Spectral multipliers on Métivier groups

TL;DR

This work proves an $L^p$ spectral multiplier theorem for sub-Laplacians on Métivier groups under the sharp regularity threshold $s> d\left|1/p-1/2\right|$, extending results beyond Heisenberg-type groups. The authors combine truncated restriction-type estimates with a dyadic multiplier reduction and a detailed spectral analysis of the first-layer structure, exploiting Radon-Hurwitz numerology to show $d_1$ typically dominates $d_2$. A central technical tool is a first-layer weighted Plancherel estimate that controls kernel localization, and the authors develop enhancements for the exceptional cases $(4,3),(8,6),(8,7)$, including directional weighted estimates that enable $p$-ranges up to $4/3$ in the primary exceptional case. Overall, the results generalize Ni14–Ni24-type spectral multiplier theory from Heisenberg-type groups to the broader Métivier class, providing Bochner–Riesz bounds and shedding light on how second-layer geometry interacts with two-step group structure in subelliptic settings.

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.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 2.1: Truncated restriction type estimates
• Remark 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.3
• Definition 4.1
• Proposition 4.2