Two weight estimates with matrix measures for well localized operators
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- by Kelly Bickel, Amalia Culiuc, Sergei Treil and Brett D. Wick PDF
- Trans. Amer. Math. Soc. 371 (2019), 6213-6240 Request permission
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 \[ \| T(\mathbf {W} f)\|_{L^2(\mathbf {V})} \le C\|f\|_{L^2(\mathbf {W})}, \] 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.References
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Additional Information
- Kelly Bickel
- Affiliation: Department of Mathematics, Bucknell University, 701 Moore Avenue, Lewisburg, Pennsylvania 17837
- MR Author ID: 997443
- 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
- MR Author ID: 232797
- 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
- MR Author ID: 766171
- ORCID: 0000-0003-1890-0608
- Email: wick@math.wustl.edu
- 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. - © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 3937322