Abstract
In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfaces; we demonstrate this connection by using our positive results to prove new restriction inequalities for a class of hypersurfaces having some additive structure.
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Beckner, W., Carbery, A., Semmes, S., Soria, F.: A note on restriction of the Fourier transform to spheres. Bull. Lond. Math. Soc. 21(4), 394–398 (1989)
Buschenhenke, S., Müller, D., Vargas, A.: A Fourier restriction theorem for a two-dimensional surface of finite type. Anal. PDE 10(4), 817–891 (2017)
Candy, T.: Multi-scale bilinear restriction estimates for general phases. Math. Ann. 375(1–2), 777–843 (2019)
Dendrinos, S., Zimmermann, E.: On \(L^p\)-improving for averages associated to mixed homogeneous polynomial hypersurfaces in \({\mathbb{R}}^3\). J. Anal. Math. 138(2), 563–595 (2019)
Drury, S.W., Guo, K.: Some remarks on the restriction of the Fourier transform to surfaces. Math. Proc. Camb. Philos. Soc. 113(1), 153–159 (1993)
Fefferman, C.: Inequalities for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)
Fefferman, C.: The multiplier problem for the ball. Ann. Math. 2(94), 330–336 (1971)
Ferreyra, E., Urciuolo, M.: Restriction theorems for anisotropically homogeneous hypersurfaces in \({\mathbb{R}}^{n+1}\). Ga. Math. J. 15(4), 643–651 (2008)
Ferreyra, E., Urciuolo, M.: Fourier restriction estimates to mixed homogeneous surfaces. JIPAM. J. Inequal. Pure Appl. Math. 10(2), 35 (2009)
Guth, L.: A restriction estimate using polynomial partitioning. J. Am. Math. Soc. 29(2), 371–413 (2016)
Guth, L.: Restriction estimates using polynomial partitioning II. Acta Math. 221(1), 81–142 (2018)
Hickman, J., Rogers, K.M.: Improved Fourier restriction estimates in higher dimensions. Camb. J. Math. 7(3), 219–282 (2019)
Schwend, J.: Near optimal \(L^p \rightarrow L^q\) estimates for Euclidean averages on prototypical hypersurfaces in \({\mathbb{R}}^3\). Preprint, arXiv:2012.15789
Tao, T.: A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal. 13(6), 1359–1384 (2003)
Tao, T., Vargas, A.: A bilinear approach to cone multipliers. I. Restriction estimates. Geom. Funct. Anal. 10(1), 185–215 (2000)
Tao, T., Vargas, A., Vega, L.: A bilinear approach to the restriction and Kakeya conjectures. J. Am. Math. Soc. 11(4), 967–1000 (1998)
Varčenko, A.N.: Newton polyhedra and estimates of oscillatory integrals. (Russian). Funkcional. Anal. i Priložen. 10(3), 13–38 (1976)
Wang, H.: A restriction estimate in \({\mathbb{R}}^3\) using brooms. Preprint, arXiv:1802.04312
Zygmund, A.: On Fourier coefficients and transforms of functions of two variables. Studia Math. 50, 189–201 (1974)
Acknowledgements
This research was supported in part by National Science Foundation Grants DMS-1653264 and DMS-1147523. The authors would like to thank the anonymous referee for comments and suggestions that significantly improved the article.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Communicated by Loukas Grafakos.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Schwend, J., Stovall, B. Fourier restriction above rectangles. Math. Ann. 381, 1807–1836 (2021). https://doi.org/10.1007/s00208-021-02202-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-021-02202-w