Details
| Original language | English |
|---|---|
| Article number | 103572 |
| Journal | Bulletin des Sciences Mathematiques |
| Volume | 199 |
| Early online date | 30 Dec 2024 |
| Publication status | Published - Mar 2025 |
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 Rn and its anisotropic generalizations.
Keywords
- Graded Lie groups, Hypoelliptic operators, Rockland condition, Shubin pseudodifferential operator, Tangent groupoid, Transformation groupoid
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Bulletin des Sciences Mathematiques, Vol. 199, 103572, 03.2025.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Shubin calculi for actions of graded Lie groups
AU - Ewert, Eske
AU - Schmitt, Philipp
N1 - Publisher Copyright: © 2024 The Author(s)
PY - 2025/3
Y1 - 2025/3
N2 - 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 Rn and its anisotropic generalizations.
AB - 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 Rn and its anisotropic generalizations.
KW - Graded Lie groups
KW - Hypoelliptic operators
KW - Rockland condition
KW - Shubin pseudodifferential operator
KW - Tangent groupoid
KW - Transformation groupoid
UR - http://www.scopus.com/inward/record.url?scp=85214123797&partnerID=8YFLogxK
U2 - 10.1016/j.bulsci.2024.103572
DO - 10.1016/j.bulsci.2024.103572
M3 - Article
AN - SCOPUS:85214123797
VL - 199
JO - Bulletin des Sciences Mathematiques
JF - Bulletin des Sciences Mathematiques
SN - 0007-4497
M1 - 103572
ER -