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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.18655 |
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| _version_ | 1866917101635108864 |
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| author | Tacy, Melissa |
| author_facet | Tacy, Melissa |
| contents | This paper develops a new framework, \emph{simultaneous saturation}, designed to quantify the size of sets whose elements are simultaneously large. The framework establishes a correspondence between the magnitude of such sets and a system of interdependent conditions linking their points. We first prove a general theorem establishing the correspondence and then apply the framework to multilinear restriction-type estimates. From this perspective, we obtain a new proof (independent of Bennett-Carbery-Tao \cite{BCT}) of the $d$-linear restriction/extension theorem, and establish the $λ^ε$ loss conjectured bounds for the $k$-linear $L^{2}\to L^{p/k}$ extension problem under mixed transversality/curvature conditions $(k<d)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18655 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The principle of simultaneous saturation: Application to the $k$-linear restriction/extension problem Tacy, Melissa Classical Analysis and ODEs Analysis of PDEs This paper develops a new framework, \emph{simultaneous saturation}, designed to quantify the size of sets whose elements are simultaneously large. The framework establishes a correspondence between the magnitude of such sets and a system of interdependent conditions linking their points. We first prove a general theorem establishing the correspondence and then apply the framework to multilinear restriction-type estimates. From this perspective, we obtain a new proof (independent of Bennett-Carbery-Tao \cite{BCT}) of the $d$-linear restriction/extension theorem, and establish the $λ^ε$ loss conjectured bounds for the $k$-linear $L^{2}\to L^{p/k}$ extension problem under mixed transversality/curvature conditions $(k<d)$. |
| title | The principle of simultaneous saturation: Application to the $k$-linear restriction/extension problem |
| topic | Classical Analysis and ODEs Analysis of PDEs |
| url | https://arxiv.org/abs/2501.18655 |