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

 

Decay rates of Fourier transforms of curves


Author: B. P. Marshall
Journal: Trans. Amer. Math. Soc. 310 (1988), 115-126
MSC: Primary 42B10
MathSciNet review: 948194
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Abstract: Let $ d\mu $ be a smooth measure on a nondegenerate curve in $ {{\mathbf{R}}^n}$. This paper examines the decay rate of spherical averages of its Fourier transform $ \widehat{d\mu }$. Thus estimates of the following form are considered:

$\displaystyle {\left( {\int_{{\sum _r}} {\vert\widehat{d\mu }(\xi {\vert^p}d\xi } } \right)^{1/p}} \leqslant C{r^{ - \sigma }}\vert\vert f\vert\vert$

where $ {\sum _r} = \{ \xi \in {{\mathbf{R}}^n}:\vert\xi \vert = r\} $.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0948194-2
PII: S 0002-9947(1988)0948194-2
Article copyright: © Copyright 1988 American Mathematical Society