On the Product of Functions in BMO and H 1

Aline Bonami[1]; Tadeusz Iwaniec[2]; Peter Jones[3]; Michel Zinsmeister[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 (2007)

  • Volume: 57, Issue: 5, page 1405-1439
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Bonami, Aline, et al. "On the Product of Functions in BMO and H$^\text{1}$." Annales de l’institut Fourier 57.5 (2007): 1405-1439. <http://eudml.org/doc/10263>.

@article{Bonami2007,
abstract = {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 $BMO$, 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.},
affiliation = {Université d’Orléans MAPMO BP 6759 45067 Orléans cedex; Syracuse University 215 Carnegie Hall Syracuse NY 13244-1150 (USA); Yale University Mathematics Dept. PO Box 208 283 New Haven CT 06520-8283 (USA); Université d’Orléans MAPMO BP 6759 45067 Orléans cedex et Ecole Polytechnique PMC 91128 Palaiseau},
author = {Bonami, Aline, Iwaniec, Tadeusz, Jones, Peter, Zinsmeister, Michel},
journal = {Annales de l’institut Fourier},
keywords = {Hardy spaces; bounded mean oscillation; Jacobian lemma; Jacobian equation; Hardy-Orlicz spaces; div-curl lemma; factorization in Hardy spaces; weak Jacobian; Hardy-Orlizc spaces},
language = {eng},
number = {5},
pages = {1405-1439},
publisher = {Association des Annales de l’institut Fourier},
title = {On the Product of Functions in BMO and H$^\text\{1\}$},
url = {http://eudml.org/doc/10263},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Bonami, Aline
AU - Iwaniec, Tadeusz
AU - Jones, Peter
AU - Zinsmeister, Michel
TI - On the Product of Functions in BMO and H$^\text{1}$
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 5
SP - 1405
EP - 1439
AB - 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 $BMO$, 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.
LA - eng
KW - Hardy spaces; bounded mean oscillation; Jacobian lemma; Jacobian equation; Hardy-Orlicz spaces; div-curl lemma; factorization in Hardy spaces; weak Jacobian; Hardy-Orlizc spaces
UR - http://eudml.org/doc/10263
ER -

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