Matrix-weighted Besov-type and Triebel-Lizorkin-type spaces I: Ap-dimensions of matrix weights and'-transform characterizations

Let , and It is well known that Besov-type spaces. B s,t p,q with and Triebel-Lizorkin-type spaces. F s,t p,q with when or with when t = 0 on Rn consist of a general family of function spaces that cover not only the well-known Besov and Triebel-Lizorkin spaces and (when t = 0) but also several other function spaces of interest, such as Morrey spaces and Q spaces. In three successive articles, the authors develop a complete real-variable theory of matrix-weighted Besov-type spaces (W) and matrix-weighted Triebel-Lizorkin-type spaces (W) on Rn, where W is a matrixvalued Muckenhoupt Ap weight. This article is the first one, whosemain novelty exists in that the authors introduce the new concept, Ap-dimensions of matrix weights, and intensively study their properties, especially those elaborate properties expressed via reducing operators. The authors then introduce the spaces (W) and (W) and, using Ap-dimensions and their nice properties, the authors establish the.-transform characterization of (W) and (W). The Ap-dimensions of matrix weights