Result 077, Real and complex analysis

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.

Proof

The bigger picture

Why it matters

Curvature can make waves cancel rather than reinforce one another. The manuscripts claim a precise bound on this effect for positively curved surfaces in three dimensions, relating the size of frequency data to the waves they generate.

What changes?

The manuscript reports that every compact smooth positively curved surface in three-dimensional space, including surfaces with smooth boundary, has bounded Fourier extension from L^p on the surface to L^p in space for every finite p greater than 3. Extension superposes plane waves with frequencies on the surface; L^p measures size using pth powers. The output norm is bounded by a constant depending on the surface and p times the input norm.

What does that help mathematicians do?

For the sphere, finite surface area means this diagonal estimate also covers bounded input data, recovering the companion manuscript's claimed bounded-data range. Its stronger content is control of inputs that can be unbounded but still have finite L^p norm. Researchers could therefore control spatial integrability without imposing a pointwise bound on frequency data, on any surface covered by the claim. No estimate at p = 3 is claimed.

Are there practical applications?

The immediate value is foundational, with a stated consequence for free Schrödinger evolution, a basic dispersive-wave model. The compact paraboloid case reportedly yields local smoothing in two spatial dimensions for every finite p greater than 3 and Sobolev regularity order s greater than 2 minus 6 divided by p. This connects the extension estimate to regularity bounds for evolving waves, rather than to a demonstrated numerical or technological application.

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.

2 manuscripts

Elliptic capacity propagation and Fourier restriction to the sphere

September 24, 2026 103 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Diagonal Fourier extension for positively curved surfaces in three dimensions

September 24, 2026 25 pages

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 $s\gt 2-6/p$ .

PDF LaTeX source

Cite (BibTeX)

@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}
}

At a glance

Subject: Real and complex analysis
Manuscripts: 2
Pages: 128
Dated: Sep 24, 2026
Lean formalization: No
Reasoning summary: No

Named problems addressed

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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By similarity of summaries and abstracts.

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