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Partitions of Flat One-Variate Functions and a Fourier Restriction Theorem for Related Perturbations of the Hyperbolic Paraboloid

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Abstract

We continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid \(z=xy\) which are of the form \(z=xy+h(y),\) where h(y) is a smooth function which is flat at the origin. The case of perturbations of finite type had already been handled before, but the flat case imposes several new obstacles. By means of a decomposition into intervals on which \(|h'''|\) is of a fixed size \({\lambda },\) we can apply methods devised in preceding papers, but since we lose control on higher order derivatives of h we are forced to rework the bilinear method for wave packets that are only slowly decaying. Another problem lies in the passage from bilinear estimates to linear estimates, for which we need to require some monotonicity of \(h'''.\)

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Authors and Affiliations

  1. Mathematisches Seminar, C.A.-Universität Kiel, Ludewig-Meyn-Strasse 4, 24118, Kiel, Germany

    Stefan Buschenhenke & Detlef Müller

  2. Departmento de Mathemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Ana Vargas

Authors
  1. Stefan Buschenhenke
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  2. Detlef Müller
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  3. Ana Vargas
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Correspondence to Detlef Müller.

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In memory of Eli Stein.

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Stefan Buschenhenke was partially supported by the ERC Grant 307617. Stefan Buschenhenke and Detlef Müller The first two authors were partially supported by the DFG Grants MU 761/11-1 and MU 761/11-2. Ana Vargas was partially supported by Grants MTM2013-40945 (MINECO) and MTM2016-76566-P and PID2019-105599GB-100 (Ministerio de Ciencia, Innovaci\(\acute{\text {o}}\)n y Universidades), Spain.

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Buschenhenke, S., Müller, D. & Vargas, A. Partitions of Flat One-Variate Functions and a Fourier Restriction Theorem for Related Perturbations of the Hyperbolic Paraboloid. J Geom Anal 31, 6941–6986 (2021). https://doi.org/10.1007/s12220-020-00587-9

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