Phase-space analysis and pseudodifferential calculus on the Heisenberg group

TL;DR

This paper develops a class of pseudodifferential operators on the Heisenberg group, establishing their algebraic properties and continuity on Sobolev spaces, with a microlocal analysis approach that extends previous work.

Contribution

It introduces a new class of pseudodifferential operators on the Heisenberg group, highlighting microlocal analysis and integrating Littlewood-Paley theory for advanced analysis.

Findings

Operators form an algebra containing differential operators

Operators act continuously on Sobolev spaces

Loss of derivatives is controlled by operator order

Abstract

A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is different, and in particular puts into light microlocal directions and completes, with the Littlewood-Paley theory developed in \cite{bgx} and \cite{bg}, a microlocal analysis of the Heisenberg group.