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Proceedings of the American Mathematical Society

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

 

 

A note on the ``hyperbolic'' Bochner-Riesz means


Author: Anthony Carbery
Journal: Proc. Amer. Math. Soc. 92 (1984), 397-400
MSC: Primary 42B15
DOI: https://doi.org/10.1090/S0002-9939-1984-0759661-0
MathSciNet review: 759661
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Abstract: We consider the $ {L^p}({{\mathbf{R}}^2})$ boundedness properties of the Fourier multiplier $ m({\xi _1},{\xi _2}) = (1 - \xi _1^2\xi _2^2)_ + ^\alpha {\text{ for }}\alpha > 0$. We prove that if $ \alpha \geqslant \frac{1}{2}$, then $ m$ is bounded on $ {L^p}$, $ 1 < p < \infty $, and that if $ \alpha > 0$, then $ m$ is bounded on $ {L^p}$, $ \frac{4}{3} \leqslant p \leqslant 4$.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0759661-0
Keywords: Fourier multipliers, square functions, maximal functions
Article copyright: © Copyright 1984 American Mathematical Society