For the Fourier extension operator on a well-curved curve in Rn\mathbb{R}^n (model case: the moment curve ξ↦(ξ,ξ2,…,ξn)\xi\mapsto(\xi,\xi^2,\dots,\xi^n)), Carbery–Li–Pang–Yung proved local weighted L2L^2 inequalities of Mizohata–Takeuchi type with exponent ac=n−32+2n−2n2(n+1)a_c=\frac{n-3}{2}+\frac{2}{n}-\frac{2}{n^2(n+1)} for every n≥2n\ge2, sharp when n=2n=2. Whether this exponent — and the RR-powers in the companion weaker L1L^1 (part-(b)) tube- and plank-weighted inequalities — is sharp in the genuine extension setting for n≥3n\ge3 is open: the wave-packet examples that certify sharpness in decoupling and axiomatic models do not correspond to admissible inputs g∈L2(Γ)g\in L^2(\Gamma) when n≥3n\ge3. The problem asks for matching upper and lower bounds for this family of inequalities, including whether the hyperplane-weighted inequality holds for every fixed a>0a>0 (the modified Mizohata–Takeuchi conjecture for curves) and whether the axiomatic necessary conditions can be made to match the hyperplane-functional upper bound.

Background

The Fourier extension operator is the central object of restriction theory: for a compact curve Γ⊂Rn\Gamma\subset\mathbb{R}^n with arclength measure dλ\mathrm{d}\lambda it maps g∈L2(Γ)g\in L^2(\Gamma) to g dλ^\widehat{g\,\mathrm{d}\lambda}, the Fourier transform of the measure g dλg\,\mathrm{d}\lambda. The Mizohata–Takeuchi conjecture predicts that the weighted L2L^2 mass of an extension is controlled by the largest amount of weight on straight lines: for hypersurfaces Σ\Sigma with surface measure dσ\mathrm{d}\sigma it asserts ∫Rn∣g dσ^∣2w≲(sup⁡ℓXw(ℓ))∫Σ∣g∣2 dσ\int_{\mathbb{R}^n}|\widehat{g\,\mathrm{d}\sigma}|^2w\lesssim(\sup_\ell Xw(\ell))\int_\Sigma|g|^2\,\mathrm{d}\sigma, where Xw(ℓ)=∫ℓwXw(\ell)=\int_\ell w is the X-ray transform of ww along the line ℓ\ell; such a bound would also locate where extension mass concentrates. Cairo (arXiv:2502.06137) disproved the endpoint by constructing a log⁡R\log R-loss counterexample for every curved C2C^2 hypersurface, so attention moved to local quantitative variants ∫BR∣g dσ^∣2w≲Ra(sup⁡ℓXw(ℓ))∫Σ∣g∣2 dσ\int_{B_R}|\widehat{g\,\mathrm{d}\sigma}|^2w\lesssim R^a(\sup_\ell Xw(\ell))\int_\Sigma|g|^2\,\mathrm{d}\sigma with the exponent a≥0a\ge0 taken as small as possible; Carbery–Iliopoulou–Wang achieved every a>(n−1)/(n+1)a>(n-1)/(n+1) for strictly convex hypersurfaces, and an enemy-scenario argument presented in that paper shows that wave-packet and decoupling tools alone cannot beat that threshold. For curves the natural formulation replaces line masses by hyperplane masses: ∫Rn∣g dλ^∣2w≲(sup⁡H∫Hw dλn−1)∫Γ∣g∣2 dλ,\int_{\mathbb{R}^n}|\widehat{g\,\mathrm{d}\lambda}|^2w\lesssim\Big(\sup_H\int_H w\,\mathrm{d}\lambda^{n-1}\Big)\int_\Gamma|g|^2\,\mathrm{d}\lambda, the supremum over hyperplanes HH whose normal is parallel to the tangent of Γ\Gamma at some point. The classical Agmon–Hörmander trace inequality, which gives the local estimate ∫BR∣g dλ^∣2≲Rn−1∫Γ∣g∣2 dλ\int_{B_R}|\widehat{g\,\mathrm{d}\lambda}|^2\lesssim R^{n-1}\int_\Gamma|g|^2\,\mathrm{d}\lambda, yields the local weighted version with a=n−1a=n-1. Carbery–Li–Pang–Yung made the first substantial improvement for well-curved curves — Cn+1C^{n+1} curves with nonvanishing torsion, i.e. ∣Γ(1)∧⋯∧Γ(n)∣≳1|\Gamma^{(1)}\wedge\cdots\wedge\Gamma^{(n)}|\gtrsim1, the model case being the moment curve: with r=n(n+1)n(n+1)−2r=\frac{n(n+1)}{n(n+1)-2} they proved, for every ϵ>0\epsilon>0, ∫BR∣g dλ^∣2w≲Rac+ϵ(sup⁡H∫Hwr dλn−1)1/r∫Γ∣g∣2 dλ,ac=n−32+2n−2n2(n+1),\int_{B_R}|\widehat{g\,\mathrm{d}\lambda}|^2w\lesssim R^{a_c+\epsilon}\Big(\sup_H\int_H w^r\,\mathrm{d}\lambda^{n-1}\Big)^{1/r}\int_\Gamma|g|^2\,\mathrm{d}\lambda,\qquad a_c=\frac{n-3}{2}+\frac{2}{n}-\frac{2}{n^2(n+1)}, equivalently with (sup⁡S∫Swr)1/r(\sup_S\int_S w^r)^{1/r} over 1-neighbourhoods SS of such hyperplanes; replacing the LrL^r norm by the L1L^1 norm sup⁡S∫Sw\sup_S\int_S w at the same exponent gives a weaker corollary rather than an equivalent formulation. For n=2n=2 this gives ac=1/3a_c=1/3, shown sharp by Carbery–Iliopoulou–Wang. The proof uses a weighted formulation of refined decoupling for curves — decoupling estimates, reweighted by where the input mass actually sits, for functions with Fourier support in the anisotropic curvature sleeve of Γ\Gamma (boxes of side R−1/n×R−2/n×⋯×R−1R^{-1/n}\times R^{-2/n}\times\cdots\times R^{-1}), built on the Bourgain–Demeter–Guth decoupling for the moment curve. A key structural phenomenon for n≥3n\ge3: decomposing 1BRg dλ^1_{B_R}\widehat{g\,\mathrm{d}\lambda} into wave packets produces pieces frequency-localized to boxes with substantial mass outside the R−1R^{-1}-neighbourhood of Γ\Gamma, so these pieces are not extensions of any function on Γ\Gamma. The same paper proves companion inequalities with tube/plank weights (tubes parallel to the final Serret–Frenet direction, i.e. the last vector of the moving orthonormal frame of Γ\Gamma): the power (n−2)+2n(n+1)(n-2)+\frac{2}{n(n+1)} is certified sharp by an elementary construction in the LrL^r tube-weighted version, whereas for the weaker L1L^1 part-(b) tube and plank versions its sharpness in the extension setting remains open (for the tube version the power is in fact not expected to be sharp), and the paper studies sharpness in an axiomatic decoupling framework for curves, where enemy constructions give necessary conditions on exponents.

