Revisiting Riesz transforms on Heisenberg groups

TL;DR

This paper characterizes higher order Riesz transforms on the Heisenberg group, establishing dimension-free bounds and extending boundedness results to related groups and expansions, advancing harmonic analysis in non-commutative settings.

Contribution

It provides a comprehensive characterization of higher order Riesz transforms on the Heisenberg group and proves dimension-free bounds under certain conditions, extending to related groups and expansions.

Findings

Dimension-free bounds for Riesz transforms established

Boundedness results extended to reduced Heisenberg group

Connections made to Hermite and Laguerre expansions

Abstract

We characterise higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. Using transfer- ence theorems, we deduce boundedness theorems for Riesz trans- forms on the reduced Heisenberg group and hence also for the Riesz transforms associated to multiple Hermite and Laguerre ex- pansions.