Commutators, Little BMO and Weak Factorization
[Commutateurs, Little BMO et Factorisation Faible]
Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 109-129.

Dans ce papier, nous donnons une preuve directe et constructive de la factorisation faible de h 1 (×) (le prédual de l’espace little BMO bmo(×) étudié par Cotlar–Sadosky et Ferguson–Sadosky), i.e., pour chaque fh 1 (×) il existe des suites {α j k } 1 et des fonctions g j k ,h j k L 2 ( 2 ) telles que

f=k=1j=1αjkhjkH1H2gjk-gjkH1H2hjk

au sens de h 1 (×), où H 1 et H 2 sont les transformées de Hilbert dans la première et la seconde variable, respectivement. De plus, la norme f h 1 (×) est donnée en termes de g j k L 2 ( 2 ) et h j k L 2 ( 2 ) . Par dualité, ceci implique directement une borne inférieure de la norme du commutateur [b,H 1 H 2 ] en termes de b bmo(×) .

Notre méthode contourne l’utilisation de l’analyticité et de la transformée de Fourier, et peut donc être étendue en dimension supérieure dans le cadre de n-paramètres arbitraires, pour les transformées de Riesz.

In this paper, we provide a direct and constructive proof of weak factorization of h 1 (×) (the predual of little BMO space bmo(×) studied by Cotlar–Sadosky and Ferguson–Sadosky), i.e., for every fh 1 (×) there exist sequences {α j k } 1 and functions g j k ,h j k L 2 ( 2 ) such that

f=k=1j=1αjkhjkH1H2gjk-gjkH1H2hjk

in the sense of h 1 (), where H 1 and H 2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm f h 1 (×) is given in terms of g j k L 2 ( 2 ) and h j k L 2 ( 2 ) . By duality, this directly implies a lower bound on the norm of the commutator [b,H 1 H 2 ] in terms of b bmo(×) .

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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DOI : 10.5802/aif.3153
Classification : 42B30, 42B20, 42B35
Keywords: $\protect \operatorname{bmo}(\protect \mathbb{R}\times \protect \mathbb{R})$, $h^1(\protect \mathbb{R}\times \protect \mathbb{R})$, commutator, weak factorization, Hilbert transform
Mot clés : $\protect \operatorname{bmo}(\protect \mathbb{R}\times \protect \mathbb{R})$, $h^1(\protect \mathbb{R}\times \protect \mathbb{R})$, commutateur, factorisation faible, transformée de Hilbert
Duong, Xuan Thinh 1 ; Li, Ji 1 ; Wick, Brett D. 2 ; Yang, Dongyong 3

1 Macquarie University Department of Mathematics NSW, 2109 (Australia)
2 Department of Mathematics Washington University – St. Louis One Brookings Drive St. Louis, MO 63130 (USA)
3 School of Mathematical Sciences Xiamen University Xiamen 361005 (China)
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Duong, Xuan Thinh; Li, Ji; Wick, Brett D.; Yang, Dongyong. Commutators, Little BMO and Weak Factorization. Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 109-129. doi : 10.5802/aif.3153. http://www.numdam.org/articles/10.5802/aif.3153/

[1] Astala, Kari; Iwaniec, Tadeusz; Saksman, Eero Beltrami operators in the plane, Duke Math. J., Volume 107 (2001) no. 1, pp. 27-56 | DOI | MR | Zbl

[2] Coifman, Ronald Raphaël; Lions, Pierre-Louis; Meyer, Yves F.; Semmes, Stephen W. Compensated compactness and Hardy spaces, J. Math. Pures Appl., Volume 72 (1993) no. 3, pp. 247-286 | MR | Zbl

[3] Coifman, Ronald Raphaël; Rochberg, Richard; Weiss, Guido Factorization theorems for Hardy spaces in several variables, Ann. Math., Volume 103 (1976) no. 3, pp. 611-635 | DOI | MR | Zbl

[4] Coifman, Ronald Raphaël; Weiss, Guido Extensions of Hardy spaces and their use in analysis, Bull. Am. Math. Soc., Volume 83 (1977) no. 4, pp. 569-645 | DOI | MR | Zbl

[5] Cotlar, Mischa; Sadosky, Cora Two distinguished subspaces of product BMO and Nehari-AAK theory for Hankel operators on the torus, Integral Equations Oper. Theory, Volume 26 (1996) no. 3, pp. 273-304 | DOI | MR | Zbl

[6] Duong, Xuan Thinh; Li, Ji; Wick, Brett D.; Yang, Dongyong Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting, Indiana Univ. Math. J., Volume 66 (2017) no. 4, pp. 1081-1106 | DOI | Zbl

[7] Fefferman, Charles Louis; Stein, Elias M. H p spaces of several variables, Acta Math., Volume 129 (1972) no. 3-4, pp. 137-193 | DOI | MR | Zbl

[8] Ferguson, Sarah H.; Lacey, Michael T. A characterization of product BMO by commutators, Acta Math., Volume 189 (2002) no. 2, pp. 143-160 | DOI | MR | Zbl

[9] Ferguson, Sarah H.; Sadosky, Cora Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures, J. Anal. Math., Volume 81 (2000), pp. 239-267 | DOI | MR | Zbl

[10] Grafakos, Loukas Classical Fourier analysis, Graduate Texts in Mathematics, 249, Springer, 2008, xvi+489 pages | MR | Zbl

[11] Journé, Jean-Lin Calderón-Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón, Lecture Notes in Mathematics, 994, Springer, 1983, vi+128 pages | DOI | MR | Zbl

[12] Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Am. Math. Soc., Volume 4 (1991) no. 2, pp. 323-347 | DOI | MR | Zbl

[13] Lacey, Michael T. Lectures on Nehari’s theorem on the polydisk, Topics in harmonic analysis and ergodic theory (Contemporary Mathematics), Volume 444, American Mathematical Society, 2007, pp. 185-213 | DOI | MR | Zbl

[14] Lacey, Michael T.; Petermichl, Stefanie; Pipher, Jill C.; Wick, Brett D. Multiparameter Riesz commutators, Am. J. Math., Volume 131 (2009) no. 3, pp. 731-769 | DOI | MR | Zbl

[15] Nehari, Zeev On bounded bilinear forms, Ann. Math., Volume 65 (1957), pp. 153-162 | DOI | MR | Zbl

[16] Ou, Yumeng; Petermichl, Stefanie; Strouse, Elizabeth Higher order Journé commutators and characterizations of multi-parameter BMO, Adv. Math., Volume 291 (2016), pp. 24-58 | DOI | MR | Zbl

[17] Uchiyama, Akihito The factorization of H p on the space of homogeneous type, Pac. J. Math., Volume 92 (1981) no. 2, pp. 453-468 http://projecteuclid.org/euclid.pjm/1102736804 | DOI | MR | Zbl

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