Glasnik Matematicki, Vol. 57, No. 2 (2022), 313-319. \( \)

A NOTE ON MAXIMAL FOURIER RESTRICTION FOR SPHERES IN ALL DIMENSIONS

Marco Vitturi

School of Mathematical Sciences, University College Cork, Western Gateway Building, Western Road, Cork, Ireland
e-mail:marco.vitturi@ucc.ie

Abstract.   We prove a maximal Fourier restriction theorem for hypersurfaces in \(\mathbb{R}^{d}\) for any dimension \(d\geq 3\) in a restricted range of exponents given by the Tomas-Stein theorem (spheres being the most canonical example). The proof consists of a simple observation. When \(d=3\) the range corresponds exactly to the full Tomas-Stein one, but is otherwise a proper subset when \(d>3\). We also present an application regarding the Lebesgue points of functions in \(\mathcal{F}(L^p)\) when \(p\) is sufficiently close to 1.

2020 Mathematics Subject Classification.   42B10, 42B25

Key words and phrases.   Fourier restriction, maximal operators

https://doi.org/10.3336/gm.57.2.10

References:

1. W. Beckner, A. Carbery, S. Semmes and F. Soria, A note on restriction of the Fourier transform to spheres, Bull. London Math. Soc. 21 (1989), 394–398.
   MathSciNet    CrossRef

2. C. Bilz, Large sets without Fourier restriction theorems, Trans. Amer. Math. Soc. 375 (2022), 6983–7000.
   MathSciNet    CrossRef

3. M. Fraccaroli, Uniform Fourier restriction for convex curves, preprint.
   Link

4. M. Jerusum, Maximal operators and Fourier restriction on the moment curve, Proc. Amer. Math. Soc. 150 (2022), 3863–3873.
   MathSciNet    CrossRef

5. V. Kovač, Fourier restriction implies maximal and variational Fourier restriction, J. Funct. Anal. 277 (2019), 3355–3372.
   MathSciNet    CrossRef

6. V. Kovač and D. Oliveira e Silva, A variational restriction theorem, Arch. Math. (Basel) 117 (2021), 65–78.
   MathSciNet    CrossRef

7. D. Müller, F. Ricci and J. Wright, A maximal restriction theorem and Lebesgue points of functions in \(\mathcal{F}(L^p)\), Rev. Mat. Iberoam. 35 (2019), 693–702.
   MathSciNet    CrossRef

8. D. Oberlin, A uniform Fourier restriction theorem for surfaces in \(\mathbb{R}^3\), Proc. Amer. Math. Soc. 132 (2004), 1195–1199.
   MathSciNet    CrossRef

9. J. P. G. Ramos, Maximal restriction estimates and the maximal function of the Fourier transform, Proc. Amer. Math. Soc. 148 (2020), 1131–1138.
   MathSciNet    CrossRef

10. J. P. G. Ramos, Low-dimensional maximal restriction principles for the Fourier transform, Indiana Univ. Math. J. 71 (2022), 339–357.
    MathSciNet    CrossRef

11. P. Sjölin, Fourier multipliers and estimates of the Fourier transform of measures carried by smooth curves in \(\mathbb{R}^2\), Studia Math. 51 (1974), 169–182.
    MathSciNet    CrossRef

12. E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, 1993.
    MathSciNet