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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Oscillatory integrals and Fourier transforms of surface carried measures


Authors: Michael Cowling and Giancarlo Mauceri
Journal: Trans. Amer. Math. Soc. 304 (1987), 53-68
MSC: Primary 42B10; Secondary 42B25
DOI: https://doi.org/10.1090/S0002-9947-1987-0906805-0
MathSciNet review: 906805
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Abstract: We suppose that $ S$ is a smooth hypersurface in $ {{\mathbf{R}}^{n + 1}}$ with Gaussian curvature $ \kappa $ and surface measure $ dS$, $ w$ is a compactly supported cut-off function, and we let $ {\mu _\alpha }$ be the surface measure with $ d{\mu _\alpha } = w{\kappa ^\alpha }\,dS$. In this paper we consider the case where $ S$ is the graph of a suitably convex function, homogeneous of degree $ d$, and estimate the Fourier transform $ {\hat \mu _\alpha }$. We also show that if $ S$ is convex, with no tangent lines of infinite order, then $ {\hat \mu _\alpha }(\xi )$ decays as $ \vert\xi {\vert^{ - n / 2}}$ provided $ \alpha \geqslant [(n + 3)/2]$. The techniques involved are the estimation of oscillatory integrals; we give applications involving maximal functions.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0906805-0
Article copyright: © Copyright 1987 American Mathematical Society