Sogge's local smoothing conjecture

Sogge's local smoothing conjecture concerns the gain in regularity obtained by averaging solutions of the wave equation over a time interval. For the half-wave evolution Uf = e^it√−Δf on Rⁿ, Sogge's Euclidean conjecture predicts ‖Uf‖_{L^p(Rⁿ×[1,2])} ≲ ‖(1 − Δ)^α/2f‖_L^p(Rⁿ) for 2 < p < ∞ and every α > max{0, (n − 1)/2 − n/p}. Two manuscripts in OpenAI's public mathematics collection (github.com/openai/math) concern it. One, dated 24 September 2026, claims the critical L³ estimate with every positive Sobolev loss and hence the conjecture in three spatial dimensions. The other, dated 25 October 2026 and added to the collection on 26 October 2026, claims the conjecture in every spatial dimension n ≥ 2, together with the Bochner–Riesz conjecture, Fourier restriction estimates in the interior of the conjectured range and the Kakeya maximal conjecture. This page summarizes the published claims 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.

As recounted in the history section of the October manuscript: Bourgain and Demeter's cone decoupling gives the conjectured gain for p ≥ 2(n + 1)/(n − 1) in every dimension; Heo, Nazarov and Seeger obtained the exact gain 1/p for n ≥ 4 and p > 2(n − 1)/(n − 3) by radial multiplier methods; Cushman, Demeter and Wu gave the sharp logarithmic power in Guth, Wang and Zhang's cone square-function inequality.

The original results

Claim (three dimensions, September). Theorem 1.1 of Critical local smoothing for the three-dimensional wave equation states that for every ε > 0 there is C_ε with ‖Uf‖_{L³(R³×[1,2])} ≤ C_ε‖(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.
Claim (every dimension, October). Theorem 1.1 of Local smoothing for the Euclidean wave equation states that for n ≥ 2, 2 ≤ p < ∞ and ε > 0 the half-wave evolution satisfies the estimate above with Sobolev order max{0, (n − 1)/2 − n/p} + ε, and with ε = 0 at p = 2. The proof rests on a single estimate at the critical exponent p = 2n/(n − 1) for data on one frequency annulus, with a loss of a power 2/p of log(2 + N) at frequency N; the manuscript states that for unrestricted data the optimal power lies between 1/p and 2/p at that exponent.
Consequences stated in the October manuscript. Uniform L^p bounds for the Bochner–Riesz means for every 1 ≤ p ≤ ∞ and every order α > max{0, n|1/p − 1/2| − 1/2}, as conjectured, with bounds for the maximal operator over all radii when 2 ≤ p < ∞; the conjectured bounds for Stein's square function for 2 ≤ p < ∞; the full interior range of global Fourier extension estimates for the sphere and, by slicing and rescaling, for the truncated cone and the compact paraboloid; local smoothing for Schrödinger waves; and the Kakeya and Nikodym maximal conjectures, so that every subset of Rⁿ containing a unit segment in every direction has Hausdorff dimension n.
Method, as described: a positive semidefinite tensor whose columns have zero divergence, built from quadratic fluxes of the symmetric first-order wave system; imaginary translations along light rays filter the waves by direction, with two auxiliary time variables and circular means of the resulting energy. The collection's release notes describe the method as the use of an auxiliary partial differential equation and state that the model output is released directly.
Sources. The manuscripts Critical local smoothing for the three-dimensional wave equation at commit 1581d4c and Local smoothing for the Euclidean wave equation at commit 7ef5c0c; release notes in history.md under 26 October 2026.
Formalization, as described by the repository. The scope note lean/docs/079.md, added on 26 October 2026, states that the Lean formalization bounds the half-wave evolution in L^p(Rⁿ×[1,2]) for every n ≥ 2 and 2 ≤ p < ∞ with the Sobolev order above; that it also gives uniform Bochner–Riesz multiplier bounds, spherical extension estimates and Stein square-function bounds in their strict ranges, an Lⁿ bound for the Kakeya maximal operator with loss δ^−ε, and full Hausdorff dimension of Kakeya sets; and that in dimension three it proves a logarithmic lower bound for the union volume of direction-separated tubes. The comparator statements are in LocalSmoothing.lean. Before 26 October 2026 the collection listed no Lean formalization for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 20 October 2026; Latent Seminar row 26 October 2026)
2026-10-06	Critical local smoothing for the three-dimensional wave equation, an article of the Latent Seminar blog at lowrank.github.io	Reading (exposition)	Explains the manuscript's claim of critical local smoothing for the three-dimensional wave equation, an L³ space-time estimate losing arbitrarily little regularity, and the resulting full Sogge range for 2 < p < ∞; notes that the lossless estimate is not asserted.	Blog article; it states that its source review covers statements, introductions and the proof mechanisms it describes and is not an independent verification of the complete proofs.
2026-10-08	F. Temur, The square function estimate for the cone in four dimensions, arXiv:2610.12681	Use (conditional on the manuscript)	f_θ\|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 λ^1−3/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).
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.
The collection's separate manuscript on the Bochner–Riesz multiplier is described on The Bochner–Riesz conjecture in three dimensions.
This page was titled Sogge's local smoothing conjecture in three dimensions until 27 October 2026; the old title redirects here.