Two weight estimates with matrix measures for well localized operators

In this paper, we give necessary and sufficient conditions for weighted L 2 L^2 estimates with matrix-valued measures of well localized operators. Namely, we seek estimates of the form \[ ‖ T ( W f ) ‖ L 2 ( V ) ≤ C ‖ f ‖ L 2 ( W ) , \| T(W f)\|L^2(V) ≤ C\|f\|L^2(W), \] where T T is formally an integral operator with additional structure, W , V W, 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.