Quantitative Uniqueness Properties for L 2 Functions with Fast Decaying, or Sparsely Supported, Fourier Transform
Quantitative Uniqueness Properties for L 2 Functions with Fast Decaying, or Sparsely Supported, Fourier Transform
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具有快速衰减或稀疏支持的傅立叶变换的 L 2 函数的定量唯一性属性
DOI：
10.1093/imrn/rnab075
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发表时间：
2021
影响因子：
1
通讯作者：
Mitkovski, Mishko
中科院分区：
文献类型：
--
作者：
Jaye, Benjamin;Mitkovski, Mishko
本文基于 Ahlfors 正则集中支持傅立叶变换的函数的 Bourgain-Dyatlov 定量唯一性定理背后的两个关键原理。我们首先描述了定量唯一性定理何时适用于具有快速衰减傅里叶变换的函数，从而提供了经典 Paneah-Logvinenko-Sereda 定理的扩展。其次，我们推导出一个转移结果，它将快速衰减傅里叶变换函数的定量唯一性定理转换为分形集上支持的傅里叶变换函数的定量唯一性定理。除了恢复 Bourgain-Dyatlov 的结果之外，我们还获得了密集分形的类似唯一性结果。
This paper builds upon two key principles behind the Bourgain–Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah–Logvinenko–Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on a fractal set. In addition to recovering the result of Bourgain–Dyatlov, we obtain analogous uniqueness results for denser fractals.