integral operators on stratified Lie groups

Michael G. Cowling, Ming-Yi Lee, Ji Li, Jill Pipher

TL;DR

This work extends endpoint harmonic analysis to the product of two stratified Lie groups by proving hyperweak bounds for area, square, maximal, and Calderón–Zygmund operators on $\mathbf{G}=G_1\times G_2$ with a global $\mathsf{L\ log\ L}$ structure. The authors develop a robust framework: (i) a Journé-type covering lemma in spaces of homogeneous type, (ii) a Calderón reproducing formula and an atomic decomposition for $\mathsf{L\ log\ L}(\mathbf{G})$ via the Poisson kernel, and (iii) endpoint estimates for radial and non-tangential maximal operators, area and square functions, and double Riesz transforms. The main result is a global endpoint bound of the form $|\{g: |\mathcal{T}f(g)|>\lambda\}| \lesssim F_{\Phi}(f/\lambda)$ with $F_{\Phi}(f)=\iint_{\mathbf{G}} |f|\log(\mathrm{e}+|f|)$, establishing hyperweak boundedness beyond the local setting. The results sharpen the multiparameter product theory on Lie groups and provide tools for sharp $L\log L$–type control of a broad class of product singular integrals with potential applications in higher-step noncommutative analysis.

Abstract

We establish hyperweak boundedness of area functions, square functions, maximal operators and Calderón--Zygmund operators on products of two stratified Lie groups.

An endpoint estimate for product singular integral operators on stratified Lie groups

TL;DR

This work extends endpoint harmonic analysis to the product of two stratified Lie groups by proving hyperweak bounds for area, square, maximal, and Calderón–Zygmund operators on with a global structure. The authors develop a robust framework: (i) a Journé-type covering lemma in spaces of homogeneous type, (ii) a Calderón reproducing formula and an atomic decomposition for via the Poisson kernel, and (iii) endpoint estimates for radial and non-tangential maximal operators, area and square functions, and double Riesz transforms. The main result is a global endpoint bound of the form with , establishing hyperweak boundedness beyond the local setting. The results sharpen the multiparameter product theory on Lie groups and provide tools for sharp –type control of a broad class of product singular integrals with potential applications in higher-step noncommutative analysis.

Abstract

We establish hyperweak boundedness of area functions, square functions, maximal operators and Calderón--Zygmund operators on products of two stratified Lie groups.
Paper Structure (17 sections, 16 theorems, 153 equations)

This paper contains 17 sections, 16 theorems, 153 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Preliminaries
  3. Stratified nilpotent Lie groups
  4. Functional calculus for the sub-Laplacian
  5. The heat and Poisson kernels
  6. Systems of pseudodyadic cubes
  7. Products of stratified groups
  8. Journé's covering lemma
  9. The Orlicz space $\mathsf{L\,log\,L}(\mathbf{G})$
  10. The strong maximal function and Proof of Theorem \ref{['thm main']}
  11. The atomic decomposition
  12. Hyperweak boundedness and the area function $\mathcal{S}_{\slashed{\nabla} p,1} (f)$
  13. The atomic decomposition for $\mathsf{L\,log\,L}$
  14. Proof of Theorem \ref{['thm main']}
  15. Proof of Theorem \ref{['thm main']} for area functions $\mathcal{S}_{\psi,\eta }(f)$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\mathcal{T}$ be a maximal operator $\mathcal{M}_{\varphi,\eta }$ or a Littlewood--Paley operator $\mathcal{S}_{\psi,\eta }$, where $\eta \geq 0$, or a double Riesz transformation $\mathcal{R}^{[1]}_{j_1} \otimes \mathcal{R}^{[2]}_{j_2}$. Then for all $f \in \mathsf{L\,log\,L}(\mathbf{G})$, where $\mathsf{L\,log\,L}(\mathbf{G})$ is the Orlicz space associated to the functional $F_\Phi$, given

Theorems & Definitions (28)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2: HK
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • ...and 18 more