Fourier restriction for positively curved surfaces

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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.

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The paper states

Theorem 1.1

In Diagonal Fourier extension for positively curved surfaces in three dimensions, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Four other possible matches are listed below the card, among them Theorem 1.2 (Uniform propagation of finite packet arrays) in Elliptic capacity propagation and Fourier restriction to the sphere and Theorem 1.1 in Elliptic capacity propagation and Fourier restriction to the sphere.

Let \Sigma\subset\R^3 be a compact smooth positively curved surface, possibly with smooth boundary. For every 3<p<\infty, there is a finite constant C_{\Sigma,p} such that

\|E_\Sigma f\|_{L^p(\R^3)} \le C_{\Sigma,p}\|f\|_{L^p(\Sigma)}

for every complex-valued f\in L^p(\Sigma).

Four more possible matches

Theorem 1.2 (Uniform propagation of finite packet arrays) in Elliptic capacity propagation and Fourier restriction to the sphere, 24 September 2026.
Theorem 1.1 in Elliptic capacity propagation and Fourier restriction to the sphere, 24 September 2026.
Corollary 1.2 (Free Schrödinger local smoothing) in Diagonal Fourier extension for positively curved surfaces in three dimensions, 24 September 2026.
Corollary 1.3 in Elliptic capacity propagation and Fourier restriction to the sphere, 24 September 2026.

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Papers and explanations: Two current papers, each readable here as a page beside its PDF.

Papers

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Elliptic capacity propagation and Fourier restriction to the sphere

24 September 2026, current edition.

Read it here 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.

Diagonal Fourier extension for positively curved surfaces in three dimensions

24 September 2026, current edition.

Read it here 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 \(s\gt 2-6/p\).

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Within this family

Papers elsewhere in the release that this family’s papers cite

The Kakeya maximal conjecture in three dimensionsCurrent edition; no accompanying Lean statement Family Kakeya in three and four dimensionsCited by Elliptic capacity propagation and Fourier restriction to the sphere and Diagonal Fourier extension for positively curved surfaces in three dimensions.
Bochner–Riesz multipliers in three dimensionsCurrent edition; no accompanying Lean statement Family The three-dimensional Bochner–Riesz conjectureCited by Elliptic capacity propagation and Fourier restriction to the sphere.

Papers elsewhere in the release that cite this family’s papers

Bochner–Riesz multipliers in three dimensionsCurrent edition; no accompanying Lean statement Family The three-dimensional Bochner–Riesz conjectureCites Elliptic capacity propagation and Fourier restriction to the sphere and Diagonal Fourier extension for positively curved surfaces in three dimensions.
The Kakeya maximal conjecture in three dimensionsCurrent edition; no accompanying Lean statement Family Kakeya in three and four dimensionsCites Elliptic capacity propagation and Fourier restriction to the sphere.

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