Besov–Triebel–Lizorkin-type spaces with matrix A_∞ weights

Introduced by A. Volberg (1997), matrix A_p,∞ weights provide a suitable generalization of Muckenhoupt A_∞ weights from the classical theory. In our previous work, we established new characterizations of these weights. Here, we use these results to study inhomogeneous Besov-type and Triebel–Lizorkin-type spaces with such weights. In particular, we characterize these spaces, in terms of the φ-transform, molecules, and wavelets, and obtain the boundedness of almost diagonal operators, pseudo-differential operators, trace operators, pointwise multipliers, and Calderón–Zygmund operators on these spaces. This is the first systematic study of inhomogeneous Besov–Triebel–Lizorkin-type spaces with A_p,∞-matrix weights, but some of the results are new even when specialized to the scalar unweighted case.