On the Product of Functions in <i>BMO</i> and <i>H</i>$^\text{1}$

Bonami, Aline; Iwaniec, Tadeusz; Jones, Peter; Zinsmeister, Michel

doi:10.5802/aif.2299

On the Product of Functions in BMO and H

^{1}

[Produits de fonctions de

H^{1}

B M O

]

Bonami, Aline ¹ ; Iwaniec, Tadeusz ² ; Jones, Peter ³ ; Zinsmeister, Michel ⁴

¹ Université d’Orléans MAPMO BP 6759 45067 Orléans cedex
² Syracuse University 215 Carnegie Hall Syracuse NY 13244-1150 (USA)
³ Yale University Mathematics Dept. PO Box 208 283 New Haven CT 06520-8283 (USA)
⁴ Université d’Orléans MAPMO BP 6759 45067 Orléans cedex et Ecole Polytechnique PMC 91128 Palaiseau

Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1405-1439.

Résumé
Abstract

Le produit d’une fonction à oscillation moyenne bornée avec une fonction de l’espace de Hardy $H^{1}$ n’est pas intégrable en général. Nous montrons toutefois qu’on peut lui donner un sens en tant que distribution tempérée, ceci grâce à la dualité $H^{1}$ , $B M O$ . Cette distribution peut de plus s’écrire comme la somme d’une fonction intégrable et d’une distribution appartenant à un espace de Hardy-Orlicz adapté. Lorsqu’on considère un tel produit pour les fonctions holomorphes du disque unité, cet énoncé possède une réciproque : toute fonction holomorphe de l’espace de Hardy-Orlicz considéré peut s’écrire comme un tel produit.

The point-wise product of a function of bounded mean oscillation with a function of the Hardy space $H^{1}$ is not locally integrable in general. However, in view of the duality between $H^{1}$ and $B M O$ , we are able to give a meaning to the product as a Schwartz distribution. Moreover, this distribution can be written as the sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. When dealing with holomorphic functions in the unit disc, the converse is also valid: every holomorphic of the corresponding Hardy-Orlicz space can be written as a product of a function in the holomorphic Hardy space $H^{1}$ and a holomorphic function with boundary values of bounded mean oscillation.

Zbl

DOI : 10.5802/aif.2299

Classification : 42B25, 42B30, 30H
Keywords: Hardy spaces, bounded mean oscillation, Jacobian lemma, Jacobian equation, Hardy-Orlicz spaces, div-curl lemma, factorization in Hardy spaces, weak Jacobian.
Mot clés : Espaces de Hardy, fonctions à oscillation moyenne bornée, lemme du Jacobien, équation du Jacobien, espaces de hardy-Orlicz, lemme div-curl, factorisation dans les classes de hardy, Jacobien faible.

Affiliations des auteurs :

Bonami, Aline ¹ ; Iwaniec, Tadeusz ² ; Jones, Peter ³ ; Zinsmeister, Michel ⁴

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     title = {On the {Product} of {Functions} in {\protect\emph{BMO}} and {\protect\emph{H}}$^\text{1}$},
     journal = {Annales de l'Institut Fourier},
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Bonami, Aline; Iwaniec, Tadeusz; Jones, Peter; Zinsmeister, Michel. On the Product of Functions in BMO and H$^\text{1}$. Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1405-1439. doi : 10.5802/aif.2299. https://www.numdam.org/articles/10.5802/aif.2299/

