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Sogge's local smoothing conjecture for the wave equation in three spatial dimensions

For the half-wave propagator eit−Δe^{it\sqrt{-\Delta}} on Rn\mathbb R^n, 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 for p≥2n/(n−1)p\ge 2n/(n-1): ∥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 σ>(n−1)(1/2−1/p)−1/p\sigma>(n-1)(1/2-1/p)-1/p, and σ>0\sigma>0 for 2<p≤2n/(n−1)2<p\le 2n/(n-1). It implies the Bochner-Riesz conjecture and was proved for n=2n=2 by Guth, Wang and Zhang (2020). In three spatial dimensions the critical exponent is p=3p=3; Bourgain-Demeter decoupling gave the conjecture for p≥4p\ge4, Gan-He-Li-Wu for p≥10/3p\ge10/3, and at p=3p=3 only σ>1/12\sigma>1/12 was known. Does the estimate hold in R3\mathbb R^3 for every 2<p<∞2<p<\infty and every σ>max⁡{0,1−3/p}\sigma>\max\{0,1-3/p\}, in particular at p=3p=3 with every positive loss?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Harmonic analysis; dispersive and wave equations
Posed by
Christopher D. Sogge
Year posed
1991
Years open
35y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
63 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every ε>0\varepsilon>0, ∥eit−Δf∥L3(R3×[1,2])≤Cε∥Jεf∥L3\|e^{it\sqrt{-\Delta}}f\|_{L^3(\mathbb R^3\times[1,2])}\le C_\varepsilon\|J^\varepsilon f\|_{L^3}. With the L2L^2 and L∞L^\infty-kernel endpoints, Corollary 10.6 gives the full strict range 2<p<∞2<p<\infty, σ>max⁡{0,1−3/p}\sigma>\max\{0,1-3/p\}. Section 11, through Beltran-Hickman-Sogge, Tao and Wolff, derives the strict three-dimensional Bochner-Riesz range, maximal Bochner-Riesz bounds for p≥3p\ge3, the diagonal sphere restriction estimate Lr(S2)→Lr(R3)L^r(S^2)\to L^r(\mathbb R^3) for r>3r>3, and the three-dimensional Kakeya maximal estimate. Not shown: the endpoint σ=0\sigma=0 at p=3p=3, dimensions n≥4n\ge4, or the conjecture's general-nn form.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript uses two external published inputs, the planar Furstenberg theorem of Ren-Wang and the cone wave-envelope theorem of Guth-Wang-Zhang.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 10.6 were read against Sogge's conjecture in dimension three; the stated range matches the conjectured strict range. The 160-page proof was not refereed. No Lean formalization exists for this family (no lean/docs/079.md at the pinned commit, nothing in lean/formalization.yaml). The proof relies on Ren-Wang's Furstenberg theorem and Guth-Wang-Zhang's wave-envelope estimate as published inputs. The paper states the lossless endpoint σ=0\sigma=0 at p=3p=3 is not addressed, and the result is Euclidean: manifold and variable-coefficient versions, which are false or different in this range, are outside it.

Sources

Changelog1 change

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