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


Multipliers for eigenfunction expansions of some Schrödinger operators

Author: Krzysztof Stempak
Journal: Proc. Amer. Math. Soc. 93 (1985), 477-482
MSC: Primary 43A80; Secondary 22E30, 33A75
MathSciNet review: 774006
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Abstract: Let $ G$ be a graded nilpotent Lie group and let $ L$ be a positive Rockland operator on $ G$. Let $ {E_\lambda }$ denote the spectral resolution of $ L$ on $ {L^2}(G)$. A sufficient condition is given under which a function $ m$ on $ {{\mathbf{R}}^ + }$ is a $ {L^p}$-multiplier for $ L$, $ 1 < p < \infty $; that is $ {\left\Vert {\int_0^\infty {m(\lambda )d{E_\lambda }f} } \right\Vert _p} \leqslant {C_p}{\left\Vert f \right\Vert _p}$ for a constant $ {C_p}$, $ f \in {L^p}(G) \cap {L^2}(G)$. Then the same is done for an operator $ \pi (L)$, where $ \pi $ is a unitary representation of $ G$ induced from a unitary character of a normal connected subgroup $ H$ of $ G$. Hence the case of the Hermite operator $ - {d^2}/d{x^2} + {x^2}$ is covered and an $ {L^p}$-multiplier theorem for classical Hermite expansions is obtained.

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Additional Information

PII: S 0002-9939(1985)0774006-9
Keywords: Multiplier, eigenfunction expansion, Rockland operator, Hermite function, homogenous type space, maximal function, Riesz means
Article copyright: © Copyright 1985 American Mathematical Society

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