Shubin calculi for actions of graded Lie groups
Eske Ewert, Philipp Schmitt
TL;DR
The paper develops a Shubin-type pseudodifferential calculus on graded Lie groups using a tangent groupoid framework, enabling analysis on non-compact spaces. It constructs fibred distributions and a Shubin tangent groupoid to connect operators to their principal cosymbols, establishing asymptotic completeness and parametrix theory for elliptic elements. Ellipticity is characterized by a Rockland condition under a structural assumption, yielding mapping properties on Sobolev-type spaces and Fredholm/spectral results, including discreteness of spectrum for self-adjoint elliptic operators. The framework encompasses Rockland operators with potentials and harmonic-oscillator-type operators on the Heisenberg group and provides connections to the classical Shubin calculus and anisotropic variants.
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.