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2018 Characterization of a two-parameter matrix valued BMO by commutator with the Hilbert transform
Dario Mena
Rocky Mountain J. Math. 48(2): 529-550 (2018). DOI: 10.1216/RMJ-2018-48-2-529

Abstract

In this paper, we prove that the space of two parameter matrix-valued BMO functions can be char\-ac\-terized by considering iterated commutators with the Hilbert transform. Specifically, we prove that $$ \| B \|_{BMO} \!\lesssim \! \| [[M_B, H_1],H_2] \|_{L^2(\mathbb{R} ^2;\mathbb{C} ^d) \rightarrow L^2(\mathbb{R} ^2;\mathbb{C} ^d)} \!\lesssim \! \| B \|_{BMO}. $$ The upper estimate relies on Petermichl's representation of the Hilbert transform as an average of dyadic shifts and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and Lacey's proof for the scalar case.

Citation

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Dario Mena. "Characterization of a two-parameter matrix valued BMO by commutator with the Hilbert transform." Rocky Mountain J. Math. 48 (2) 529 - 550, 2018. https://doi.org/10.1216/RMJ-2018-48-2-529

Information

Published: 2018
First available in Project Euclid: 4 June 2018

zbMATH: 06883480
MathSciNet: MR3810208
Digital Object Identifier: 10.1216/RMJ-2018-48-2-529

Subjects:
Primary: 42B20

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 2 • 2018
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