Uniform Estimates for the Fourier Transform of Surface Carried Measures in ℝ3 and an Application to Fourier Restriction

Let S be a hypersurface in which is the graph of a smooth, finite type function φ , and let μ = ρ   dσ be a surface carried measure on S , where dσ denotes the surface element on S and ρ a smooth density with sufficiently small support. We derive uniform estimates for the Fourier transform of μ , wh...

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Bibliographic Details
Published in:The Journal of fourier analysis and applications 2011-12, Vol.17 (6), p.1292-1332
Main Authors: Ikromov, Isroil A., Müller, Detlef
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Approximations and Expansions
Fourier Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Partial Differential Equations
Signal,Image and Speech Processing
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Summary:Let S be a hypersurface in which is the graph of a smooth, finite type function φ , and let μ = ρ   dσ be a surface carried measure on S , where dσ denotes the surface element on S and ρ a smooth density with sufficiently small support. We derive uniform estimates for the Fourier transform of μ , which are sharp except for the case where the principal face of the Newton polyhedron of φ , when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp L p - L 2 Fourier restriction theorem for S in the case where the original coordinates are adapted to φ . This improves on earlier joint work with M. Kempe.
ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-011-9191-4