Result 079, Real and complex analysis

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<p<∞2\lt p\lt \infty, with Sobolev regularity above max⁡{0,1−3/p}\max\{0,1-3/p\}. In particular, the critical L3 estimate holds with every positive Sobolev loss.

Proof

The bigger picture

Why it matters

A wave can be easier to control when measured over time than at a single instant. This manuscript reports a sharp description of that advantage for the Euclidean wave equation in three spatial dimensions.

What changes?

The manuscript reports resolving Sogge's local smoothing conjecture in three-dimensional Euclidean space. Local smoothing measures improved control after averaging a wave's size over a bounded time interval. The estimates cover every exponent p strictly between 2 and infinity; p specifies how strongly the measurement weights large amplitudes. They require Sobolev regularity, a measure of spatial smoothness, strictly above the larger of zero and 1 minus 3 divided by p. At p = 3, every positive Sobolev loss is allowed.

What does that help mathematicians do?

The critical L3 estimate supplies control at the point where the stated regularity threshold changes form. Researchers can therefore use arbitrarily small positive smoothness assumptions throughout the range from p greater than 2 through p = 3, while larger p requires regularity above 1 minus 3 divided by p. The strict inequalities matter: the claim does not supply a zero-loss L3 estimate or include p = 2 or infinity.

Are there practical applications?

The immediate value is foundational: the result specifies how much spatial regularity is sufficient to control a three-dimensional Euclidean wave over time. Such estimates provide tools for mathematical analysis of wave equations. The supplied sources establish no numerical method or physical deployment, and the stated result does not automatically extend to curved spaces or other dimensions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Critical local smoothing for the three-dimensional wave equation

September 24, 2026 165 pages

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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Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.