The X-ray endpoint conjecture for the Fourier extension operator on the sphere (Bennett–Nakamura, Conjecture 1.3)

For the Fourier extension operator on the unit sphere, Bennett and Nakamura (2021) posed a conjectural endpoint bound stating that the X-ray transform of the truncated $|\widehat{gd\sigma}|^{2/(n-1)}$ satisfies $\|X(1_{B_R}|\widehat{gd\sigma}|^{2/(n-1)})\|_{L_\omega^n L_v^\infty}\lesssim_\varepsilon R^\varepsilon\|g\|_{L^{2n/(n-1)}(\mathbb{S}^{n-1})}^{2/(n-1)}$ for all $R>0$. They proved that the classical restriction conjecture implies this bound, and that this bound implies the Kakeya maximal conjecture, making it a precise conjectural bridge between the two most central open problems of Euclidean restriction theory. The conjecture remains open in dimensions $n\ge3$ (as of 2026), while the classical Mizohata–Takeuchi weighted conjecture, the other main open question highlighted in the same paper, was refuted by Cairo (2025) via a $\log R$-loss counterexample. The problem asked here is Conjecture 1.3 in its original generality; resolving it positively in any $n\ge3$ would settle the Kakeya maximal conjecture in that dimension, while refuting it would refute the restriction conjecture.

Background

The Fourier extension operator on the unit sphere $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ maps a density $g$ on the sphere to $\widehat{gd\sigma}$, the Fourier transform of the surface-measure density $g\,d\sigma$; it is the central object of Fourier restriction theory, which seeks $L^p$ bounds on $\widehat{gd\sigma}$ in physical space. The Stein restriction conjecture is equivalent to the localized endpoint bound $\|\widehat{gd\sigma}\|_{L^{2n/(n-1)}(B_R)}\lesssim_\varepsilon R^\varepsilon\|g\|_{L^{2n/(n-1)}(\mathbb{S}^{n-1})}$ on balls $B_R$, and remains open for $n\ge3$; the Kakeya maximal conjecture, an equivalent-side problem about averaging a function over unit-cross-section $1\times R$ tubes pointing in many directions, is likewise open for $n\ge3$ (the companion Kakeya set dimension conjecture was settled in $\mathbb{R}^3$ by Wang and Zahl in 2025, but the maximal-function conjecture itself remains open in all dimensions $n\ge3$). The X-ray transform $Xf(\omega,v)$ of a function $f$ on $\mathbb{R}^n$ is its integral along the doubly-infinite line through $v$ in direction $\omega$, where $\omega\in\mathbb{S}^{n-1}$ and $v$ ranges over the hyperplane $\langle\omega\rangle^\perp$; the mixed norm $L_\omega^n L_v^\infty$ takes the $L^\infty$ norm in $v$ followed by the $L^n$ norm in $\omega$. Bennett and Nakamura, building on identities of Planchon–Vega and Beltran–Vega that exploit Plancherel's theorem on lines, launched a tomographic approach to restriction: they proved sharp bounds on $X(|\widehat{gd\sigma}|^2)$ and $\mathcal{R}(|\widehat{gd\sigma}|^2)$ (line and hyperplane averages of the squared extension), including a complete $(p,q)$ characterization for $X(|\widehat{gd\sigma}|^2)$ when $n=3$ (Theorem 1.6) and a sharp threshold $p_n=4(n-1)/(2n-3)$ for fractional X-ray norms (Theorem 7.2). Motivated by the integrability structure of these identities, they isolated a conjectural non-quadratic endpoint bound — Conjecture 1.3, the subject of this problem — and proved (Proposition 1.4) the chain of implications: Restriction Conjecture $\Rightarrow$ Conjecture 1.3 $\Rightarrow$ Kakeya Maximal Conjecture. The same paper studied the classical Mizohata–Takeuchi conjecture $\int_{\mathbb{R}^n}|\widehat{gd\sigma}|^2w\lesssim\|Xw\|_\infty\int_{\mathbb{S}^{n-1}}|g|^2$ for arbitrary weights $w$, proving localised variants (e.g. Theorem 1.7 in the plane, optimal in the range $1\le q\le2$ with a $\log R$ loss); that weighted conjecture was subsequently refuted by Cairo (2025), who constructed a $\log R$-loss counterexample for every non-flat $C^2$ hypersurface — a development that reshapes the weighted side of the program but leaves Conjecture 1.3, which permits an $R^\varepsilon$ loss and concerns the X-ray transform of the extension itself rather than weighted control by $\|Xw\|_\infty$, untouched.

