Two weight estimates with matrix measures for well localized operators

Publication Date

2019

Description

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.

Journal

Transactions of the American Mathematical Society

Volume

371

Issue

9

First Page

6213

Last Page

6240

Department

Mathematics

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