Abstract
In this note, 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 of finite type. Our results build on previous joint work in which we have studied the case h(y) = y 3∕3 by means of the bilinear method. As it turns out, the understanding of that special case becomes also crucial for the treatment of arbitrary finite type perturbation terms h(y).
To Fulvio at the occasion of his seventieth birthday
The first author was partially supported by the ERC grant 307617. The first two authors were partially supported by the DFG grant MU 761/11-2. The third author was partially supported by the grants PID2019-105599GB-I00/ AEI / 10.13039/501100011033 (Ministerio de Ciencia e Innovación) and MTM2016-76566-P (Ministerio de Ciencia, Innovación y Universidades), Spain.
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Notes
- 1.
We don’t need to distinguish precisely the two cases δ > 1 and δ ≤ 1 from the Theorem, since the desired bounds are comparable for δ ∼ 1.
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The authors would like to express their sincere gratitude to the referee for many valuable suggestions which have greatly helped to improve the presentation of the material in this article.
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Buschenhenke, S., Müller, D., Vargas, A. (2021). On Fourier Restriction for Finite-Type Perturbations of the Hyperbolic Paraboloid. In: Ciatti, P., Martini, A. (eds) Geometric Aspects of Harmonic Analysis. Springer INdAM Series, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-030-72058-2_5
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