Spectral multipliers on Heisenberg-Reiter and related groups

TL;DR

This paper extends spectral multiplier theorems for sublaplacians on 2-step stratified Lie groups, showing that a lower smoothness order suffices for boundedness on L^p spaces, broadening previous results.

Contribution

It proves that a smoothness condition of order s > d/2 ensures boundedness of spectral multipliers on a wider class of 2-step groups, improving earlier conditions.

Findings

Spectral multipliers are bounded on L^p under s > d/2 smoothness.

Extension of results to larger classes of 2-step groups.

Generalization of previous theorems for Heisenberg and related groups.

Abstract

Let $L$ be a homogeneous sublaplacian on a 2-step stratified Lie group $G$ of topological dimension $d$ and homogeneous dimension $Q$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is bounded on $L^p$ for $1 < p < \infty$ if $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. Under suitable assumptions on $G$ and $L$, here we show that a smoothness condition of order $s > d/2$ is sufficient. This extends to a larger class of 2-step groups the results for the Heisenberg and related groups by M\"uller and Stein and by Hebisch, and for the free group $N_{3,2}$ by M\"uller and the author.