Bilinear Bochner-Riesz Means on Métivier groups

TL;DR

This work extends bilinear Bochner-Riesz multiplier theory to Métivier groups, a broad class of two-step nilpotent Lie groups, by establishing $L^{p_1}(G)\times L^{p_2}(G)\to L^{p}(G)$ boundedness for the sub-Laplacian-based bilinear means whenever $\alpha>\alpha(p_1,p_2)$ under the Hölder relation $1/p=1/p_1+1/p_2$. Central to the approach are a dyadic multiplier decomposition, kernel-size and off-diagonal estimates, and weighted Plancherel inequalities that shift the effective dimension from the homogeneous $Q$ to the topological $d$ for $p>1$, with nuanced endpoint handling via bilinear interpolation and restriction-type ideas. The paper also obtains mixed-norm bounds and proves restricted-input results that parallel Euclidean restricted Bochner-Riesz theorems, while highlighting open questions about Assumption A for Métivier groups. Collectively, these results advance the understanding of spectral multipliers in non-Euclidean settings and underscore dimension-shift phenomena in two-step groups. The findings have potential implications for harmonic analysis on broad classes of nilpotent Lie groups and related PDE problems on sub-Riemannian manifolds.

Abstract

In this paper, we study the $L^{p_1}(G) \times L^{p_2}(G)$ to $L^{p}(G)$ boundedness of the bilinear Bochner-Riesz means associated with the sub-Laplacian on Métivier group $G$ under the Hölder's relation $1/p = 1/p_1 + 1/p_2$, $1\leq p_1, p_2 \leq \infty$. Our objective is to obtain boundedness results, analogous to the Euclidean setting, where the Euclidean dimension in the smoothness threshold is possibly replaced by the topological dimension of the underlying Métivier group $G$.

Figures (2)

• Figure 1: Here $O=(0,0)$, and $\alpha>\alpha(p_1, p_2)$ represents that $B^{\alpha}_R$ is bounded on $L^{p_1}(\mathbb{R}^n) \times L^{p_2}(\mathbb{R}^n) \to L^p(\mathbb{R}^n)$ for $\alpha>\alpha(p_1, p_2)$ (see Theorem \ref{['Theorem: Euclidean bilinear Bochner-Riesz for Grafakos']}).
• Figure 2: Here $O=(0,0)$, and $\alpha>\alpha(p_1, p_2)$ represents that $\mathcal{B}^{\alpha}$ is bounded on $L^{p_1}(G) \times L^{p_2}(G) \to L^p(G)$ for $\alpha>\alpha(p_1, p_2)$. The left picture is described by Theorem \ref{['Bilinear Bochner-Riesz Main theorem']}, while right picture is described by Theorem \ref{['Bilinear Bochner-Riesz theorem with restricted f and g']}.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof