Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates

Barron, Alex

doi:10.4171/rmi/1310

Rev. Mat. Iberoam.

2021

DOI: 10.4171/rmi/1310

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Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates

Alex Barron

Abstract: Let H be a .d 1/-dimensional hyperbolic paraboloid in R d and let Ef be the Fourier extension operator associated to H, with f supported in B d 1 .0; 2/. We prove that kEf k L p .B.0;R// Ä C " R " kf k L p for all p 2.d C 2/=d whenever d=2 m C 1, where m is the minimum between the number of positive and negative principal curvatures of H. Bilinear restriction estimates for H proved by S. Lee and Vargas play an important role in our argument.

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Cited by 3 publications

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Fourier restriction for smooth hyperbolic 2-surfaces

2022

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We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in $$\mathbb {R}^3.$$ R 3 . Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding “exceptional” sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.

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Fourier restriction for smooth hyperbolic 2-surfaces

2022

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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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Restriction estimates for hyperbolic cone in high dimensions

Zhong

Zheng

2023

Proc. Amer. Math. Soc.

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In this paper, we obtain the L p L^p restriction estimates for the truncated conic surface Σ = { ( ξ ′ , ξ n , − ξ n − 1 ⟨ ξ ′ , N ξ ′ ⟩ ) : ( ξ ′ , ξ n ) ∈ B n − 1 ( 0 , 1 ) × [ 1 , 2 ] } \begin{equation*} \Sigma =\big \{(\xi ’,\xi _n,-\xi _n^{-1}\langle \xi ’,N\xi ’\rangle ): (\xi ’,\xi _n)\in B^{n-1}(0,1)\times [1,2]\big \} \end{equation*} with N = I n − 1 − m ⊕ ( − I m ) N=I_{n-1-m}\oplus (-I_m) for m ≤ ⌊ n − 3 2 ⌋ m\leq \lfloor \tfrac {n-3}2\rfloor provided p > 2 ( n + 3 ) n + 1 p>\tfrac {2(n+3)}{n+1} . The main ingredients of the proof are the bilinear estimates of strongly separated property and a geometric distribution about caps.

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Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates

Cited by 3 publications

References 19 publications

Fourier restriction for smooth hyperbolic 2-surfaces

Fourier restriction for smooth hyperbolic 2-surfaces

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

Restriction estimates for hyperbolic cone in high dimensions

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