Defect measures on graded lie groups

TL;DR

This paper extends microlocal defect measures to graded nilpotent Lie groups by developing a pseudo-differential calculus adapted to their structure, enabling analysis of oscillating sequences and compactness properties.

Contribution

It introduces a generalized framework for defect measures on graded Lie groups, including the development of homogeneous symbols and pseudo-differential calculus based on group representations.

Findings

Defined microlocal defect measures in graded Lie groups

Computed defect measures for oscillating sequences

Analyzed compactness methods in this setting

Abstract

In this article, we define a generalisation of microlocal defect measures (also known as H-measures) to the setting of graded nilpotent Lie groups. This requires to develop the notions of homogeneous symbols and classical pseudo-differential calculus adapted to this setting and defined via the representations of the groups. Our method relies on the study of the C *-algebra of 0-homogeneous symbols. Then, we compute microlocal defect measures for concentrating and oscillating sequences, which also requires to investigate the notion of oscillating sequences in graded Lie groups. Finally, we discuss compacity compactness approaches in the context of graded nilpotent Lie groups.