Phase-space analysis and pseudodifferential calculus on the Heisenberg group

TL;DR

This paper introduces a new class of pseudodifferential operators on the Heisenberg group, establishing their algebraic properties and continuity on Sobolev spaces, enhancing microlocal analysis in this setting.

Contribution

It defines a novel pseudodifferential calculus on the Heisenberg group that forms an algebra and integrates microlocal analysis with Littlewood-Paley theory.

Findings

The class of pseudodifferential operators forms an algebra.

Operators act continuously on Sobolev spaces.

Loss of derivatives is controlled by operator order.

Abstract

This paper has been withdrawn by the authors. 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.