New Characterizations and Properties of Matrix $A_{\infty }$ Weights

Abstract

We provide several new characterizations of $A_{p,\infty }$-matrix weights, originally introduced by A. Volberg as matrix-valued substitutes of the classical $A_{\infty }$ weights. In analogy with the notion of $A_p$-dimension of matrix weights introduced in our previous work, we introduce the concepts of the lower and the upper dimensions of $A_{p,\infty }$-matrix weights, which enable us to obtain sharp estimates related to their reducing operators. In a follow-up work, these results will play a key role in the study of function spaces with $A_{p,\infty }$-matrix weights, which extends earlier results in the more restricted class of $A_p$-matrix weights.

39 pages. arXiv admin note: text overlap with arXiv:2304.00292