Commutators, little bmo and weak factorization

Xuan Thinh Duong; Ji Li; Brett D. Wick; Dongyong Yang

doi:10.5802/aif.3153

Commutators, little bmo and weak factorization

Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang

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Abstract

In this paper, we provide a direct and constructive proof of weak factorization of h¹ (R × R) (the predual of little BMO space bmo(R × R) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f ∈ h¹ (R × R) there exist sequences {α^k _j } ∈ and functions g_j ^k, h^k _j ∈ L² (R² ) such that ∞ ∞ f = α^k _j ^k _j H₁ H₂ g_j ^k − g_j ^kH₁ H₂ h^k k=1 j=1 in the sense of h¹ (R × R), where H₁ and H₂ are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm f_h1₍R×R₎ is given in terms of g_j ^k _L2_(R2₎ and h^k _{j L}2_(R2₎. By duality, this directly implies a lower bound on the norm of the commutator [b, H₁ H₂ ] in terms of b_bmo(R×R). Our method bypasses the use of analyticity and the Fourier transform, and hence can be extended to the higher dimension case in an arbitrary n-parameter setting for the Riesz transforms.

Original language	English
Pages (from-to)	109-129
Number of pages	21
Journal	Annales de l'Institut Fourier
Volume	68
Issue number	1
DOIs	https://doi.org/10.5802/aif.3153
Publication status	Published - 1 Jan 2018

Bibliographical note

Copyright Association des Annales de l’institut Fourier, 2018. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

Keywords

Bmo(R × R)
Commutator
H1(R × R)
Hilbert transform
Weak factorization

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10.5802/aif.3153Licence: CC BY-ND

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title = "Commutators, little bmo and weak factorization",

abstract = "In this paper, we provide a direct and constructive proof of weak factorization of h1 (R × R) (the predual of little BMO space bmo(R × R) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f ∈ h1 (R × R) there exist sequences {αk j } ∈ and functions gj k, hk j ∈ L2 (R2 ) such that ∞ ∞ f = αk j k j H1 H2 gj k − gj kH1 H2 hk k=1 j=1 in the sense of h1 (R × R), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm fh1(R×R) is given in terms of gj k L2(R2) and hk j L2(R2). By duality, this directly implies a lower bound on the norm of the commutator [b, H1 H2 ] in terms of bbmo(R×R). Our method bypasses the use of analyticity and the Fourier transform, and hence can be extended to the higher dimension case in an arbitrary n-parameter setting for the Riesz transforms.",

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N2 - In this paper, we provide a direct and constructive proof of weak factorization of h1 (R × R) (the predual of little BMO space bmo(R × R) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f ∈ h1 (R × R) there exist sequences {αk j } ∈ and functions gj k, hk j ∈ L2 (R2 ) such that ∞ ∞ f = αk j k j H1 H2 gj k − gj kH1 H2 hk k=1 j=1 in the sense of h1 (R × R), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm fh1(R×R) is given in terms of gj k L2(R2) and hk j L2(R2). By duality, this directly implies a lower bound on the norm of the commutator [b, H1 H2 ] in terms of bbmo(R×R). Our method bypasses the use of analyticity and the Fourier transform, and hence can be extended to the higher dimension case in an arbitrary n-parameter setting for the Riesz transforms.

AB - In this paper, we provide a direct and constructive proof of weak factorization of h1 (R × R) (the predual of little BMO space bmo(R × R) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f ∈ h1 (R × R) there exist sequences {αk j } ∈ and functions gj k, hk j ∈ L2 (R2 ) such that ∞ ∞ f = αk j k j H1 H2 gj k − gj kH1 H2 hk k=1 j=1 in the sense of h1 (R × R), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm fh1(R×R) is given in terms of gj k L2(R2) and hk j L2(R2). By duality, this directly implies a lower bound on the norm of the commutator [b, H1 H2 ] in terms of bbmo(R×R). Our method bypasses the use of analyticity and the Fourier transform, and hence can be extended to the higher dimension case in an arbitrary n-parameter setting for the Riesz transforms.

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