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.

Manuscripts (2)

Elliptic capacity propagation and Fourier restriction to the sphere

September 24, 2026 · source · cite as \cite{OAI:Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-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.

Diagonal Fourier extension for positively curved surfaces in three dimensions

September 24, 2026 · source · cite as \cite{OAI:Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-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.

Result 077 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.