$L^p$-multipliers: a new proof of an old theorem
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- by Tomas P. Schonbek
- Proc. Amer. Math. Soc. 102 (1988), 361-364
- DOI: https://doi.org/10.1090/S0002-9939-1988-0921000-3
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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 \[ |\xi {|^{ - \beta }}\psi (\xi )\exp (i|\xi {|^\alpha })\] is a multiplier in ${L^p}({\mathbb {R}^n})$ for $|1/p - 1/2| < \beta /n\alpha$; is not a multiplier in ${L^p}\left ( {{\mathbb {R}^n}} \right )$ for $|1/p - 1/2| > \beta /n\alpha$.References
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Bibliographic Information
- © Copyright 1988 American Mathematical Society
- 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