Spectral multiplier theorems of Euclidean type on new classes of 2-step stratified groups

TL;DR

This paper extends spectral multiplier theorems for sublaplacians on 2-step stratified groups, reducing smoothness requirements for boundedness on L^p spaces under certain dimensional constraints.

Contribution

It improves existing spectral multiplier conditions for 2-step stratified groups, lowering the smoothness threshold from scale-invariant to topological dimension-based bounds in specific cases.

Findings

Smoothness condition reduced to s > d/2 for certain groups

Boundedness results hold for groups with topological dimension d ≤ 7

Results apply when the derived algebra has dimension ≤ 2

Abstract

From a theorem of Christ and Mauceri and Meda it follows that, for a homogeneous sublaplacian $L$ on a $2$-step stratified group $G$ with Lie algebra $\mathfrak{g}$, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for $1 < p < \infty$ if the spectral multiplier $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$, where $Q = \dim \mathfrak{g} + \dim[\mathfrak{g},\mathfrak{g}]$ is the homogeneous dimension of $G$. Here we show that the condition can be pushed down to $s > d/2$, where $d = \dim \mathfrak{g}$ is the topological dimension of $G$, provided that $d \leq 7$ or $\dim [\mathfrak{g},\mathfrak{g}] \leq 2$.