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Transactions of the American Mathematical Society

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

 
 

 

Criteria for solvability of left invariant operators on nilpotent Lie groups


Author: Lawrence Corwin
Journal: Trans. Amer. Math. Soc. 280 (1983), 53-72
MSC: Primary 22E25; Secondary 22E30, 35A99, 58G99
DOI: https://doi.org/10.1090/S0002-9947-1983-0712249-3
MathSciNet review: 712249
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Abstract: We define a special nilpotent Lie group $ N$ to be one which has a $ 1$-dimensional center, dilations, square-integrable representations, and a maximal subordinate algebra common to almost all functionals on the Lie algebra $ \mathfrak{N}$. Every nilpotent Lie group with dilations imbeds in such a special group so that the dilations extend. Let $ L$ be a homogeneous left invariant differential operator on $ N$. We give a representation-theoretic condition on $ L$ which $ L$ must satisfy if it has a tempered fundamental solution and which implies global solvability of $ L$. (The sufficiency is a corollary of a more general theorem, valid on all nilpotent $ N$.) For the Heisenberg group, the condition is equivalent to having a tempered fundamental solution.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0712249-3
Article copyright: © Copyright 1983 American Mathematical Society

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