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Zhang, Pu; Wu, Jianglong; Sun, Jie Commutators of Some Maximal Functions with Lipschitz Function on Orlicz Spaces, Mediterranean Journal of Mathematics, Volume 15 (2018) no. 6 | DOI:10.1007/s00009-018-1263-0
Qi, Chunyan; Zhang, Hui; Li, Baode New real-variable characterizations of anisotropic weak Hardy spaces of Musielak-Orlicz type, Rocky Mountain Journal of Mathematics, Volume 48 (2018) no. 2 | DOI:10.1216/rmj-2018-48-2-607
Cao, Jun; Liu, Liguang; Yang, Dachun; Yuan, Wen Intrinsic Structures of Certain Musielak–Orlicz Hardy Spaces, The Journal of Geometric Analysis, Volume 28 (2018) no. 4, p. 2961 | DOI:10.1007/s12220-017-9943-8
Sawano, Yoshihiro Various Function Spaces, Theory of Besov Spaces, Volume 56 (2018), p. 709 | DOI:10.1007/978-981-13-0836-9_6
AǦCAYAZI, Müjdat; GOGATISHVILI, Amiran; MUSTAFAYEV, Rza Weak-type Estimates in Morrey Spaces for Maximal Commutator and Commutator of Maximal Function, Tokyo Journal of Mathematics, Volume 41 (2018) no. 1 | DOI:10.3836/tjm/1502179258
Bonami, Aline; Feuto, Justin; Grellier, Sandrine; Ky, Luong Dang Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms, Bulletin des Sciences Mathématiques, Volume 141 (2017) no. 7, p. 676 | DOI:10.1016/j.bulsci.2017.07.004
Zhang, Pu Characterization of Lipschitz spaces via commutators of the Hardy–Littlewood maximal function, Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, p. 336 | DOI:10.1016/j.crma.2017.01.022
Zhang, Hui; Qi, Chunyan; Li, Baode Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications, Frontiers of Mathematics in China, Volume 12 (2017) no. 4, p. 993 | DOI:10.1007/s11464-016-0546-7
Ponce, Augusto C.; Wilmet, Nicolas Schrödinger operators involving singular potentials and measure data, Journal of Differential Equations, Volume 263 (2017) no. 6, p. 3581 | DOI:10.1016/j.jde.2017.04.039
Fu, Xing; Yang, Dachun; Liang, Yiyu Products of Functions in $BMO (X)$ BMO ( X ) and $H_{at}^{1} (X)$ H at 1 ( X ) via Wavelets Over Spaces of Homogeneous Type, Journal of Fourier Analysis and Applications, Volume 23 (2017) no. 4, p. 919 | DOI:10.1007/s00041-016-9483-9
Fan, XingYa; He, JianXun; Li, BaoDe; Yang, DaChun Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces, Science China Mathematics, Volume 60 (2017) no. 11, p. 2093 | DOI:10.1007/s11425-016-9024-2
Fu, Xing; Yang, Dachun Products of Functions in $H_{ρ}^{1} (X)$ H ρ 1 ( X ) and ${BMO}_{ρ} (X)$ BMO ρ ( X ) over RD-Spaces and Applications to Schrödinger Operators, The Journal of Geometric Analysis, Volume 27 (2017) no. 4, p. 2938 | DOI:10.1007/s12220-017-9789-0
Li, Jinxia; Sun, Ruirui; Li, Baode Anisotropic interpolation theorems of Musielak-Orlicz type, Journal of Inequalities and Applications, Volume 2016 (2016) no. 1 | DOI:10.1186/s13660-016-1184-z
Liang, Yiyu; Yang, Dachun; Jiang, Renjin Weak Musielak–Orlicz Hardy spaces and applications, Mathematische Nachrichten, Volume 289 (2016) no. 5-6, p. 634 | DOI:10.1002/mana.201500152
Yang, Sibei Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators, Czechoslovak Mathematical Journal, Volume 65 (2015) no. 3, p. 747 | DOI:10.1007/s10587-015-0206-1
Yang, Dachun; Yang, Dongyong Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators, Frontiers of Mathematics in China, Volume 10 (2015) no. 5, p. 1203 | DOI:10.1007/s11464-015-0432-8
Betancor, Jorge J.; Castro, Alejandro J.; Rodríguez-Mesa, Lourdes UMD-Valued Square Functions Associated with Bessel Operators in Hardy and BMO Spaces, Integral Equations and Operator Theory, Volume 81 (2015) no. 3, p. 319 | DOI:10.1007/s00020-014-2202-5
Ky, Luong Dang On the product of functions in BMO and H1 over spaces of homogeneous type, Journal of Mathematical Analysis and Applications, Volume 425 (2015) no. 2, p. 807 | DOI:10.1016/j.jmaa.2014.12.057
Hoepfner, G.; Hounie, J.; Picon, T. Div–curl type estimates for elliptic systems of complex vector fields, Journal of Mathematical Analysis and Applications, Volume 429 (2015) no. 2, p. 774 | DOI:10.1016/j.jmaa.2015.04.054
AGCAYAZI, Mujdat; GOGATISHVILI, Amiran; KOCA, Kerim; MUSTAFAYEV, Rza A note on maximal commutators and commutators of maximal functions, Journal of the Mathematical Society of Japan, Volume 67 (2015) no. 2 | DOI:10.2969/jmsj/06720581
