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Critical local smoothing for the three-dimensional wave equation
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Abstract
We prove the critical L3 local smoothing estimate for the wave equation in three spatial dimensions, with every positive Sobolev loss. This resolves Sogge's Euclidean local smoothing conjecture in dimension three.
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Follow-up works
- Faruk Temur, The square function estimate for the cone in four dimensions (arXiv:2610.12681, 2026-10-08). Proves an $L^3$ reverse square function estimate for the light cone in four dimensions, with consequences for spheres and elliptic paraboloids in three dimensions, taking as stated external inputs this paper's Theorem 1.1, Definitions 8.1–8.2, Lemma 8.4, Theorem 9.1 (round-cone propagation) and equation (10.22); it says it "does not supply an independent proof of those statements".
Related prior literature mentioned by others
- Zipeng Wang, $L^p$-boundedness of the Bochner-Riesz operator (arXiv:2501.12742, v1 2025-01-22, v6 2026-06-09). The VibeMathed entry for the companion Bochner–Riesz family (078) names this as a proposed all-dimensional proof. It bears on this paper's Bochner–Riesz Corollary (Section 10), which covers the same strict three-dimensional range.
- Francesco Di Plinio and Ioannis Parissis, Maximal subspace averages (Advances in Mathematics 410, 108749, 2022; arXiv:2107.02109). Temur's follow-up combines the codimension-one plate maximal estimate in Theorem C with this paper's scalar local-smoothing estimate to obtain a spacetime averaging bound used in the square-function argument.
Possibly relevant literature (found by the agent, not checked)
- Hong Wang and Shukun Wu, Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities (arXiv:2411.08871, 2024). Proves restriction in $\mathbb R^3$ for $p>22/7$, which was the prior state of the art for the spherical extension estimate $3<r<\infty$ that Section 10.1 derives from the Bochner–Riesz corollary.
- Xiangyu Wang, Wave Packet Density in Local Smoothing: Refinements and Limitations (arXiv:2609.34675, v1 2026-09-28, v4 2026-10-16). Refines the Gan–He–Li–Wu broad-incidence argument, which the introduction cites as the previous best result at $p=3$, and discusses geometric limits of wave-packet density methods in the Euclidean setting.
Digests and expositions
- AI Proved What?, Smoother on average? Local smoothing for the wave equation in three dimensions (Episode 079, October 2026). Video and transcript explaining the claimed positive-loss critical estimate, the packet-window and entropy strategy, and the Bochner–Riesz consequences, while distinguishing exposition from verification. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 079
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026,
author = {{OpenAI}},
title = {{Critical local smoothing for the three-dimensional wave equation}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf}{OAI:Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026}},
year = {2026}
}Family 079: Local smoothing in three dimensions
Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range $2\lt p\lt \infty$, with Sobolev regularity above $\max\{0,1-3/p\}$. In particular, the critical L3 estimate holds with every positive Sobolev loss.
This is the only paper in this family.