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Restriction type estimates on general two-step stratified Lie groups

Lars Niedorf

TL;DR

This work establishes restriction-type estimates for sub-Laplacians on arbitrary two-step stratified Lie groups by exploiting a joint functional calculus that reduces the sub-Laplacian to a family of anisotropic twisted Laplacians on the first layer. The authors derive spectral-cluster estimates for these twisted operators via two routes—heat-subordination and Koch–Tataru dispersive methods—and then combine them with a sphere-restriction analysis on the second layer to obtain truncated restriction-type bounds in terms of Cowling–Sikora norms. The main result provides sharp $L^p o L^2$ bounds for operators of the form $F(L)\,oldsymbol{1}_{A}(L)\,oldsymbol{1}_{(0, au]}(U)$ with a precise interpolation parameter $ heta_p$, applicable for $1 ext{ } \le p ext{ } ext{to } ext{min}igigigigigigig{p_{d_1},p_{d_2}igigigigig}$, and illuminates the interplay between the first and second layers through spectral-block analysis. The results extend restriction-type phenomena beyond Heisenberg-type settings and lay the groundwork for spectral multiplier theory on Métivier and related two-step groups. The techniques have potential applications to subelliptic PDEs and harmonic analysis on nilpotent Lie groups, linking spectral cluster bounds to multiplier estimates in non-Euclidean geometries.

Abstract

We prove restriction type estimates for sub-Laplacians on general two-step stratified Lie groups. The core of our approach is to use spectral cluster estimates to effectively control the eigenvalue distribution of a family of anisotropic twisted Laplacians.

Restriction type estimates on general two-step stratified Lie groups

TL;DR

This work establishes restriction-type estimates for sub-Laplacians on arbitrary two-step stratified Lie groups by exploiting a joint functional calculus that reduces the sub-Laplacian to a family of anisotropic twisted Laplacians on the first layer. The authors derive spectral-cluster estimates for these twisted operators via two routes—heat-subordination and Koch–Tataru dispersive methods—and then combine them with a sphere-restriction analysis on the second layer to obtain truncated restriction-type bounds in terms of Cowling–Sikora norms. The main result provides sharp bounds for operators of the form with a precise interpolation parameter , applicable for , and illuminates the interplay between the first and second layers through spectral-block analysis. The results extend restriction-type phenomena beyond Heisenberg-type settings and lay the groundwork for spectral multiplier theory on Métivier and related two-step groups. The techniques have potential applications to subelliptic PDEs and harmonic analysis on nilpotent Lie groups, linking spectral cluster bounds to multiplier estimates in non-Euclidean geometries.

Abstract

We prove restriction type estimates for sub-Laplacians on general two-step stratified Lie groups. The core of our approach is to use spectral cluster estimates to effectively control the eigenvalue distribution of a family of anisotropic twisted Laplacians.
Paper Structure (16 sections, 12 theorems, 288 equations)

This paper contains 16 sections, 12 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Restriction estimates
  3. Statement of the main results
  4. Structure of the paper
  5. Notation
  6. Acknowledgments
  7. Sketch of the proof
  8. Sub-Laplacians on two-step stratified Lie groups
  9. Joint functional calculus
  10. Decomposition into blocks of twisted Laplacians
  11. Spectral properties of anisotropic twisted Laplacians
  12. Convolution kernels
  13. Spectral cluster estimates
  14. First proof via subordination by the heat semigroup
  15. Second proof via the dispersive estimates of Koch and Tataru
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $G$ be a two-step stratified Lie group, and, as in def:sub-Laplacian, let $L$ be a sub-Laplacian on $G$. Suppose that $1\le p\le \min\{p_{d_1},p_{d_2}\}$ with dimensions $d_1,d_2$ as in eq:dimensions. If $F:\mathbb{R}\to\mathbb{C}$ is a bounded Borel function supported in a compact subset $A\sub where $\theta_p\in [0,1]$ satisfies $1/p = (1-\theta_p) + \theta_p/\min\{p_{d_1},p_{d_2}\}$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • Remark 3.5
  • proof
  • Remark 3.6
  • Definition 3.7
  • ...and 22 more