Criteria for solvability of left invariant operators on nilpotent Lie groups
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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.References
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Additional Information
- © Copyright 1983 American Mathematical Society
- 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