Spectral multipliers on M\'etivier groups

TL;DR

This paper establishes sharp $L^p$ spectral multiplier theorems for sub-Laplacians on Métivier groups, leveraging their Lie algebra structure and a restriction estimate, advancing harmonic analysis on these groups.

Contribution

It proves a sharp spectral multiplier theorem for sub-Laplacians on Métivier groups using a novel restriction estimate and structural properties of their Lie algebras.

Findings

Proves $L^p$ spectral multiplier theorem with sharp regularity condition.

Utilizes a restriction estimate surprisingly effective for sharp results.

Exploits Lie algebra stratification related to Radon-Hurwitz numbers.

Abstract

We prove an $L^p$-spectral multiplier theorem under the sharp regularity condition $s > d\left|1/p - 1/2\right|$ for sub-Laplacians on M\'etivier groups. The proof is based on a restriction type estimate which, at first sight, seems to be suboptimal for proving sharp spectral multiplier results, but turns out to be surprisingly effective. This is achieved by exploiting the structural property that for any M\'etivier group the first layer of any stratification of its Lie algebra is typically much larger than the second layer, a phenomenon closely related to Radon-Hurwitz numbers.