Generalized sign Fourier uncertainty

Abstract

We consider a generalized version of the sign uncertainty principle for the Fourier transform, first proposed by Bourgain, Clozel and Kahane in 2010 and revisited by Cohn and Gonçalves in 2019. In our setup, the signs of a function and its Fourier transform resonate with a generic given function $P$ outside of a ball. One essentially wants to know if and how soon this resonance can happen, when facing a suitable competing weighted integral condition. The original version of the problem corresponds to the case $P\equiv 1$. Surprisingly, even in such a rough setup, we are able to identify sharp constants in some cases.

arXiv comment27 pages. To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze