∑math20

Real and complex analysis · Result 077

Fourier restriction for positively curved surfaces

2 papers

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.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

First page of Elliptic capacity propagation and Fourier restriction to the sphere

Elliptic capacity propagation and Fourier restriction to the sphere

OpenAI · September 24, 2026

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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@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}
}
First page of Diagonal Fourier extension for positively curved surfaces in three dimensions

Diagonal Fourier extension for positively curved surfaces in three dimensions

OpenAI · September 24, 2026

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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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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.