A fractal uncertainty principle for Bergman spaces and analytic wavelets

Abstract

Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor sets and by similar results of Knutsen for joint time-frequency representations (i.e., the short-time Fourier transform (STFT) with a Gaussian window, equivalent to Fock spaces), we suggest a general setting relating localization and uncertainty and prove, within this context, an uncertainty principle for Cantor sets in Bergman spaces on the unit disk, where the Cantor set is defined as a union of annuli that are equidistributed in the hyperbolic measure. The result can be written in terms of analytic Cauchy wavelets. As in the case of the STFT considered by Knutsen, our result consists of a two-sided bound for the norm of a localization operator involving the fractal dimension log⁡2/log⁡3 in the exponent. As in the STFT case and in Dyatlov's fractal uncertainty principle, the (hyperbolic) measure of the dilated iterates of the Cantor set in the disk tends to infinity, while the corresponding norm of the localization operator tends to zero.