Restriction Theorems On M\'etiver Groups Associated to Joint Functional Calculus

TL;DR

This paper develops restriction theorems for joint functional calculus operators on Métivier groups, providing spectral solutions and mix-norm bounds for specific classes of functions, extending classical harmonic analysis results.

Contribution

It introduces spectral solutions for joint operators on Métivier groups and establishes new restriction theorems analogous to Thomas-Stein results for these operators.

Findings

Spectral solutions for operators $m( abla, abla_z)$ on Métivier groups.

Restriction theorems with mix-norm bounds for specific classes of functions.

Extension of classical restriction results to the setting of Métivier groups.

Abstract

In this article, we get the spectral solution $\mathcal{P}_{\mu}^{m}$ of operators $m(\mathcal{L}, -\Delta_\mathfrak{z})$, the joint functional calculus of the sub-Laplacian and Laplacian on the centre of M\'etivier group. Then, we give some group-analogues of the Thomas-Stein-type restriction theorem, asserting the mix-norm boundness of the restriction operators $\mathcal{P}_{\mu}^{m}$ for two classes of functions $m=(a^\alpha+b^\beta)^\gamma$ and $m=(1+a^\alpha+b^\beta)^\gamma$ with $\alpha, \beta>0, \gamma\neq0$.