# Spectral Multipliers on $2$-Step Stratified Groups, II

## TL;DR

This paper investigates spectral multipliers on 2-step stratified groups, establishing conditions under which the associated convolution kernels imply the multiplier functions are continuous or Schwartz, extending classical harmonic analysis results.

## Contribution

It extends the Riemann-Lebesgue and Schwartz multiplier theorems to a class of 2-step stratified groups with specific operator conditions.

## Key findings

- If the convolution kernel is in L^1, the multiplier is almost everywhere continuous and vanishes at infinity.
- If the convolution kernel is Schwartz, the multiplier function is also Schwartz.
- Results generalize classical Fourier analysis to non-abelian, stratified Lie groups.

## Abstract

Given a graded group $G$ and commuting, formally self-adjoint, left-invariant, homogeneous differential operators $\mathcal{L}_1,\dots, \mathcal{L}_n$ on $G$, one of which is Rockland, we study the convolution operators $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ and their convolution kernels, with particular reference to the case in which $G$ is abelian and $n=1$, and the case in which $G$ is a $2$-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and $\mathcal{L}_1,\dots,\mathcal{L}_n$ are either sub-Laplacians or central elements of the Lie algebra of $G$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ 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 $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ is a Schwartz function, then $m$ equals almost everywhere a Schwartz function.