具有快速衰减或稀疏支撑的傅里叶变换的$L^2$函数的定量唯一性
Quantitative uniqueness properties for $L^2$ functions with fast decaying, or sparsely supported, Fourier transform
摘要：
本文建立在两个关键原则背后的Bourgain-Dyatlov定量唯一性定理的功能与傅里叶变换支持Ahlfors正则集。我们首先提供了一个表征时的定量唯一性定理的功能非常迅速衰减傅里叶变换，从而提供了经典的Paneah-Logvinenko-Sereda定理的扩展。其次，我们得到了一个转移结果，它将一个关于快速衰减傅里叶变换的函数的定量唯一性定理转化为一个关于支撑在分形集上的傅里叶变换的函数的定量唯一性定理。以及恢复Bourgain-Dyatlov的结果，我们得到类似的唯一性结果密集的分形。
英文摘要：
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. As well as recovering the result of Bourgain-Dyatlov, we obtain analogous uniqueness results for denser fractals.
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