Matrix-Weighted Besov–Triebel–Lizorkin Spaces with Logarithmic Smoothness

In this article, the authors study the matrix-weighted Besov–Triebel–Lizorkin spaces with logarithmic smoothness. Via first obtaining the Lp(Rn)-boundedness and the Fefferman–Stein type vector-valued inequality of matrix-weighted Peetre-type maximal functions with the exquisite ranges of indices in terms of the Ap dimension of matrix weights under consideration, the authors establish an equivalent characterization of these spaces in terms of the matrix-weighted Peetre-type maximal functions, which further implies that these spaces are well defined. As an application, the authors obtain the boundedness of some pointwise multipliers on these spaces and, even back to classical Besov–Triebel–Lizorkin spaces, some of them are also new.