Geodesic
Unconditional bases, the matrix Muckenhoupt condition, and Carleson series in the spectrum
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 27, Tome 262 (1999), pp. 90-126

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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. 
@article{ZNSL_1999_262_a3,
     author = {G. M. Gubreev and E. I. Olefir},
     title = {Unconditional bases, the matrix {Muckenhoupt} condition, and {Carleson} series in the spectrum},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {90--126},
     publisher = {mathdoc},
     volume = {262},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_262_a3/}
}
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G. M. Gubreev; E. I. Olefir. Unconditional bases, the matrix Muckenhoupt condition, and Carleson series in the spectrum. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 27, Tome 262 (1999), pp. 90-126. http://geodesic.mathdoc.fr/item/ZNSL_1999_262_a3/