Spectral multipliers on $2$-step groups: topological versus homogeneous dimension

TL;DR

This paper investigates spectral multipliers on 2-step stratified groups, establishing that the optimal smoothness threshold for boundedness is strictly below the homogeneous dimension divided by two, but not below the topological dimension divided by two.

Contribution

It proves that the sharp smoothness threshold for spectral multipliers on all 2-step groups lies strictly between the topological and homogeneous dimensions.

Findings

The threshold is strictly less than Q/2 for all 2-step groups.

The threshold is not less than d/2, where d is the topological dimension.

This refines previous results by establishing a universal lower bound.

Abstract

Let $G$ be a $2$-step stratified group of topological dimension $d$ and homogeneous dimension $Q$. Let $L$ be a homogeneous sub-Laplacian on $G$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ whenever the multiplier $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. It is known that, for several $2$-step groups and sub-Laplacians, the threshold $Q/2$ in the smoothness condition is not sharp and in many cases it is possible to push it down to $d/2$. Here we show that, for all $2$-step groups and sub-Laplacians, the sharp threshold is strictly less than $Q/2$, but not less than $d/2$.