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

We continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid z = x y which are of the form z = x y + h ( y ) , where h ( y ) is a smooth function which is flat at the origin. The case of perturbations of finite type had already...

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Bibliographic Details
Published in:The Journal of geometric analysis 2021-07, Vol.31 (7), p.6941-6986
Main Authors: Buschenhenke, Stefan, Müller, Detlef, Vargas, Ana
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Decay rate
Differential Geometry
Dynamical Systems and Ergodic Theory
Elias M. Stein: In Memoriam
Estimates
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
Parabolic bodies
Perturbation
Wave packets
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Summary:We continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid z = x y which are of the form z = x y + 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 λ , 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 ′ ′ ′ .
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-020-00587-9