OP-00718 Solution claimed

Sharp Fourier extension from the sphere in three dimensions

Let $\sigma$ be surface area measure on $S^2 = \{\omega \in \mathbb{R}^3 : |\omega| = 1\}$ and define $$Eg(x) = \int_{S^2} e^{2\pi i x \cdot \omega} g(\omega)\, d\sigma(\omega), \qquad x \in \mathbb{R}^3.$$ By the factorisation theory of Maurey, Nikishin and Pisier, the estimate below for all $p > 3$ is equivalent to Stein's restriction conjecture for the sphere in $\mathbb{R}^3$. Guth proved it for $p > 3.25$. Wang and Wu proved the $L^p(\sigma_S) \to L^p(\mathbb{R}^3)$ extension estimate, $\sigma_S$ the surface measure of $S$, for every compact $C^2$ surface $S$ with strictly positive second fundamental form and every $p > 22/7$, which gives the estimate below for those $p$.

Problem. Is it true that for every $p > 3$ there exists $C_p < \infty$ such that $$\|Eg\|_{L^p(\mathbb{R}^3)} \le C_p \|g\|_{L^\infty(S^2, \sigma)}$$ for every bounded measurable $g \colon S^2 \to \mathbb{C}$? The constant must not depend on $g$.

Source: Stein's restriction conjecture for the sphere in three dimensions, in the dual (extension) form of Guo, Oh, Wang, Wu and Zhang (2025), Conjecture 1.2, with n = 3. Stated by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: CA — Classical Analysis and ODEs
Keywords: AIM Open Applied Problems, extension operator, Fourier restriction, sphere

References

@article{guo2025,
  author        = {Guo, Shaoming and Oh, Changkeun and Wang, Hong and Wu, Shukun and Zhang, Ruixiang},
  title         = {The {B}ochner--{R}iesz problem: an old approach revisited},
  journal       = {Peking Math. J.},
  volume        = {8},
  number        = {2},
  year          = {2025},
  pages         = {201--270},
  doi           = {10.1007/s42543-023-00082-4},
  eprint        = {2104.11188},
  archivePrefix = {arXiv},
}
@article{guth2016,
  author        = {Guth, Larry},
  title         = {A restriction estimate using polynomial partitioning},
  journal       = {J. Amer. Math. Soc.},
  volume        = {29},
  number        = {2},
  year          = {2016},
  pages         = {371--413},
  doi           = {10.1090/jams827},
  eprint        = {1407.1916},
  archivePrefix = {arXiv},
}
@misc{wang2024,
  author        = {Wang, Hong and Wu, Shukun},
  title         = {Restriction estimates using decoupling theorems and two-ends {F}urstenberg inequalities},
  year          = {2024},
  eprint        = {2411.08871},
  archivePrefix = {arXiv},
}
@misc{aim-sphere-fourier-restriction,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Sharp Fourier extension from the sphere in three dimensions}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/208-sphere-fourier-restriction.md},
  note         = {Entry 208 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Elliptic capacity propagation and Fourier restriction to the sphere v1 GitHub ↗

OpenAI

Yes: for every p > 3 there is C_p, depending only on p, with ||Eg||_{L^p(R^3)} <= C_p ||g||_{L^infinity(S^2, sigma)} for every bounded measurable complex g on S^2..

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/paper.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00718, "Sharp Fourier extension from the sphere in three dimensions", clopen.solutions.