OpenAI · 077-01 · 2026-09-24 · Real and complex analysis

Elliptic capacity propagation and Fourier restriction to the sphere

preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026

Family 077. Fourier restriction for positively curved surfaces.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface $Σ \subset R^{3}$ , including surfaces with boundary, the extension operator is bounded from $L^{p} (Σ)$ to $L^{p} (R^{3})$ for every $p > 3$ .

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 $L^{p} (R^{3})$ for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.