Stein restriction conjecture for the paraboloid in ℝ³: the extension operator maps L^∞ to L^p for all p > 3
Fifty years of work and the accepted exponent 22/7 is still short of 3, so 72: above Kakeya in ℝ⁴ (62), since restriction is strictly stronger than Kakeya, and below the Lindelöf exponent problem (74). The unrefereed full claim is not credited. Application value 8 is indirect (dispersive PDE thresholds). Leverage 44.5: Bochner–Riesz, Strichartz-type estimates and the restriction hierarchy depend on it.
The problem
How well can a wave pattern in 3D be controlled by data on a curved surface? The sharp range of exponents is still unproved.
- 1Waves built from data on a curved surface
- 2Ask how data size controls wave size
- 3Conjectured range is every p above three
- 4Best proven range stops at about 3.14
Let be the for the truncated paraboloid in . It takes a function on the unit disc (the data on the surface) and returns the wave on obtained by summing plane waves with frequencies on the paraboloid. This is, up to rescaling, the solution of the free Schrödinger equation in two space dimensions.
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Harmonic analysisTheory · host
- Then
- Stein restriction conjecture for the paraboloid in ℝ³: the extension operator maps L^∞ to L^p for all p > 3The problem
State of the art
Open in the accepted literature; a full claim is unverified. In ℝ³ the exponent was lowered by Tao's bilinear method (p > 10/3), Bourgain–Guth (p > 3.3), Guth's polynomial partitioning (p > 3.25, arXiv:1407.1916, J. Amer. Math. Soc. 2016), Hong Wang's brooms (p > 3 + 3/13, arXiv:1802.04312) and Wang–Wu (p > 22/7, arXiv:2411.08871), who combine refined decoupling with a two-ends Furstenberg inequality and pose a two-ends Furstenberg conjecture that would imply the full restriction conjecture. Demeter and Wu (arXiv:2505.09037) reach p > 22/7 for the hyperbolic paraboloid. The Kakeya set conjecture in ℝ³ is now a theorem (Wang–Zahl), which removes one obstruction but does not imply restriction. OpenAI's 6 October 2026 release (github.com/openai/math, family 077) includes a model-produced manuscript claiming the full range p > 3 for every compact positively curved surface in ℝ³ (and a companion claim of the 3D Bochner–Riesz conjecture); it has no listed Lean formalization, is unrefereed, and had not been independently assessed by 15 October 2026.
Realistic applications
No direct application. Restriction estimates give sharp dispersive estimates for Schrödinger and wave equations and control Bochner–Riesz means; improved exponents sharpen local well-posedness thresholds for nonlinear dispersive PDEs.
What counts as solved
A proof of the L^p bound for all p > 3, or a counterexample, refereed and accepted. Improvement of the exponent below the current record is a partial result, not a resolution.
How it gets verified
Expert reviewOnly expert review of the argument is possible today.
The statement is an L^p bound for the Fourier extension operator of the paraboloid in ℝ³ for all p > 3. Mathlib has the Fourier transform but not the surface measures and multiscale (Bourgain–Guth, polynomial partitioning) machinery, and Formal Conjectures (Formal Conjectures pinned at commit e754a78 (17 October 2026)) has no statement of it. Referees check the proof; improved exponents below the current record are partial results.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
The work is purely analytic, new incidence-geometry and Fourier-analytic input.
Bottlenecks
ranked by votes- #1TheoryBeyond the two-ends and bilinear regimes
Polynomial partitioning, brooms and two-ends Furstenberg estimates stop at p = 22/7; the remaining configurations concentrate wave packets near low-degree varieties in ways the known induction-on-scales arguments do not control.
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Connected problems
- This problem is related to Kakeya sets in ℝ⁴ have Hausdorff dimension 4, and the Kakeya maximal function in ℝ⁴
Progress
- Oct 6, 2026Unrefereed OpenAI claim of the full rangesource
Model-produced manuscript claims p > 3 for positively curved surfaces; not yet assessed.
- Nov 13, 2024Wang–Wu: p > 22/7source
Refined decoupling plus a planar two-ends Furstenberg estimate.
- Feb 12, 2018Brooms: p > 3 + 3/13source
Hong Wang combines Wolff’s two-ends argument with polynomial partitioning.
- Jul 8, 2014Polynomial partitioning: p > 3.25source
Guth (arXiv July 2014; J. Amer. Math. Soc. 2016).