Fourier restriction constant for the 2-sphere ($C_{46}$)

• Many papers work with the paraboloid model surface (or a bounded subset thereof); by localization and rescaling, the best-known exponents for compact strictly convex surfaces (including $S^2$) track the paraboloid results up to standard $\varepsilon$-losses that can often be removed by "epsilon removal lemmas".

• For most of the results in the literature, the $L^\infty(S^2)$ norm on the right-hand side can be replaced with $L^q(S^2)$ for various $q$; for instance, in the Tomas-Stein theorem one can take $q=2$. There are also bilinear and multilinear variants of the conjecture. See for instance [Tao2004] for more discussion.

• Stein's restriction conjecture $C_{46}=3$ implies the Kakeya conjecture in ${\mathbb R}^3$ (see, e.g., [Tao2004]), which was recently proven in [WZ2025].

References (from the repository page)

• [Bo1991] Bourgain, J. Besicovitch-type maximal operators and applications to Fourier analysis. Geom. Funct. Anal. 1 (2) (1991), 147–187.
• [BG2011] Bourgain, J.; Guth, L. Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct. Anal. 21 (6) (2011), 1239–1295.
• [Fefferman1970] Fefferman, C. Inequalities for strongly singular convolution operators. Acta Math. 124 (1970), 9–36.
• [Gut2016] Guth, L. A restriction estimate using polynomial partitioning. J. Amer. Math. Soc. 29 (2) (2016), 371–413. arXiv:1407.1916.
• [MVV1996] Moyua, A.; Vargas, A.; Vega, L. Schrödinger maximal function and restriction properties of the Fourier transform. Int. Math. Res. Not. 16 (1996), 793–815.
• [Ste1979] Stein, E. M. Some problems in harmonic analysis. In: Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Vol. XXXV, Part 1), Amer. Math. Soc., Providence, RI, 1979, 3–20.
• [Stein1993] Stein, E. M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, 1993. (Contains the Stein–Tomas theorem and background.)
• [Tao2003] Tao, T. A sharp bilinear restriction estimate for paraboloids. Geom. Funct. Anal. 13 (6) (2003), 1359–1384. arXiv:math/0210084.
• [Tao2004] Tao, T. Some recent progress on the restriction conjecture. In: Applied and numerical harmonic analysis (Birkhäuser Boston, Boston, MA, 2004), 217–243. arXiv:math/0307275.
• [TVV1998] Tao, T.; Vargas, A.; Vega, L. A bilinear approach to the restriction and Kakeya conjectures. J. Amer. Math. Soc. 11 (1998), 967–1000.
• [TV2000] Tao, T.; Vargas, A. A bilinear approach to cone multipliers I. Restriction Estimates. Geom. Funct. Anal. 10 (2000), 185–215.
• [Tom1975] Tomas, P. A. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81 (1975), 477–478.
• [Wan2022] Wang, H. A restriction estimate in (\mathbb{R}^3) using brooms. Duke Math. J. 171 (8) (2022), 1749–1822. arXiv:1802.04312.
• [WW2022] Wang, H.; Wu, S. An improved restriction estimate in (\mathbb{R}^3). arXiv:2210.03878.
• [WW2024] Wang, H.; Wu, S. Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities. arXiv:2411.08871 (v3: 19 Dec 2024).
• [WZ2025] Wang, H.; Zahl, J. Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions. arXiv:2502.17655 (v2: 17 Feb 2025).
• [Wo1995] Wolff, T. An improved bound for Kakeya type maximal functions. Revista Mat. Iberoamericana 11 (1995), 651–674.

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 3; best known upper bound $\frac{22}{7}\approx 3.142857$. Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds