A weighted formulation of refined decoupling and inequalities of Mizohata-Takeuchi-type for the moment curve
Anthony Carbery, Zane Kun Li, Yixuan Pang, Po-Lam Yung
TL;DR
This work advances the Mizohata–Takeuchi program for one-dimensional manifolds (curves) in $\mathbb{R}^n$ by developing a weighted refined decoupling theory for well-curved curves and translating it into local extension-operator estimates. The main achievement is a sharp-like local bound with exponent $a > (n-3)/2 + 2/n - 2/(n^2(n+1)) + \varepsilon$, together with a weighted formulation that ties decoupling to hyperplane- and tube-based weight norms. The authors establish an axiomatic decoupling framework that isolates the core geometric-analytic mechanisms and prove sharpness results in that setting, alongside full appendices detailing the delicate wave-packet and incidence arguments. By passing from the weighted decoupling bounds to Mizohata–Takeuchi-type inequalities for the Fourier extension operator, the paper yields new partial progress on the conjecture for curves in dimensions $n \ge 3$, with explicit exponents and structural comparisons to the hypersurface case. The results thus provide a robust toolkit—refined decoupling, weighted formulations, and an axiomatic perspective—that deepens our understanding of local Fourier extension phenomena for curved manifolds and their weight-sensitive behavior.
Abstract
Let $Γ$ be a compact patch of a well-curved $C^{n+1}$ curve in $\mathbb{R}^n$ with induced Lebesgue measure ${\rm d} λ$, and let $g \mapsto \widehat{g \,{\rm d}λ}$ be the Fourier extension operator for $Γ$. Then we have, for arbitrary non-negative weights $w$, \begin{equation*} \int_{B_R} |\widehat{g \,{\rm d}λ}|^2w \leq C_{n,a} R^{a} \sup_S \left(\int_S w\right)\int_Γ|g|^2 \, {\rm d} λ \end{equation*} for any $a> \frac{n-3}{2} + \frac{2}{n} - \frac{2}{n^2(n+1)}$, where the $\sup$ is over all $1$-neighbourhoods $S$ of hyperplanes whose normals are parallel to the tangent at some point of $Γ$. This represents partial progress on the Mizohata-Takeuchi conjecture for curves in dimensions $n \geq 3$, improving upon the exponent $a=n-1$ which can be obtained as a consequence of the Agmon-Hörmander trace inequality. Our main tool in establishing this inequality will be a weighted formulation of refined decoupling for well-curved curves. We also discuss the sharpness of the exponents we obtain in this and in auxiliary results, and further explore this in the context of axiomatic decoupling for curves.