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$ L^p$-multipliers: a new proof of an old theorem


Author: Tomas P. Schonbek
Journal: Proc. Amer. Math. Soc. 102 (1988), 361-364
MSC: Primary 42B15,; Secondary 46E30,47B38
DOI: https://doi.org/10.1090/S0002-9939-1988-0921000-3
MathSciNet review: 921000
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Abstract | References | Similar Articles | Additional Information

Abstract: New proofs are given for the following results of Hirschman and Wainger: Let $ \psi \in {C^\infty }({\mathbb{R}^n})$ vanish in a neighborhood of the origin; $ \psi (\xi ) = 1$ for large $ \xi $. Then

$\displaystyle \vert\xi {\vert^{ - \beta }}\psi (\xi )\exp (i\vert\xi {\vert^\alpha })$

is a multiplier in $ {L^p}({\mathbb{R}^n})$ for $ \vert 1/p - 1/2\vert < \beta /n\alpha $; is not a multiplier in $ {L^p}\left( {{\mathbb{R}^n}} \right)$ for $ \vert 1/p - 1/2\vert > \beta /n\alpha $.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0921000-3
Article copyright: © Copyright 1988 American Mathematical Society

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