Besov--Triebel--Lizorkin-Type Spaces with Matrix $A_{\mathrm{\infty }}$ Weights

Introduced by A. Volberg, matrix $A_{p,\mathrm{\infty }}$ weights provide a suitable generalization of Muckenhoupt $A_{\mathrm{\infty }}$ 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 $\phi$-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 $A_{p,\mathrm{\infty }}$-matrix weights, but some of the results are new even when specialized to the scalar unweighted case.