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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Fourier restriction and the Newton polygon


Author: Ákos Magyar
Journal: Proc. Amer. Math. Soc. 137 (2009), 615-625
MSC (2000): Primary 42B10; Secondary 43A32
Published electronically: August 26, 2008
MathSciNet review: 2448583
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Abstract: Local $ L^p\to L^2$ bounds are proved for the restriction of the Fourier transform to analytic surfaces of the form $ S=(x,f(x))$ in $ \mathbb{R}^3$. It is found that the range of exponents is determined by the so-called distance of the Newton polygon, associated to $ f$, except when the principal quasi-homogeneous part of $ f(x)$ contains a factor of high multiplicity. The proofs are based on the method of Phong-Stein and Rychkov, adapted to scalar oscillatory integrals.


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Additional Information

Ákos Magyar
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Address at time of publication: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Room 121, Vancouver, British Columbia V6T 1Z2, Canada
Email: amagyar2000@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09510-5
PII: S 0002-9939(08)09510-5
Keywords: Fourier transform, oscillatory integrals, Newton polygon
Received by editor(s): August 20, 2007
Received by editor(s) in revised form: January 25, 2008
Published electronically: August 26, 2008
Additional Notes: This research was supported in part by NSF Grant DMS-0456490
Communicated by: Andreas Seeger
Article copyright: © Copyright 2008 American Mathematical Society