Quantitative uniqueness properties for $L^2$ functions with fast decaying, or sparsely supported, Fourier transform
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.
Keywords
Cite
@article{arxiv.1808.02149,
title = {Quantitative uniqueness properties for $L^2$ functions with fast decaying, or sparsely supported, Fourier transform},
author = {Benjamin Jaye and Mishko Mitkovski},
journal= {arXiv preprint arXiv:1808.02149},
year = {2020}
}
Comments
27 pages. Some further discussion around quantitative estimates has been added