A pseudo-differential calculus on graded Lie groups

TL;DR

This paper develops a symbolic pseudo-differential calculus on graded Lie groups, defining symbol classes via difference operators and establishing an algebra of operators with properties similar to classical calculus.

Contribution

It introduces a new pseudo-differential calculus framework on graded Lie groups using difference operators and representation theory, extending classical calculus to this setting.

Findings

Defined symbol classes using difference operators on graded Lie groups

Constructed an algebra of operators with properties akin to Hormander calculus

Established a natural quantisation process via representation theory

Abstract

In this paper, we present first results of our investigation regarding symbolic pseudo-differential calculi on nilpotent Lie groups. On any graded Lie group, we define classes of symbols using difference operators. The operators are obtained from these symbols via the natural quantisation given by the representation theory. They form an algebra of operators which shares many properties with the usual Hormander calculus.