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Elliptic capacity propagation and Fourier restriction to the sphere
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Abstract
We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps $L^\infty(S^2)$ boundedly into $L^p(\mathbb R^3)$ for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Related prior literature mentioned by others
- Hong Wang and Shukun Wu, Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities (arXiv:2411.08871v3, 19 December 2024). The Post-Cutoff discussion identifies its $p>22/7$ estimate as a prior comparison for this family’s claimed $p>3$ bounds. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Terence Tao, Recent progress on the restriction conjecture (arXiv:math/0311181, November 2003). A survey of the sphere restriction problem and its links to the Kakeya and Bochner–Riesz conjectures, relevant to the paper's background and to its sphere-to-Kakeya-maximal transfer (Corollary 1.3); not in the paper's bibliography.
News and blogs
- Post-Cutoff, OpenAI claims 3D restriction and Bochner–Riesz conjectures (19 October 2026). Summarizes both restriction claims and the diagonal manuscript’s dependence on the sphere packet theorem and the separate Kakeya maximal manuscript. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 077
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026,
author = {{OpenAI}},
title = {{Elliptic capacity propagation and Fourier restriction to the sphere}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf}{OAI:Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026}},
year = {2026}
}Family 077: Fourier restriction for positively curved surfaces
Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface $\Sigma\subset\mathbb R^3$, including surfaces with boundary, the extension operator is bounded from $L^p(\Sigma)$ to $L^p(\mathbb R^3)$ for every p > 3.
Other papers in this family:
- Diagonal Fourier extension for positively curved surfaces in three dimensions · September 24, 2026