Shubin calculi for actions of graded Lie groups

TL;DR

This paper develops a pseudodifferential calculus for actions of graded Lie groups on graded spaces, generalizing classical operators and establishing properties like hypoellipticity and spectral analysis.

Contribution

It introduces a new calculus of Shubin type pseudodifferential operators on non-compact spaces using a groupoid approach, extending classical and harmonic oscillator operators.

Findings

Operators form an asymptotically complete calculus

Elliptic operators have parametrices and are hypoelliptic

Spectral properties are analyzed on Sobolev spaces

Abstract

In this article, we develop a calculus of Shubin type pseudodifferential operators on certain non-compact spaces, using a groupoid approach similar to the one of van Erp and Yuncken. More concretely, we consider actions of graded Lie groups on graded vector spaces and study pseudodifferential operators that generalize fundamental vector fields and multiplication by polynomials. Our two main examples of elliptic operators in this calculus are Rockland operators with a potential and the generalizations of the harmonic oscillator to the Heisenberg group due to Rottensteiner-Ruzhansky. Deforming the action of the graded group, we define a tangent groupoid which connects pseudodifferential operators to their principal (co)symbols. We show that our operators form a calculus that is asymptotically complete. Elliptic elements in the calculus have parametrices, are hypoelliptic, and can be characterized in terms of a Rockland condition. Moreover, we study the mapping properties as well as the spectra of our operators on Sobolev spaces and compare our calculus to the Shubin calculus on $\mathbb R^n$ and its anisotropic generalizations.