Spectral multipliers on Métivier groups

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 Métivier 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 Métivier 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.

This paper is Part 2 of the paper previously submitted as arXiv:2304.12960. The original submission has been split into two separate papers to provide a clearer presentation of the results. Part 1 is available as a revised submission at arXiv:2304.12960. Added a funding acknowledgment in the latest version