Two weight estimates with matrix measures for well localized operators

TL;DR

This paper characterizes when well localized operators are bounded between matrix-weighted L^2 spaces using Sawyer-type testing conditions, extending classical results to a more general setting with matrix measures and filtrations.

Contribution

It provides necessary and sufficient conditions for matrix-weighted L^2 estimates of well localized operators, including novel modifications of T1 proof strategies in a general setting.

Findings

Characterization of boundedness via testing conditions

Extension to matrix measures and arbitrary filtrations

Polynomial estimates on Haar shift operator complexity

Abstract

In this paper, we give necessary and sufficient conditions for weighted $L^2$ estimates with matrix-valued measures of well localized operators. Namely, we seek estimates of the form: \[ \| T(\mathbf{W} f)\|_{L^2(\mathbf{V})} \le C\|f\|_{L^2(\mathbf{W})} \] where $T$ is formally an integral operator with additional structure, $\mathbf{W}, \mathbf{V}$ are matrix measures, and the underlying measure space possesses a filtration. The characterization we obtain is of Sawyer-type; in particular we show that certain natural testing conditions obtained by studying the operator and its adjoint on indicator functions suffice to determine boundedness. Working in both the matrix weighted setting and the setting of measure spaces with arbitrary filtrations requires novel modifications of a T1 proof strategy; a particular benefit of this level of generality is that we obtain polynomial estimates on the complexity of certain Haar shift operators.