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

Abstract

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

84 pages; Submitted. arXiv admin note: text overlap with arXiv:2304.00292