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

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Abstract

Let S be a hypersurface in \({\Bbb{R}}^{3}\) 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 \(\hat{\mu}\) 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.

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Authors and Affiliations

  1. Department of Mathematics, Samarkand State University, University Boulevard 15, 140104, Samarkand, Uzbekistan

    Isroil A. Ikromov

  2. Mathematisches Seminar, C.A.-Universität Kiel, Ludewig-Meyn-Straße 4, 24098, Kiel, Germany

    Detlef Müller

Authors
  1. Isroil A. Ikromov
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  2. Detlef Müller
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Correspondence to Detlef Müller.

Additional information

Communicated by R. Strichartz.

We acknowledge the support for this work by the Deutsche Forschungsgemeinschaft.

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Ikromov, I.A., Müller, D. Uniform Estimates for the Fourier Transform of Surface Carried Measures in ℝ3 and an Application to Fourier Restriction. J Fourier Anal Appl 17, 1292–1332 (2011). https://doi.org/10.1007/s00041-011-9191-4

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  • DOI: https://doi.org/10.1007/s00041-011-9191-4

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