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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)

L^p(\mathbb R^3)

for every p > 3.

Manuscripts in this family · 2

Elliptic capacity propagation and Fourier restriction to the sphere

Sep 24, 2026·Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.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.

Full abstract

PDF on GitHub LaTeX source

Diagonal Fourier extension for positively curved surfaces in three dimensions

Sep 24, 2026·Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.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

Full abstract

PDF on GitHub LaTeX source