Abstract
The main result of this note is the strengthening of a quite arbitrary a priori Fourier restriction estimate to a multi-parameter maximal estimate of the same type. This allows us to discuss a certain multi-parameter Lebesgue point property of Fourier transforms, which replaces Euclidean balls by standard ellipsoids or axes-parallel rectangles. Along the lines of the same proof, we also establish a d-parameter Menshov–Paley–Zygmund-type theorem for the Fourier transform on \({\mathbb {R}}^d\). Such a result is interesting for \(d\geqslant 2\) because, in a sharp contrast with the one-dimensional case, the corresponding endpoint \({\text {L}}^2\) estimate (i.e., a Carleson-type theorem) is known to fail since the work of C. Fefferman in 1970. Finally, we show that a Strichartz estimate for a given homogeneous constant-coefficient linear dispersive PDE can sometimes be strengthened to a certain pseudo-differential version.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benedek, A., Panzone, R.: The space \(L^{p}\), with mixed norm. Duke Math. J. 28, 301–324 (1961)
Bilz, C.: Large sets without Fourier restriction theorems. Trans. Amer. Math. Soc. 375(10), 6983–7000 (2022)
Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)
Carleson, L., Sjölin, P.: Oscillatory integrals and a multiplier problem for the disc. Studia Math. 44, 287–299 (1972)
Christ, M., Kiselev, A.: Maximal functions associated to filtrations. J. Funct. Anal. 179(2), 409–425 (2001)
Cordero, E., Zucco, D.: Strichartz estimates for the Schrödinger equation. Cubo 12(3), 213–239 (2010)
Fefferman, C.: On the divergence of multiple Fourier series. Bull. Amer. Math. Soc. 77, 191–195 (1971)
Fraccaroli, M.: Uniform Fourier restriction for convex curves (preprint), (2021). Available at: arXiv:2111.06874
Jesurum, M.: Maximal operators and Fourier restriction on the moment curve. Proc. Amer. Math. Soc. 150(9), 3863–3873 (2022)
Koch, H., Tataru, D., Vişan, M.: Dispersive Equations and Nonlinear Waves. Generalized Korteweg de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps, volume 45 of Oberwolfach Seminars. Birkhäuser/Springer, Basel (2014)
Kovač, V.: Fourier restriction implies maximal and variational Fourier restriction. J. Funct. Anal. 277(10), 3355–3372 (2019)
Kovač, V., Silva, D.O.: A variational restriction theorem. Arch. Math. (Basel) 117(1), 65–78 (2021)
Krause, B., Mirek, M., Trojan, B.: Two-parameter version of Bourgain’s inequality: rational frequencies. Adv. Math. 323, 720–744 (2018)
Müller, D., Ricci, F., Wright, J.: A maximal restriction theorem and Lebesgue points of functions in \(\cal{F} (L^p)\). Rev. Mat. Iberoam. 35(3), 693–702 (2019)
Muscalu, C., Pipher, J., Tao, T., Thiele, C.: Bi-parameter paraproducts. Acta Math. 193(2), 269–296 (2004)
Ramos, J.P.G.: Maximal restriction estimates and the maximal function of the Fourier transform. Proc. Amer. Math. Soc. 148(3), 1131–1138 (2020)
Ramos, J.P.G.: Low-dimensional maximal restriction principles for the Fourier transform. Indiana Univ. Math. J. 71(1), 339–357 (2022)
Sjölin, P.: Fourier multipliers and estimates of the Fourier transform of measures carried by smooth curves in \(\mathbb{R} ^{2}\). Studia Math. 51, 169–182 (1974)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, No. 43. Princeton University Press, Princeton, N.J. (1993)
Strichartz, R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(3), 705–714 (1977)
Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2006)
Tao, T.: Math 247a: Fourier Analysis (lecture notes, University of California, Los Angeles), (2006). Available at: http://www.math.ucla.edu/~tao/247a.1.06f/ (accessed on July 21, 2022)
Vitturi, M.: Almost Originality, a Mathematical Journal. Post from April 22, (2019). Available at: https://almostoriginality.wordpress.com/2019/04/22/ (accessed on July 21, 2022)
Vitturi, M.: A note on maximal Fourier Restriction for spheres in all dimensions. Glas. Mat. Ser. III 57(2), 313–319 (2022)
Acknowledgements
The authors are grateful to the anonymous referee for numerous suggestions, which have greatly improved the presentation.
Funding
This work was supported in part by the Croatian Science Foundation project UIP-2017-05-4129 (MUNHANAP).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yoshihiro Sawano.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bulj, A., Kovač, V. Multi-parameter Maximal Fourier Restriction. J Fourier Anal Appl 30, 26 (2024). https://doi.org/10.1007/s00041-024-10083-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-024-10083-1
Keywords
- Fourier transform
- Fourier restriction operator
- Maximal estimate
- Multi-parameter estimate
- Convergence almost everywhere
- Christ–Kiselev lemma