Restriction exponent for the 2-sphere (Stein’s $L^\infty$ extension problem)
Description of constant
$C_{46}$ is the infimal exponent $p$ such that one has the global bound \(\|\widehat{f\,d\sigma}\|_{L^p(\mathbb{R}^3)} \;\lesssim_p\; \|f\|_{L^\infty(S^2)} \qquad\text{for all }f\in L^\infty(S^2).\)
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $\infty$ | Trivial | |
| $6$ | Stein (1967) | Appears in [Fefferman1970] |
| $4$ | [Tom1975], [Stein1993] | From the Stein–Tomas theorem; limit of $L^2(S^2)$ methods |
| $4-\frac{2}{15} \approx 3.86667$ | [Bo1991] | |
| $4-\frac{2}{11} \approx 3.81818$ | [Wo1995], [MVV1996] | |
| $4-\frac{2}{9} \approx 3.77778$ | [TVV1998] | Bilinear methods |
| $4-\frac{2}{7} \approx 3.71429$ | [TV2000] | |
| $\frac{10}{3}\approx 3.33333$ | [Tao2003], [BG2011] | |
| $\frac{13}{4}=3.25$ | [Gut2016] | Used polynomial partitioning |
| $3+\frac{3}{13}\approx 3.23077$ | [Wan2022] | Introduced “Brooms” |
| $3+\frac{3}{14}\approx 3.21429$ | [WW2022] | |
| $\frac{22}{7}\approx 3.142857$ | [WW2024] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $3$ | Stationary phase / explicit computation | Take $f=1$. Conjectured to be sharp [Ste1979] |
Further remarks
-
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”.
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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 [Ta2004] for more discussion.
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Stein’s restriction conjecture $C_{46}=3$ implies the Kakeya conjecture in ${\mathbb R}^3$ (see, e.g., [Ta2004]), which was recently proven in [WZ2025].
References
- [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.
Contribution notes
Prepared with assistance from ChatGPT.