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Level B · Reproducible Analysis P-constant-46a-restriction-exponent-for-the-2-sphere-stein-s-l-extension-problem

Restriction exponent for the 2-sphere (Stein's L^∞ extension problem)

C_46 is the infimal exponent p such that one has the global bound ‖widehatf dσ‖_L^p(ℝ^3) lesssim_p ‖f‖_L^∞(S^2) qquadfor all f∈ L^∞(S^2).

From the catalogue. Imported from Terence Tao and contributors (optimizationproblems repository) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-constant-46a-restriction-exponent-for-the-2-sphere-stein-s-l-extension-problem,
  title        = {Restriction exponent for the 2-sphere (Stein's L^∞ extension problem)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-46a-restriction-exponent-for-the-2-sphere-stein-s-l-extension-problem}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
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Disputed
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

Description of constant

is the infimal exponent such that one has the global bound

Known upper bounds

BoundReferenceComments
Trivial
Stein (1967)Appears in [Fefferman1970]
[Tom1975], [Stein1993]From the Stein--Tomas theorem; limit of methods
[Bo1991]
[Wo1995], [MVV1996]
[TVV1998]Bilinear methods
[TV2000]
[Tao2003], [BG2011]
[Gut2016]Used polynomial partitioning
[Wan2022]Introduced “Brooms”
[WW2022]
[WW2024]

Known lower bounds

BoundReferenceComments
Stationary phase / explicit computationTake . 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 ) track the paraboloid results up to standard -losses that can often be removed by "epsilon removal lemmas".
  • For most of the results in the literature, the norm on the right-hand side can be replaced with for various ; for instance, in the Tomas-Stein theorem one can take . There are also bilinear and multilinear variants of the conjecture. See for instance [Tao2004] for more discussion.
  • Stein's restriction conjecture implies the Kakeya conjecture in (see, e.g., [Tao2004]), 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.

What counts as progress

  • A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
  • A formal proof (Lean) of a known bound, or a precise error in a claimed one.
  • New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.