Abstract
We prove new Fourier restriction estimates to the unit sphere \({\mathbb{S}^{d-1}}\) on the class of O(d − k) × O(k)-symmetric functions, for every d ≥ 4 and 2 ≤ k ≤ d − 2. As an application, we establish the existence of maximizers for the endpoint Stein–Tomas inequality within that class. Moreover, we construct examples showing that the range of Lebesgue exponents in our estimates is sharp.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer, New York, 2001.
W. Beckner, Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc. 136 (2008), 62–65.
W. Beckner, Pitt’s inequality and the fractional Laplacian: sharp error estimates, Forum Math. 24 (2012), 62–65.
P. Bégout and A. Vargas, Mass concentration phenomena for the L2-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc. 359 (2007), 62–65.
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction. Springer, Berlin–New York, 1976.
C. Biswas and B. Stovall, Existence of extremizers for Fourier restriction to the moment curve, arXiv:2012.01528 [math.CA]
S. Bloom and G. Sampson, Weighted spherical restriction theorems for the Fourier transform, Illinois J. Math. 36 (1992), 62–65.
H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 62–65.
L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 62–65.
E. Carneiro, D. Oliveira e Silva and M. Sousa, Extremizers for Fourier restriction on hyperboloids, Ann. Inst. H. Poincaré C Anal. Non Linéaire 36 (2019), 62–65.
E. Carneiro, D. Oliveira e Silva, M. Sousa and B. Stovall, Extremizers for adjoint Fourier restriction on hyperboloids: the higher dimensional case, Indiana Univ. Math. J. 70 (2021), 62–65.
M. Christ and S. Shao, Existence of extremals for a Fourier restriction inequality, Anal. PDE. 5 (2012), 62–65.
L. de Carli and L. Grafakos, On the restriction conjecture, Michigan Math. J. 52 (2004), 62–65.
L. Fanelli, L. Vega and N. Visciglia, On the existence of maximizers for a family of restriction theorems, Bull. Lond. Math. Soc. 43 (2011), 62–65.
C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 62–65.
D. Foschi, Global maximizers for the sphere adjoint Fourier restriction inequality, J. Funct. Anal. 268 (2015), 62–65.
D. Foschi and D. Oliveira e Silva, Some recent progress on sharp Fourier restriction theory, Anal. Math. 43 (2017), 62–65.
R. Frank, E. H. Lieb and J. Sabin, Maximizers for the Stein–Tomas inequality, Geom. Funct. Anal. 26 (2016), 62–65.
B. Green, Roth’s theorem in the primes, Ann. of Math. (2) 161 (2005), 62–65.
L. Guth, Restriction estimates using polynomial partitioning II, Acta Math. 221 (2018), 62–65.
J. Hickman and K. Rogers, Improved Fourier restriction estimates in higher dimensions, Camb. J. Math. 7 (2019), 62–65.
C. Kenig, A. Ruiz and C. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), 62–65.
D. Lass Fernandez, Lorentz spaces, with mixed norms, J. Funct. Anal. 25 (1977), 62–65.
E. H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, 2001.
P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 62–65.
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 62–65.
R. Mandel, Real interpolation for mixed Lorentz spaces and Minkowski’s inequality, arXiv:2303.00607.
C. Miao, J. Zhang and J. Zheng, A note on the cone restriction conjecture, Proc. Amer. Math. Soc. 140 (2012), 62–65.
C. Miao, J. Zhang and J. Zheng, Linear adjoint restriction estimates for paraboloid, Math. Z. 292 (2019), 62–65.
G. Mockenhaupt, Salem sets and restriction properties of Fourier transforms, Geom. Funct. Anal. 10 (2000), 62–65.
R. O’Neil, Convolution operators and L(p, q) spaces, Duke Math. J. 30 (1963), 62–65.
R. Quilodrán, On extremizing sequences for the adjoint restriction inequality on the cone, J. Lond. Math. Soc. (2) 87 (2013), 62–65.
R. Quilodrán, Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid, J. Anal. Math. 125 (2015), 62–65.
J. Ramos, Arefinement of the Strichartz inequality for the wave equation with applications, Adv. Math. 230 (2012), 62–65.
S. Shao, A note on the cone restriction conjecture in the cylindrically symmetric case, Proc. Amer. Math. Soc. 137 (2009), 62–65.
S. Shao, Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case, Rev. Mat. Iberoam. 25 (2009), 62–65.
S. Shao, On existence of extremizers for the Tomas-Stein inequality for S1, J. Funct. Anal. 270 (2016), 3996–4038.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
E. M. Stein and R. Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis, Princeton University Press, Princeton, NJ, 2011.
E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 (1958), 62–65.
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971.
B. Stovall, Waves, spheres, and tubes: a selection of Fourier restriction problems, methods, and applications, Notices Amer. Math. Soc. 66 (2019), 62–65.
B. Stovall, Extremizability of Fourier restriction to the paraboloid, Adv. Math. 360 (2020), 62–65.
R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 62–65.
T. Tao, Some recent progress on the restriction conjecture, in Fourier Analysis and Convexity, Birkhäuser, Boston, MA, 2004, pp. 217–243.
P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 62–65.
G. N. Watson, A Treatise on the theory of Bessel functions. Cambridge University Press, Cambridge, 1995.
T. Weth and T. Yeşil, Fourier extension estimates for symmetric functions and applications to nonlinear Helmholtz equations, Ann. Mat. Pura Appl. (4) 200 (2021), 62–65.
A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 62–65.
Acknowledgements
RM is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Project-ID 258734477–SFB 1173. DOS is supported by the EPSRC New Investigator Award “Sharp Fourier Restriction Theory”, grant no. EP/T001364/1, and the DFG under Germany’s Excellence Strategy–EXC-2047/1–390685813, and is grateful to Giuseppe Negro, Andreas Seeger and Mateus Sousa for valuable discussions during the preparation of this work. The authors thank the anonymous referee for carefully reading the manuscript and valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution and reproduction in any medium, provided the appropriate credit is given to the original authors and the source, and a link is provided to the Creative Commons license, indicating if changes were made (https://creativecommons.org/licenses/by/4.0/)
About this article
Cite this article
Mandel, R., Oliveira e Silva, D. The Stein–Tomas inequality under the effect of symmetries. JAMA 150, 547–582 (2023). https://doi.org/10.1007/s11854-023-0282-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-023-0282-3