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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Two weight estimates with matrix measures for well localized operators


Authors: Kelly Bickel, Amalia Culiuc, Sergei Treil and Brett D. Wick
Journal: Trans. Amer. Math. Soc. 371 (2019), 6213-6240
MSC (2010): Primary 42B20; Secondary 60G42, 60G46, 47G10
DOI: https://doi.org/10.1090/tran/7400
Published electronically: February 1, 2019
MathSciNet review: 3937322
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Abstract | References | Similar Articles | Additional Information

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

$\displaystyle \Vert T(\mathbf {W} f)\Vert _{L^2(\mathbf {V})} \le C\Vert f\Vert _{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.

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Additional Information

Kelly Bickel
Affiliation: Department of Mathematics, Bucknell University, 701 Moore Avenue, Lewisburg, Pennsylvania 17837
Email: kelly.bickel@bucknell.edu

Amalia Culiuc
Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
Email: amalia@math.gatech.edu

Sergei Treil
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Email: treil@math.brown.edu

Brett D. Wick
Affiliation: Department of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 63130-4899
Email: wick@math.wustl.edu

DOI: https://doi.org/10.1090/tran/7400
Received by editor(s): January 6, 2017
Received by editor(s) in revised form: August 24, 2017
Published electronically: February 1, 2019
Additional Notes: The first author’s research is supported in part by National Science Foundation DMS grant #1448846.
The third author’s research is supported in part by National Science Foundation DMS grants #1301579, #1600139.
The fourth author’s research is supported in part by National Science Foundation DMS grant #1500509.
Article copyright: © Copyright 2019 American Mathematical Society