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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups
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by Kenneth G. Miller PDF
Trans. Amer. Math. Soc. 280 (1983), 721-736 Request permission

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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 721-736
  • MSC: Primary 58G15; Secondary 22E25, 22E30, 35S05
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0716847-2
  • MathSciNet review: 716847