Characterizations of $A_2$ Matrix Power Weights

Abstract

In the scalar setting, the power functions $|x|^{\gamma}$, for $-1 < \gamma<1$, are the canonical examples of $A_2$ weights. In this paper, we study two types of power functions in the matrix setting, with the goal of obtaining canonical examples of $A_2$ matrix weights. We first study Type 1 matrix power functions, which are $n\times n$ matrix functions whose entries are of the form $a|x|^{\gamma}.$ Our main result characterizes when these power functions are $A_2$ matrix weights and has two extensions to Type $1$ power functions of several variables. We also study Type 2 matrix power functions, which are $n\times n$ matrix functions whose eigenvalues are of the form $a|x|^{\gamma}.$ We find necessary conditions for these to be $A_2$ matrix weights and give an example showing that even nice functions of this form can fail to be $A_2$ matrix weights.