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Cho, Chu-Hee; Lee, Jungjin

doi:10.1016/j.jfa.2017.04.015

Cited by 17 publications

(18 citation statements)

References 15 publications

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“…Stovall [27] recently proved the endpoint case. Also, Cho and Lee [11] and Kim [20] improved the range. Perhaps surprisingly at the first thought, their methods did not give the desired result for any other surface with negative Gaussian curvature.…”

Section: Introductionmentioning

confidence: 99%

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

Buschenhenke

Müller

Vargas

2019

Proc. London Math. Soc.

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In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non‐vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for 2‐surfaces, the only case of a hyperbolic surface for which Fourier restriction estimates could be established that are analogous to the ones known for elliptic surfaces is the hyperbolic paraboloid or ‘saddle’ z=xy. The bilinear method gave here sharp results for p>10/3, and this result was recently improved to p>3.25. This paper aims to be the first step in extending those results to more general hyperbolic surfaces. We consider a specific cubic perturbation of the saddle and obtain the sharp result, up to the endpoint, for p>10/3. In the application of the bilinear method, we show that the behavior at small scales in our surface is drastically different from the saddle. Indeed, as it turns out, in some regimes the perturbation term assumes a dominant role, which necessitates the introduction of a number of new techniques that should also be useful for the study of more general hyperbolic surfaces. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of perturbations.

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Section: Introductionmentioning

confidence: 99%

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

Buschenhenke

Müller

Vargas

2019

Proc. London Math. Soc.

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“…It is conjectured [33] that the operators E Q (and, in fact, extension operators associated to any surface of non-vanishing Gaussian curvature) are L p (B n−1 ) → L p (R n ) bounded for p > 2 • n n−1 , regardless of the signature. Restriction theory for hyperbolic parabolae involves a number of novel considerations compared with that of the elliptic case, and has been investigated in a variety of works [1,12,18,25,35,38]. There has also been a recent programme [13][14][15][16] to investigate L p -boundedness of extension operators associated to negatively-curved surfaces given by smooth perturbations of the hyperbolic paraboloid H 2,0 from Example 1.4; this turns out to be a rather subtle problem for p < 4.…”

Section: Non-sharpnessmentioning

confidence: 99%

Sharp $$L^p$$ estimates for oscillatory integral operators of arbitrary signature

Hickman

Iliopoulou

2022

Math. Z.

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The sharp range of $$L^p$$ L p -estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a general signature assumption on the phase. This simultaneously generalises earlier work of the authors and Guth, which treats the maximal signature case, and also work of Stein and Bourgain–Guth, which treats the minimal signature case.

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“…Recently, Stovall [28] was able to include also the end-point case. Moreover, Cho and Lee [12], and Kim [21], improved the range by adapting ideas by Guth [14,15] which are based on the polynomial partitioning method. Results on higher dimensional hyperbolic paraboloids have just been reported by Barron [1].…”

Section: Introductionmentioning

confidence: 99%

A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning

2022

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We consider a surface with negative curvature in $${{\mathbb {R}}}^3,$$ R 3 , which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of one-variate perturbations, and we hope that the present paper will further help to pave the way for the study of general perturbations of the saddle by means of the polynomial partitioning method.

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Improved restriction estimate for hyperbolic surfaces in R3

Abstract: Abstract. Recently, L. Guth improved the restriction estimate for the surfaces with strictly positive Gaussian curvature in R 3 . In this paper we extend his restriction estimate to the surfaces with strictly negative Gaussian curvature.

Cited by 17 publications

References 15 publications

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

Sharp $$L^p$$ estimates for oscillatory integral operators of arbitrary signature

A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning

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