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

TL;DR

This paper establishes new $L^p$ spectral multiplier estimates for sub-Laplacians on two-step stratified Lie groups with degenerate structures, extending previous results to more complex group geometries.

Contribution

It introduces sharp regularity conditions for spectral multipliers on degenerate two-step groups, utilizing a refined spectral decomposition and geometric analysis.

Findings

Proved $L^p$ spectral multiplier estimates under sharp regularity conditions.

Extended results to degenerate group structures like Heisenberg-Reiter groups.

Developed a novel spectral decomposition technique using caps on the sphere.

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.