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arXiv · 1808.02149

Quantitative uniqueness properties for $L^2$ functions with fast decaying, or sparsely supported, Fourier transform

Benjamin Jaye·Mishko Mitkovski
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Abstract

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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Benjamin Jaye, Mishko Mitkovski. 2018-08-06. Quantitative uniqueness properties for $L^2$ functions with fast decaying, or sparsely supported, Fourier transform. https://arxiv.org/abs/1808.02149

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