| 苗长兴.调和分析及相关领域中的公开问题[J].数学年刊A辑,2025,46(1):1~54 |
| 调和分析及相关领域中的公开问题 |
| Open Problems in Harmonic Analysis and Related Fields |
| Received:March 20, 2025 Revised:April 01, 2025 |
| DOI：10.16205/j.cnki.cama.2025.0001 |
| 中文关键词: 猜想 限制性猜想 振荡积分算子 Fourier 积分算子 解耦方法 |
| 英文关键词:akeya conjecture Restriction conjecture Oscillatory integral operators Fourier integral operators Decoupling theory |
| 基金项目:国家重点研发项目 (No.2022YFA1005700) 和国家自然科学基金 (No.12371095) |
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| 中文摘要: |
| 适逢Wang-Zahl [Wang H, Zahl J. Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions [J/OL].arXiv: 2502.17655, 2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领域中的公开问题.围绕Kakeya猜想 (源于几何测度论, 分析版本对应Kakeya极大猜想)、限制性猜想、Bochner-Riesz猜想、局部光滑性猜想等四大猜想的研究, 发展了诸如解析插值方法、正交性与双线性方法, Heisenberg不确定原理与局部化方法、 微局部分析与驻相分析, 催生了波包分解与尺度归纳,多线性理论、Bourgain-Guth的broad-narrow分析、关联几何及多项式方法,特别是Wolff 及Bourgain-Demeter等发展的解耦方法,不仅推动了调和分析中四大猜想的研究,同时也为解决其他数学领域的重要问题提供了一系列强有力工具. |
| 英文摘要: |
| On the occasion of Wang-Zahl’s [Wang H and Zahl J. Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions. arXiv: 2502.17655,2025] announcement of a solution to the three-dimensional Kakeya set conjecture, this survey introduces open problems in harmonic analysis and related fields. In order to study orsolve the four conjectures of Kakeya conjecture (originating from geometric measure theory,whose analytic version is the Kakeya maximal conjecture), the restriction conjecture, theBochner-Riesz conjecture and the local smoothness conjecture, mathematicians have developed a variety of powerful techniques, such as analytic interpolation method, orthogonalityand bilinear method, Heisenberg uncertainty principle and localization method, microlocaland stationary phase analysis, wave packet decomposition and induction on scales, multilinear theory, Bourgain-Guth’s broad-narrow analysis, incidence geometry and polynomialsmethods, as well as the decoupling methods developed by Wolff and Bourgain-Demeter.These tools not only facilitate the study of the four conjectures in harmonic analysis, butalso provide a series of powerful tools for solving important problems in other mathematicalfields. |
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