Two weight estimates with matrix measures for well localized operators
In this paper, we give necessary and sufficient conditions for weighted L2 estimates with matrix-valued measures of well localized operators. Namely, we seek estimates of the form
T(Wf)L2(V) ≤ CfL2(W),
where T is formally an integral operator with additional structure, 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.
Transactions of the American Mathematical Society
Kelly Bickel, Amalia Culiuc, Sergei Treil and Brett D. Wick. Trans. Amer. Math. Soc. 371 (2019), 6213-6240.