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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Spectral multipliers on Lie groups of polynomial growth


Author: G. Alexopoulos
Journal: Proc. Amer. Math. Soc. 120 (1994), 973-979
MSC: Primary 22E30
MathSciNet review: 1172944
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Abstract: Let $ L$ be a left invariant sub-Laplacian on a connected Lie group $ G$ of polynomial volume growth, and let $ \{ {E_\lambda },\lambda \geqslant 0\} $ be the spectral resolution of $ L$ and $ m$ a bounded Borel measurable function on $ [0,\infty )$. In this article we give a sufficient condition on $ m$ for the operator $ m(L) = \smallint _0^\infty m(\lambda )d{E_\lambda }$ to extend to an operator bounded on $ {L^p}(G),\;1 < p < \infty $, and also from $ {L^1}(G)$ to weak-$ {L^1}(G)$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1172944-4
PII: S 0002-9939(1994)1172944-4
Keywords: Lie group, volume growth, multiplier, sub-Laplacian, wave equation
Article copyright: © Copyright 1994 American Mathematical Society