Problem Statement

Conjecture 1.3 of Bennett–Nakamura (Tomography bounds for the Fourier extension operator and applications, Math. Ann. 380 (2021)): for every integer $n\ge2$ and every $\varepsilon>0$, does there exist a constant $C_\varepsilon<\infty$, depending only on $\varepsilon$ and $n$, such that for all $R>0$ and all $g\in L^{2n/(n-1)}(\mathbb{S}^{n-1})$ the inequality

$$\Big\|X\bigl(1_{B_R}|\widehat{g\,d\sigma}|^{\frac{2}{n-1}}\bigr)\Big\|_{L_\omega^n L_v^\infty}\le C_\varepsilon R^\varepsilon\|g\|_{L^{\frac{2n}{n-1}}(\mathbb{S}^{n-1})}^{\frac{2}{n-1}}$$

holds? Here $\widehat{g\,d\sigma}$ is the Fourier extension operator on the unit sphere, $Xf(\omega,v)=\int_{\mathbb{R}}f(v+s\omega)\,ds$ is the X-ray transform on the manifold of lines parametrized by $(\omega,v)\in\mathbb{S}^{n-1}\times\langle\omega\rangle^\perp$, $L_\omega^n L_v^\infty$ is the corresponding mixed norm, and $B_R$ is the ball of radius $R$ about the origin. The $n=2$ case follows from the classical restriction theorem in the plane via the authors' Proposition 1.4, so the open content is the conjecture in dimensions $n\ge3$, where it sits between the restriction conjecture (which implies it) and the Kakeya maximal conjecture (which it implies). The conjecture fails if there exist $g$'s for which the X-ray norm grows faster than $R^\varepsilon\|g\|_{L^{2n/(n-1)}}^{2/(n-1)}$ for some fixed $\varepsilon>0$; note that $\log R$-type growth does not refute the conjecture, since the constant may depend on $\varepsilon$.

References

1. Jonathan Bennett and Shohei Nakamura, "Tomography bounds for the Fourier extension operator and applications", Mathematische Annalen 380 (2021), 119–159. DOI: 10.1007/s00208-020-02131-0. https://arxiv.org/abs/2001.01674 (source paper)
2. Hannah M. Cairo, "A Counterexample to the Mizohata–Takeuchi Conjecture", preprint (2025). arXiv:2502.06137. https://arxiv.org/abs/2502.06137
3. Hannah M. Cairo and Ruixiang Zhang, "Power loss for the Mizohata–Takeuchi conjecture on $C^k$ convex hypersurfaces", preprint (2025). arXiv:2512.08064. https://arxiv.org/abs/2512.08064
4. Anthony Carbery, Marina Iliopoulou, and Hong Wang, "Some sharp inequalities of Mizohata–Takeuchi-type", Revista Matemática Iberoamericana 40 (2024), 1387–1418. DOI: 10.4171/rmi/1463. https://arxiv.org/abs/2302.11877
5. Jonathan Bennett, Susana Gutiérrez, Shohei Nakamura, and Itamar Oliveira, "A phase-space approach to weighted Fourier extension inequalities", Forum of Mathematics, Sigma 13 (2025). DOI: 10.1017/fms.2025.10127. https://arxiv.org/abs/2406.14886
6. Jonathan Bennett, Vjekoslav Kovač, Shohei Nakamura, and Itamar Oliveira, "Rectangles, triangles and Schrödinger waves", preprint (2026). arXiv:2606.30178. https://arxiv.org/abs/2606.30178
7. Jonathan Bennett, Shohei Nakamura, and Shobu Shiraki, "Tomographic Fourier extension identities for submanifolds of $\mathbb{R}^n$", Selecta Mathematica 30 (2024). DOI: 10.1007/s00029-024-00970-2. https://arxiv.org/abs/2212.12348
8. Alexander Ortiz, "A sharp weighted Fourier extension estimate for the cone in $\mathbb{R}^3$ based on circle tangencies", Journal d'Analyse Mathématique (2026). DOI: 10.1007/s11854-026-0448-x. https://arxiv.org/abs/2307.11731
9. Bassam Shayya, "Mizohata–Takeuchi estimates in the plane", preprint (2022). arXiv:2208.10305. https://arxiv.org/abs/2208.10305
10. Shukun Wu, "Weighted $L^2$ estimates with applications to $L^p$ problems", preprint (2025). arXiv:2506.02650. https://arxiv.org/abs/2506.02650
11. Inbo Gottlieb Fenves, "Cusp Excursions, Lattice Points on Manifolds, and the Mizohata–Takeuchi Conjecture", preprint (2026). arXiv:2606.27020. https://arxiv.org/abs/2606.27020
12. Hong Wang and Joshua Zahl, "Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions", preprint (2025). arXiv:2502.17655. https://arxiv.org/abs/2502.17655