# Spectral Multipliers on 2-step Stratified Groups, I

## TL;DR

This paper investigates spectral multipliers for sub-Laplacians on certain 2-step stratified groups, establishing conditions under which the associated convolution kernels imply the multiplier functions are continuous or Schwartz functions.

## Contribution

It proves Riemann-Lebesgue and Schwartz class multiplier theorems for operators on 2-step stratified groups not satisfying a strengthened Moore-Wolf condition.

## Key findings

- Convolution kernel in L^1 implies multiplier is continuous and vanishes at infinity.
- Kernel in Schwartz class implies multiplier is Schwartz.
- Extends spectral multiplier theory to a class of 2-step groups.

## Abstract

Given a $2$-step stratified group which does not satisfy a slight strengthening of the Moore-Wolf condition, a sub-Laplacian $\mathcal{L}$ and a family $\mathcal{T}$ of elements of the derived algebra, we study the convolution kernels associated with the operators of the form $m(\mathcal{L}, -i \mathcal{T})$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L},-i \mathcal{T})$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\infty$ (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator $m(\mathcal{L},-i\mathcal{T})$ is a Schwartz function, then $m$ equals almost everywhere a Schwartz function.