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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Convolution operators on groups and multiplier theorems for Hermite and Laguerre expansions


Author: Jolanta Długosz
Journal: Proc. Amer. Math. Soc. 102 (1988), 919-924
MSC: Primary 43A22; Secondary 22E30, 42C10
DOI: https://doi.org/10.1090/S0002-9939-1988-0934868-1
MathSciNet review: 934868
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Abstract: Using harmonic analysis on nilpotent Lie groups the following theorem is proved.

Let a sequence $ \{ {a_{\text{n}}}\} $ be defined by a function $ K \in {C^N}({{\mathbf{R}}_ + })$ such that $ {\sup _{\lambda > 0}}\vert{K^{(j)}}(\lambda ){\lambda ^j}\vert < \infty ,j = 0,1, \ldots ,N$, for $ N$ sufficiently large, putting $ {a_{\text{n}}} = K(\vert{\text{n}}\vert + m/2)$. Let $ {\varphi _{\text{n}}}$ be either Hermite or Laguerre functions. Then the operator

$\displaystyle \sum\limits_{\text{n}} {(f,{\varphi _{\text{n}}}){\varphi _{\text... ...ts_{\text{n}} {{a_{\text{n}}}(f,{\varphi _{\text{n}}}){\varphi _{\text{n}}}} } $

is bounded on $ {L^p}\left( {{\mathbb{R}^m}} \right)$ or $ {L^p}(\mathbb{R}_ + ^m)$ respectively, $ 1 < p < \infty $.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0934868-1
Keywords: Multiplier, Laguerre expansions, Hermite expansions, convolution operators, Rockland operator
Article copyright: © Copyright 1988 American Mathematical Society

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