Bochner-Riesz commutators on Métivier groups: boundedness and compactness

TL;DR

The paper establishes sharp $L^p$-boundedness and compactness results for Bochner-Riesz commutators associated to the sub-Laplacian on Métivier groups, with the smoothness threshold governed by the topological dimension $d$ rather than the homogeneous dimension $Q$. By developing truncated restriction-type estimates and weighted Plancherel bounds tailored to the Métivier geometry, the authors overcome the lack of full spectral measure estimates for $\mathcal{L}$ and prove that $[b, S^{\alpha}(\mathcal{L})]$ is bounded on $L^q(G)$ for $1\le p\le p_{d_1,d_2}$ and $\alpha> d(1/p-1/2)-1/2$, with improved $p$-ranges in Heisenberg-type subcases. They also prove that when $b$ lies in the appropriate $CMO^{\varrho}(G)$ space, the commutator is compact on $L^q(G)$, using a Kolmogorov–Riesz framework and detailed kernel estimates. These results extend the spectral multiplier theory for Métivier groups by achieving sharp, topology-driven regularity thresholds and by clarifying the role of group geometry in Bochner-Riesz phenomena, offering new tools for sub-Riemannian spectral analysis.

Abstract

In this paper, we prove the boundedness and compactness properties of Bochner-Riesz commutator associated to the sub-Laplacians on Métivier groups. We show that the smoothness parameter can be expressed in terms of the topological dimension rather than the homogeneous dimension of the Métivier groups.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.1
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 2.1