Let S⊂R^n be a compact C^2 hypersurface, possibly with boundary, with strictly positive second fundamental form. Let E_S be its Fourier extension operator. Local Fourier restriction conjecture. If p≥frac2nn-1, then for every varepsilon>0 and R>1, |E_Sf|_L^p(B_R)≤ C_varepsilon R^varepsilon|f|_L^p(d sigma_S). By an epsilon-removal argument, this local estimate implies the global restriction conjecture; it is not known at the endpoint in general.

Local Fourier restriction conjecture

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Let $S\subset\mathbb{R}^n$ be a compact $C^2$ hypersurface, possibly with boundary, with strictly positive second fundamental form. Let $E_S$ be its Fourier extension operator. Local Fourier restriction conjecture. If $p\geq\frac{2n}{n-1}$ , then for every $\varepsilon>0$ and $R>1$ ,

\|E_Sf\|_{L^p(B_R)}\leq C_\varepsilon R^\varepsilon\|f\|_{L^p(d\sigma_S)}.

By an epsilon-removal argument, this local estimate implies the global restriction conjecture; it is not known at the endpoint in general.

References

Primary source

Hong Wang and Shukun Wu, “Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities”, arXiv:2411.08871 (2024).

fourier-analysis restriction-theory

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RemarkAI-assistedClaimed by OpenAI. The manuscript claims global Fourier extension from bounded functions on the two-sphere to Lp of three-space for every p > 3. This gives a related bounded-input sphere estimate. It does not by itself establish the target’s finite-Lp input bound or its p=3 endpoint.See full solutionHide full solution

Claimed by OpenAI. The manuscript claims global Fourier extension from bounded functions on the two-sphere to Lp of three-space for every p > 3. This gives a related bounded-input sphere estimate. It does not by itself establish the target’s finite-Lp input bound or its p=3 endpoint.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf

OpenAI-077-01-Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere.pdf945,868 bytesOpen

RemarkAI-assistedClaimed by OpenAI. For every compact smooth positively curved surface in three Euclidean dimensions, including smooth boundary, the manuscript claims global diagonal Fourier extension from Lp of the surface to Lp of three-space for every 3 < p < infinity. This implies the target’s local bound in that smooth dimension-three nonendpoint subcase, without an R loss; p=3 and higher dimensions are not asserted.See full solutionHide full solution

Claimed by OpenAI. For every compact smooth positively curved surface in three Euclidean dimensions, including smooth boundary, the manuscript claims global diagonal Fourier extension from Lp of the surface to Lp of three-space for every 3 < p < infinity. This implies the target’s local bound in that smooth dimension-three nonendpoint subcase, without an R loss; p=3 and higher dimensions are not asserted.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf

OpenAI-077-02-Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions.pdf503,679 bytesOpen