Hou, Shao Xiong; Yang, Da Chun; Yang, Si Bei Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity, Acta Mathematica Sinica, English Series, Volume 30 (2014) no. 11, p. 1917 | DOI:10.1007/s10114-014-3181-9
Cao, Jun; Chang, Der-Chen; Yang, Dachun; Yang, Sibei Estimates for second-order Riesz transforms associated with magnetic Schrödinger operators on Musielak-Orlicz-Hardy spaces, Applicable Analysis, Volume 93 (2014) no. 11, p. 2519 | DOI:10.1080/00036811.2014.918607
Cao, Jun; Chang, Der-Chen; Yang, Dachun; Yang, Sibei Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Communications on Pure and Applied Analysis, Volume 13 (2014) no. 4, p. 1435 | DOI:10.3934/cpaa.2014.13.1435
Bonami, Aline; Ky, Luong Dang Factorization of some Hardy-type spaces of holomorphic functions, Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, p. 817 | DOI:10.1016/j.crma.2014.09.004
Ky, Luong Dang New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators, Integral Equations and Operator Theory, Volume 78 (2014) no. 1, p. 115 | DOI:10.1007/s00020-013-2111-z
Yang, Dachun; Yang, Sibei Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications, Journal of Geometric Analysis, Volume 24 (2014) no. 1, p. 495 | DOI:10.1007/s12220-012-9344-y
Ky, Luong Dang Bilinear decompositions for the product space, Mathematische Nachrichten, Volume 287 (2014) no. 11-12, p. 1288 | DOI:10.1002/mana.201200101
Yang, Dachun; Yuan, Wen; Zhuo, Ciqiang Musielak–Orlicz Besov-type and Triebel–Lizorkin-type spaces, Revista Matemática Complutense, Volume 27 (2014) no. 1, p. 93 | DOI:10.1007/s13163-013-0120-8
Yang, Sibei SOME ESTIMATES FOR SCHRÖDINGER TYPE OPERATORS ON MUSIELAK-ORLICZ-HARDY SPACES, Taiwanese Journal of Mathematics, Volume 18 (2014) no. 4 | DOI:10.11650/tjm.18.2014.3897
Li, Baode; Yang, Dachun; Yuan, Wen Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators, The Scientific World Journal, Volume 2014 (2014), p. 1 | DOI:10.1155/2014/306214
Liang, Yiyu; Yang, Dachun Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces, Transactions of the American Mathematical Society, Volume 367 (2014) no. 5, p. 3225 | DOI:10.1090/s0002-9947-2014-06180-1
HOU, SHAOXIONG; YANG, DACHUN; YANG, SIBEI LUSIN AREA FUNCTION AND MOLECULAR CHARACTERIZATIONS OF MUSIELAK–ORLICZ HARDY SPACES AND THEIR APPLICATIONS, Communications in Contemporary Mathematics, Volume 15 (2013) no. 06, p. 1350029 | DOI:10.1142/s0219199713500296
Liang, Yiyu; Yang, Dachun Musielak–Orlicz Campanato spaces and applications, Journal of Mathematical Analysis and Applications, Volume 406 (2013) no. 1, p. 307 | DOI:10.1016/j.jmaa.2013.04.069
Bonami, Aline; Grellier, Sandrine; Ky, Luong Dang Paraproducts and products of functions in BMO(Rn) and H1(Rn) through wavelets, Journal de Mathématiques Pures et Appliquées, Volume 97 (2012) no. 3, p. 230 | DOI:10.1016/j.matpur.2011.06.002
Liang, Yiyu; Huang, Jizheng; Yang, Dachun New real-variable characterizations of Musielak–Orlicz Hardy spaces, Journal of Mathematical Analysis and Applications, Volume 395 (2012) no. 1, p. 413 | DOI:10.1016/j.jmaa.2012.05.049
Yang, DaChun; Yang, SiBei Local Hardy spaces of Musielak-Orlicz type and their applications, Science China Mathematics, Volume 55 (2012) no. 8, p. 1677 | DOI:10.1007/s11425-012-4377-z
Feuto, Justin Product of functions in BMO and H 1 in non-homogeneous spaces, Acta Mathematica Sinica, English Series, Volume 27 (2011) no. 8, p. 1535 | DOI:10.1007/s10114-011-9366-6
RUSS, EMMANUEL RACINES CARRÉES D'OPÉRATEURS ELLIPTIQUES ET ESPACES DE HARDY, Confluentes Mathematici, Volume 03 (2011) no. 01, p. 1 | DOI:10.1142/s1793744211000278
Bonami, Aline; Feuto, Justin Products of Functions in Hardy and Lipschitz or BMO Spaces, Recent Developments in Real and Harmonic Analysis (2010), p. 57 | DOI:10.1007/978-0-8176-4588-5_4
Li, Pengtao; Peng, Lizhong The decomposition of product space HL1×BMOL, Journal of Mathematical Analysis and Applications, Volume 349 (2009) no. 2, p. 484 | DOI:10.1016/j.jmaa.2008.05.024
Feuto, Justin Products of functions in BMO and H1 spaces on spaces of homogeneous type, Journal of Mathematical Analysis and Applications, Volume 359 (2009) no. 2, p. 610 | DOI:10.1016/j.jmaa.2009.06.022

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