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.
released 2026-09-24 | 6 theorems · 39 lemmas · 58 proofs · 48,960 words |
PLAY LEVEL 1 »(pdf)
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.
released 2026-09-24 | 3 theorems · 8 lemmas · 15 proofs · 9,454 words |
PLAY LEVEL 2 »(pdf)
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 $s\gt 2-6/p$.