Well-Localized Operators on Matrix Weighted $L^2$ Spaces

Abstract

Nazarov-Treil-Volberg recently proved an elegant two-weight T1 theorem for "almost diagonal" operators that played a key role in the proof of the $A_2$ conjecture for dyadic shifts and related operators. In this paper, we obtain a generalization of their T1 theorem to the setting of matrix weights. Our theorem does differ slightly from the scalar results, a fact attributable almost completely to differences between the scalar and matrix Carleson Embedding Theorems. The main tools include a reduction to the study of well-localized operators, a new system of Haar functions adapted to matrix weights, and a matrix Carleson Embedding Theorem.

v2: 26 pages, additional references added, small changes to Remarks 1.4 and 3.4, v3: referee comments addressed, additional changes to Remark 1.4

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