OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
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.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps L^∞(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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
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< p< ∞, the extension operator maps L^p(\Sigma) boundedly into L^p(\mathbb R^3). The compact paraboloid case also yields free Schrödinger local smoothing…
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< p< ∞, 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< p< ∞ and Sobolev order s> 2-6/p.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
Explore the AGI Clock ↗