The inverse, the heat semigroup, Liouville’s theorems and the spectrum for the Grushin operator

By decomposing the Grushin operator on into a family of parameterized Hermite operators, we give estimates for the inverses and the heat semigroups of these Hermite operators, which are then used to obtain Sobolev estimates for the inverse and the heat semigroup of the Grushin operator. Using the gl...

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Bibliographic Details
Published in:Journal of pseudo-differential operators and applications 2010-12, Vol.1 (4), p.377-388
Main Authors: Dasgupta, Aparajita, Molahajloo, Shahla, Wong, M. W.
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Functional Analysis
Mathematics
Mathematics and Statistics
Operator Theory
Partial Differential Equations
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Summary:By decomposing the Grushin operator on into a family of parameterized Hermite operators, we give estimates for the inverses and the heat semigroups of these Hermite operators, which are then used to obtain Sobolev estimates for the inverse and the heat semigroup of the Grushin operator. Using the global hypoellipticity of the parametrized Hermite operators, Liouville’s theorems for harmonic functions of the Grushin operator on are obtained. The spectrum of the Grushin operator is computed.
ISSN:1662-9981
1662-999X
DOI:10.1007/s11868-010-0016-z