A weighted formulation of refined decoupling and inequalities of Mizohata-Takeuchi-type for the moment curve

Abstract

Let $\Gamma$ be a compact patch of a well-curved $C^{n+1}$ curve in ${\mathbb{R}}^n$ with induced Lebesgue measure $\mathrm{d}\lambda$, and let $g\mapsto \widehat{g\mathrm{d}\lambda }$ be the Fourier extension operator for $\Gamma$. 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 $\Gamma$. This represents partial progress on the Mizohata-Takeuchi conjecture for curves in dimensions $n\ge 3$, 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.

47 pages