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Sogge's local smoothing conjecture for the wave equation in every dimension

For the half-wave propagator eit−Δe^{it\sqrt{-\Delta}} on Rn\mathbb R^n, n≥2n\ge2, the sharp fixed-time estimate on LpL^p loses (n−1)∣1/2−1/p∣(n-1)|1/2-1/p| derivatives. Sogge (1991) conjectured that averaging over t∈[1,2]t\in[1,2] gains almost 1/p1/p of a derivative: ∥eit−Δf∥Lp(Rn×[1,2])≤C∥f∥Wσ,p\|e^{it\sqrt{-\Delta}}f\|_{L^p(\mathbb R^n\times[1,2])}\le C\|f\|_{W^{\sigma,p}} for every σ>(n−1)(1/2−1/p)−1/p\sigma>(n-1)(1/2-1/p)-1/p when p≥2n/(n−1)p\ge 2n/(n-1), and every σ>0\sigma>0 when 2<p≤2n/(n−1)2<p\le 2n/(n-1). The conjecture implies the Bochner-Riesz conjecture and, through it, the Fourier restriction and Kakeya conjectures. It was proved for n=2n=2 by Guth, Wang and Zhang; in higher dimensions Bourgain-Demeter decoupling gave p≥2(n+1)/(n−1)p\ge2(n+1)/(n-1). Does it hold in every dimension?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Harmonic analysis; dispersive and wave equations
Posed by
Christopher D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991)
Year posed
1991
Years open
35y
Solved
2026-10-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version 4eaee87
Significance
72 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for n≥2n\ge2, 2≤p<∞2\le p<\infty and ε>0\varepsilon>0, ∥e−iT∣D∣f∥Lp(Rn×[1,2])≤C∥Js+(p)+εf∥Lp\|e^{-iT|D|}f\|_{L^p(\mathbb R^n\times[1,2])}\le C\|J^{s_+(p)+\varepsilon}f\|_{L^p} with s+(p)=max⁡{0,(n−1)/2−n/p}s_+(p)=\max\{0,(n-1)/2-n/p\}, and ε=0\varepsilon=0 at p=2p=2: Sogge's conjecture in its strict form in every dimension. Consequences in the paper: the Bochner-Riesz conjecture for 1≤p≤∞1\le p\le\infty with maximal bounds, Stein's square function, global sphere and truncated-cone extension estimates in the interior of the conjectured range, the compact paraboloid and Schrodinger local smoothing, and the Kakeya and Nikodym maximal conjectures, so every Kakeya set in Rn\mathbb R^n has Hausdorff dimension nn. Not shown: the endpoint σ=0\sigma=0 at p≥2n/(n−1)p\ge 2n/(n-1) or restriction endpoints. It supersedes the catalog's three-dimensional entry.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The manuscript is authored 'OpenAI' and names no human author. The release README says the manuscripts are best understood as raw research data rather than polished papers, and that unformalized results could have issues. The release history (26 October 2026) says OpenAI is releasing this model output directly, as in its 6 October release, and that the proof uses an auxiliary partial differential equation; OpenAI's announcement says the model produced it on 18 October. It extends the release's 24 September three-dimensional result (catalogued as local-smoothing-conjecture-r3) to every dimension. Lean formalizations and a Comparator configuration ship for the headline theorem and five consequences.

Verification

No independent mathematician has checked this yet. Checked here on 3 November 2026: the abstract, Section 1 and Theorem 1.1 were read against Sogge's conjecture; the exponent max⁡{0,(n−1)/2−n/p}\max\{0,(n-1)/2-n/p\} equals the conjectured (n−1)(1/2−1/p)−1/p(n-1)(1/2-1/p)-1/p above 2n/(n−1)2n/(n-1) and 00 below. The proof was not refereed. Lean: formalization.yaml lists six declarations under OAI.Analysis.Fourier.LocalSmoothing.Comparator (comparator ComparatorChallenges/LocalSmoothing.json, axioms propext, Quot.sound, Classical.choice). The challenge file was read in full: euclidean_local_smoothing states, for every n≥2n\ge2, p≥2p\ge2 and ε>0\varepsilon>0 (or p=2p=2, ε=0\varepsilon=0), the Lp(Rn×[1,2])L^p(\mathbb R^n\times[1,2]) bound by the Bessel-potential norm of order max⁡(0,(n−1)/2−n/p)+ε\max(0,(n-1)/2-n/p)+\varepsilon for Schwartz data, with Fourier transforms defined from scratch; the other five (Bochner-Riesz for all δ>max⁡(n∣1/p−1/2∣−1/2,0)\delta>\max(n|1/p-1/2|-1/2,0), spherical extension, Kakeya maximal with δ−ε\delta^{-\varepsilon} loss, dim⁡HF=n\dim_H F=n for every Kakeya set, Stein square function) match the paper's statements. GitHub code search of the solution folder found no sorry, axiom or native_decide. Not rebuilt here.

Sources

Submitted by VibeGene on

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