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Spectral multipliers on two-step stratified Lie groups with degenerate group structure

Lars Niedorf

TL;DR

The paper develops sharp $L^p$ spectral multiplier theory for sub-Laplacians on two-step stratified Lie groups with degenerate centers, extending results beyond Métivier groups under Assumptions (A) and (B). It introduces a novel cap-based spectral decomposition on the center and combines truncated restriction-type estimates with a dyadic reduction and precise first/second-layer localizations to prove $L^p$-boundedness for $s> d|1/p-1/2|$ and Bochner–Riesz-type results. The approach leverages a twisted Laplacian framework $L^\mu$ and Laguerre function analysis, enabling refined kernel control and finite propagation speed arguments. The results apply to groups like variants of the free two-step nilpotent group on three generators and Heisenberg–Reiter groups, providing a new path to spectral multiplier estimates in degenerate sub-Riemannian geometries. Overall, the work broadens the scope of sharp spectral multiplier phenomena to non-Métivier two-step groups with quantitative cap-based restriction techniques.

Abstract

Let $L$ be a sub-Laplacian on a two-step stratified Lie group $G$ of topological dimension $d$. We prove new $L^p$-spectral multiplier estimates under the sharp regularity condition $s>d\left|1/p-1/2\right|$ in settings where the group structure of $G$ is degenerate, extending previously known results for the non-degenerate case. Our results include variants of the free two-step nilpotent group on three generators and Heisenberg-Reiter groups. The proof combines restriction type estimates with a detailed analysis of the sub-Riemannian geometry of $G$. A key novelty of our approach is the use of a refined spectral decomposition into caps on the unit sphere in the center of the Lie group.

Spectral multipliers on two-step stratified Lie groups with degenerate group structure

TL;DR

The paper develops sharp spectral multiplier theory for sub-Laplacians on two-step stratified Lie groups with degenerate centers, extending results beyond Métivier groups under Assumptions (A) and (B). It introduces a novel cap-based spectral decomposition on the center and combines truncated restriction-type estimates with a dyadic reduction and precise first/second-layer localizations to prove -boundedness for and Bochner–Riesz-type results. The approach leverages a twisted Laplacian framework and Laguerre function analysis, enabling refined kernel control and finite propagation speed arguments. The results apply to groups like variants of the free two-step nilpotent group on three generators and Heisenberg–Reiter groups, providing a new path to spectral multiplier estimates in degenerate sub-Riemannian geometries. Overall, the work broadens the scope of sharp spectral multiplier phenomena to non-Métivier two-step groups with quantitative cap-based restriction techniques.

Abstract

Let be a sub-Laplacian on a two-step stratified Lie group of topological dimension . We prove new -spectral multiplier estimates under the sharp regularity condition in settings where the group structure of is degenerate, extending previously known results for the non-degenerate case. Our results include variants of the free two-step nilpotent group on three generators and Heisenberg-Reiter groups. The proof combines restriction type estimates with a detailed analysis of the sub-Riemannian geometry of . A key novelty of our approach is the use of a refined spectral decomposition into caps on the unit sphere in the center of the Lie group.

Paper Structure

This paper contains 23 sections, 17 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two-step stratified Lie groups
  4. Spectral analysis of the matrices Jmu
  5. Particular classes of non-Métiver groups
  6. Some examples
  7. Projections onto the kernel
  8. Truncated restriction type estimates on caps
  9. A decomposition into caps
  10. Restriction type estimates
  11. Proof of Theorem 3.1
  12. Reduction of Theorem 1.1 to dyadic spectral multipliers
  13. Dyadic spectral multipliers
  14. Finite propagation speed
  15. A localization on the first layer
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $L=-(X_1^2+\dots+X_{d_1})$ be a sub-Laplacian on a two-step stratified Lie group $G\cong \mathbb{R}^d = \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$, where $X_1,\dots,X_{d_1}$ are the left-invariant vector fields associated with a basis of the first layer of a stratification of $G$. Suppose that $G$ in assumptionA. Let Then, if $1\le p \le p_{d_1,d_2}$, the following statements hold:

Theorems & Definitions (33)

  • Definition
  • Theorem 1.1
  • Proposition 2.1
  • Example 2.2: Copies of $N_{3,2}$ glued along their centers
  • Example 2.3: Heisenberg--Reiter groups
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • ...and 23 more