Characterizations of $A_2$ Matrix Power Weights

TL;DR

This paper characterizes when certain matrix-valued power functions serve as $A_2$ weights, extending scalar concepts to matrix settings and identifying conditions under which these functions are valid $A_2$ matrix weights.

Contribution

It provides the first characterization of $A_2$ matrix power weights for Type 1 functions and necessary conditions for Type 2 functions, advancing the understanding of matrix $A_2$ weights.

Findings

Characterization of Type 1 matrix power functions as $A_2$ weights.

Necessary conditions for Type 2 matrix power functions to be $A_2$ weights.

Examples showing some well-behaved functions fail to be $A_2$ 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.