Problem Statement

Determine the optimal power of RR in the local Mizohata–Takeuchi-type weighted L2L^2 inequalities for the Fourier extension operator on well-curved curves in Rn\mathbb{R}^n, as left open by Carbery–Li–Pang–Yung. The common scientific objective is to establish matching upper and lower bounds for this family of inequalities in the genuine extension setting, where inputs are functions g∈L2(Γ)g\in L^2(\Gamma) rather than decoupling or axiomatic wave-packet objects; the four components below are the specific instances of that objective left open by the source. (1) Is the exponent ac=n−32+2n−2n2(n+1)a_c=\frac{n-3}{2}+\frac{2}{n}-\frac{2}{n^2(n+1)} sharp for n≥3n\ge3: either construct, for each n≥3n\ge3, families of nonnegative weights ww and inputs g∈L2(Γ)g\in L^2(\Gamma) showing that ∫BR∣g dλ^∣2w≲Ra(sup⁡H∫Hwr dλn−1)1/r∫Γ∣g∣2 dλ\int_{B_R}|\widehat{g\,\mathrm{d}\lambda}|^2w\lesssim R^a(\sup_H\int_H w^r\,\mathrm{d}\lambda^{n-1})^{1/r}\int_\Gamma|g|^2\,\mathrm{d}\lambda fails (or saturates) for any smaller exponent, or prove the inequality with a strictly smaller exponent. (2) Are the companion part-(b) inequalities — where the weight is measured by sup⁡T∫Tw\sup_T\int_Tw over 1-tubes parallel to the final Serret–Frenet direction of Γ\Gamma, or by sup⁡L∫Lw\sup_L\int_Lw over planks — sharp in their RR-powers in the extension setting for n≥3n\ge3 (for the plank version this is open even for n=2n=2)? (3) Does the hyperplane-weighted inequality hold for every fixed a>0a>0 (constants depending on aa) uniformly over all nonnegative weights — the curve version of the modified Mizohata–Takeuchi conjecture — or is a power loss necessary for curves as it is for hypersurfaces? (4) In the axiomatic decoupling framework for curves, can the necessary condition for the hyperplane (thickened-hyperplane) functional be made to match the upper bound when n≥3n\ge3, closing the gap caused by the useful scale coinciding with the canonical scale? A resolution of all four components establishes the sharp form of the weighted extension inequalities for well-curved curves.

References

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