Sogge's local smoothing conjecture in three dimensions
From OpenAI Aftermath
Sogge's local smoothing conjecture in three dimensions concerns the gain in regularity obtained by averaging solutions of the wave equation over a time interval. For the half-wave evolution Uf = eit√−Δf on R3, Sogge's Euclidean conjecture predicts ‖Uf‖Lp(R3×[1,2]) ≲ ‖(1 − Δ)α/2f‖Lp(R3) for 2 < p < ∞ and every α > max{0, 1 − 3/p}, with critical exponent p = 3. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 24 September 2026, claims the critical L3 estimate with every positive Sobolev loss and hence the conjecture in three spatial dimensions. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Sogge introduced local smoothing in his study of circular maximal operators and proposed the conjecture, measured against the sharp fixed-time estimates of Peral and Miyachi. Mockenhaupt, Seeger and Sogge developed the square-function approach. Wolff obtained the conjectured gain for large p in two dimensions, Łaba and Wolff extended the method to higher dimensions, and Garrigós and Seeger improved the plate-decomposition inequalities.
- Bourgain and Demeter's sharp cone decoupling gives the conjectured gain for p ≥ 4 in three dimensions. Beltran, Hickman and Sogge proved sharp gains for variable coefficients and showed that p ≥ 4 is sharp for the broader cinematic-curvature class; Minicozzi and Sogge's metrics show why the conjecture is Euclidean.
- In two dimensions, Guth, Wang and Zhang reached the critical exponent p = 4 through a sharp cone square-function estimate. In three dimensions, Gao, Liu, Miao and Xi obtained every α > 1/9 at p = 3, and Gan, He, Li and Wu the conjectured gain for p ≥ 10/3, which interpolates to α > 1/12 at p = 3.
The original result
- Claim. Theorem 1.1 of Critical local smoothing for the three-dimensional wave equation states that for every ε > 0 there is Cε with ‖Uf‖L3(R3×[1,2]) ≤ Cε‖(1 − Δ)ε/2f‖L3(R3) for all Schwartz functions f. A corollary gives the full range 2 < p < ∞, α > max{0, 1 − 3/p}. The manuscript states that the lossless estimate ε = 0 is not addressed. Further corollaries give a maximal estimate for the half-wave evolution for s > 1 − 2/p (3 ≤ p < ∞), maximal Bochner–Riesz bounds with almost-everywhere convergence for 3 ≤ p < ∞ and δ > 1 − 3/p, and uniform bounds for the individual Bochner–Riesz means in the strict range.
- Method, as described: the proof studies the concentration of wave packets across scales through the entropy of a finite phase-space record that retains the changing shear frames together with time and direction information. Its two external quantitative inputs are the planar Furstenberg theorem of Ren and Wang and the cone wave-envelope theorem of Guth, Wang and Zhang.
- Source. The manuscript Critical local smoothing for the three-dimensional wave equation at commit
1581d4c. - Formalization. The collection lists no Lean formalization for this result.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 20 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | F. Temur, The square function estimate for the cone in four dimensions, arXiv:2610.12681 | Use (conditional on the manuscript) | fθ|2)1/2‖L3 for functions with Fourier support in the canonical sectors of dimensions 1 × R−1/2 × R−1/2 × R−1, assuming the critical local-smoothing and round-cone statements of the manuscript (its Theorem 1.1, Definitions 8.1–8.2, Lemma 8.4, Theorem 9.1 and equation (10.22)), which the paper states as external inputs together with the plate maximal theorem of Di Plinio and Parissis. As a consequence it obtains the corresponding L3 estimates for the sphere and elliptic paraboloids in three dimensions. It also cites a lemma of the collection's manuscript on Fourier extension for positively curved surfaces as the model for one decomposition. | arXiv preprint, version 1. |
| 2026-10-13 | MColbrook/AIM, problem 210 (GitHub; AI for Mathematical Problems, checked 13 October 2026) | Catalogue | Lists Sharp local smoothing for the three-dimensional wave equation (for 3 ≤ p < ∞ and ε > 0, the frequency-localized estimate with loss λ1−3/p+ε) as solved by the manuscript, after an AI statement-level comparison of the theorem, hypotheses and reductions; it records that no corresponding complete Lean development was identified. | Open-problem collection entry, not affiliated with the American Institute of Mathematics; status unchanged at repository commit 24edaad (15 October 2026).
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| 2026-10-16 | J. Jardine, Lattice24, referee review rank_008 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap, with stated limits. The reviewer states that it read the introduction, the framework, the round-cone, physical-wave and Bochner–Riesz sections in full, but only spot-checked the parts that carry the proof of the cylinder theorem (the rest of Section 2, most of Section 3, all of Sections 4–6 and the rest of Section 7, about eight to nine thousand lines), so that its verdict "is therefore limited to the endgame and the reductions; it is not a certification of the cylinder machinery." Its one minor note concerns exponents quoted from Wolff's work in the restriction-to-Kakeya step, which it could not check against the source and describes as bookkeeping. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0. |
Notes
- The earlier partial results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
