Characterizations of A2 matrix power weights

In the scalar setting, the power functions |x|γ, for −1<γ<1, are the canonical examples of A2 weights. In this paper, we study two types of power functions in the matrix setting, with the goal of obtaining canonical examples of A2 matrix weights. We first study Type 1 matrix power functions, which are n×n matrix functions whose entries are of the form a|x|γ. Our main result characterizes when these power functions are A2 matrix weights and has two extensions to Type 1 power functions of several variables. We also study Type 2 matrix power functions, which are n×n matrix functions whose eigenvalues are of the form a|x|γ. We find necessary conditions for these to be A2 matrix weights and give an example showing that even nice functions of this form can fail to be A2 matrix weights.