Browse › Real and complex analysis › 077 Fourier restriction for positively curved surfaces
Diagonal Fourier extension for positively curved surfaces in three dimensions
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Abstract
We prove the diagonal Fourier extension conjecture for compact smooth positively curved surfaces $\Sigma\subset\mathbb R^3$, including surfaces with smooth boundary. For every $3\lt p\lt \infty$, the extension operator maps $L^p(\Sigma)$ boundedly into $L^p(\mathbb R^3)$. The compact paraboloid case also yields free Schrödinger local smoothing in two spatial dimensions for every $3\lt p\lt \infty$ and Sobolev order $s\gt 2-6/p$.
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Follow-up works
- Lyudmila Korobenko and Eric T. Sawyer, Probabilistic Fourier extension and the Schrödinger equation (arXiv:2610.22634v1, 15 October 2026). Cites this manuscript as the deterministic three-dimensional comparison for its martingale-averaged Fourier extension and Schrödinger regularity estimates.
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]
- Eric T. Sawyer, A probabilistic analogue of the Fourier extension conjecture (arXiv:2311.03145v11, 22 August 2024). Korobenko and Sawyer’s follow-up uses this smooth-Alpert-multiplier averaged extension theorem as the probabilistic counterpart to the deterministic diagonal estimate claimed here.
Possibly relevant literature (found by the agent, not checked)
- Shaoming Guo, Hong Wang, Ruixiang Zhang, A dichotomy for Hörmander-type oscillatory integral operators (arXiv:2210.05851, October 2022). Introduces "Bourgain's condition" and conjectures that phases satisfying it obey restriction-type L^p bounds; relevant to the paper's oscillatory-integral corollary under Bourgain's condition, and not in its bibliography.
- Arian Nadjimzadah, Bourgain's condition, sticky Kakeya, and new examples (arXiv:2511.10918, November 2025). Studies straightening of curved tubes for phases satisfying Bourgain's condition and gives examples where no diffeomorphism straightens them, relevant to the paper's reduction of such phases to a fixed elliptic graph under an extra phase condition.
- Terence Tao, Recent progress on the restriction conjecture (arXiv:math/0311181, November 2003). A survey linking restriction estimates to Kakeya, Bochner–Riesz and local smoothing, relevant to the paper's background and its Schrödinger local-smoothing corollary; 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
- VibeMathed: Problem record
- Emergent Mind: family 077
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026,
author = {{OpenAI}},
title = {{Diagonal Fourier extension for positively curved surfaces in three dimensions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf}{OAI:Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-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:
- Elliptic capacity propagation and Fourier restriction to the sphere · September 24, 2026