Riesz means and bilinear Riesz means on M\'{e}tivier groups

TL;DR

This paper studies the boundedness properties of Riesz and bilinear Riesz means on Métivier groups, extending harmonic analysis techniques to more general non-commutative structures with high-dimensional centers.

Contribution

It establishes boundedness results for Riesz means and bilinear Riesz means on Métivier groups, generalizing previous work on Heisenberg and H-type groups.

Findings

Proves $L^{p}$-boundedness of Riesz means on Métivier groups.

Establishes $L^{p_{1}} imes L^{p_{2}} ightarrow L^{p}$ boundedness for bilinear Riesz means.

Develops new methods to handle high-dimensional centers in Métivier groups.

Abstract

In this paper, we investigate the $L^{p}$-boundedness of the Riesz means and the $L^{p_{1}}\times L^{p_{2}}\rightarrow L^{p}$ boundedness of the bilinear Riesz means on M\'{e}tivier groups. M\'{e}tivier groups are generalization of Heisenberg groups and general H-type groups. Because general M\'{e}tivier groups only satisfy the non-degeneracy condition and have high-dimensional centre, we have to use different methods and techniques from those on Heisenberg groups and H-type groups.