Unconditional bases, the matrix Muckenhoupt condition and Carleson series in a spectrum (Q1608590)

Two families of functions are assigned to a finite system of scalar Muckenhoupt weights on \(\mathbb{R}\). The problem under consideration is if they form unconditional bases in the corresponding spaces. A criterion is obtained for the first family and a sufficient condition for the second one, the both in terms of the corresponding property for rational vector-valued functions. This is done by studying the spectral structure of perturbations of the integration operators and applying two-sided estimates for the Hilbert transform of vector-valued functions. The problem for the rational functions is handled by means of technique of Carleson series.