Unconditional bases, the matrix Muckenhoupt condition, and Carleson series in the spectrum

For two families of functions generated by a system of $n$ scalar Muckenhoupt weights, criteria are obtained for being unconditional basic sequences. From the point of view of the spectral operator theory, the problem is reduced to analyzing the structure of $n$-dimensional perturbations of the integration operator. With the help of weighted estimates for the Hilbert transform in the spaces of vector-functions, an operator is constructed that transforms the functions of the given families into vector-valued rational functions. The concept of Carleson series is used for solving the problem of being an unconditional basis.