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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the restriction of the Fourier transform to a conical surface


Author: Bartolome Barcelo Taberner
Journal: Trans. Amer. Math. Soc. 292 (1985), 321-333
MSC: Primary 42B10
MathSciNet review: 805965
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Abstract: Let $ \Gamma $ be the surface of a circular cone in $ {{\mathbf{R}}^3}$. We show that if $ 1 \leqslant p < 4/3$, $ 1/q = 3(1 - 1/p)$ and $ f \in {L^p}({{\mathbf{R}}^3})$, then the Fourier transform of $ f$ belongs to $ {L^q}(\Gamma ,d\sigma )$ for a certain natural measure $ \sigma $ on $ \Gamma $. Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint $ p = 4/3$, with logarithmic growth of the bound as the thickness of the annulus tends to zero.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0805965-8
PII: S 0002-9947(1985)0805965-8
Article copyright: © Copyright 1985 American Mathematical Society