A weighted formulation of refined decoupling and inequalities of Mizohata-Takeuchi-type for the moment curve
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| Format: | Preprint |
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2025
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| _version_ | 1866911192751013888 |
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| author | Carbery, Anthony Li, Zane Kun Pang, Yixuan Yung, Po-Lam |
| author_facet | Carbery, Anthony Li, Zane Kun Pang, Yixuan Yung, Po-Lam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A weighted formulation of refined decoupling and inequalities of Mizohata-Takeuchi-type for the moment curve Carbery, Anthony Li, Zane Kun Pang, Yixuan Yung, Po-Lam Classical Analysis and ODEs Analysis of PDEs 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. |
| title | A weighted formulation of refined decoupling and inequalities of Mizohata-Takeuchi-type for the moment curve |
| topic | Classical Analysis and ODEs Analysis of PDEs |
| url | https://arxiv.org/abs/2510.04345 |