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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups


Author: Kenneth G. Miller
Journal: Trans. Amer. Math. Soc. 280 (1983), 721-736
MSC: Primary 58G15; Secondary 22E25, 22E30, 35S05
MathSciNet review: 716847
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Abstract: Let $ P$ be a right-invariant pseudodifferential operator with principal part $ {P_0}$ on a simply connected two-step nilpotent Lie group $ G$ of type $ H$. It will be shown that if $ \pi (P_0)$ is injective in $ {\mathcal{S}_\pi }$ for every nontrivial irreducible unitary representation $ \pi $ of $ G$, then $ P$ has a pseudodifferential left parametrix. For such groups this generalizes the Rockland-Helffer-Nourrigat criterion for the hypoellipticity of a homogeneous right-invariant partial differential operator on $ G$. If, in addition, $ \pi (P)$ is injective in $ {\mathcal{S}_\pi }$ for every irreducible unitary representation of $ G$, it will be shown that $ P$ has a pseudodifferential left inverse. The constructions of the inverse and parametrix make use of the Kirillov theory, their symbols being obtained on the orbits individually and then pieced together.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0716847-2
PII: S 0002-9947(1983)0716847-2
Article copyright: © Copyright 1983 American Mathematical Society