Remarks on some subspaces of B M O ( n ) and of b m o ( n )

Gérard Bourdaud[1]

  • [1] Université Paris VII, UFR de Mathématiques, 2 place Jussieu, 75251 Paris Cedex 05 (France)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 4, page 1187-1218
  • ISSN: 0373-0956

Abstract

top
We present various characterizations of the closure of the set of functions with compact support and of the set of infinitely differentiable functions with compact support in the space B M O ( n ) and in its local version b m o ( n ) , respectively. Some of these results are novel, some others are considered as classical, although an explicit proof does not seem to have been published. By means of counterexamples, we show the differences among the various subspaces we have considered.

How to cite

top

Bourdaud, Gérard. "Remarques sur certains sous-espaces de $BMO ({\mathbb {R}}^n)$ et de $bmo({\mathbb {R}}^n)$." Annales de l’institut Fourier 52.4 (2002): 1187-1218. <http://eudml.org/doc/116007>.

@article{Bourdaud2002,
abstract = {On décrit de diverses façons les fermetures respectives, dans l’espace $BMO(\{\mathbb \{R\}\}^n)$ et dans sa version locale $bmo(\{\mathbb \{R\}\}^n)$, de l’ensemble des fonctions à support compact et de l’ensemble des fonctions $C^\infty $ à support compact. Certains de ces résultats sont nouveaux; d’autres, considérés comme classiques, ne semblent pas avoir fait l’objet de publication. Des contre-exemples permettent de vérifier la diversité des sous-espaces considérés.},
affiliation = {Université Paris VII, UFR de Mathématiques, 2 place Jussieu, 75251 Paris Cedex 05 (France)},
author = {Bourdaud, Gérard},
journal = {Annales de l’institut Fourier},
keywords = {bounded mean oscillations; vanishing mean oscillations; continuous mean oscillations},
language = {fre},
number = {4},
pages = {1187-1218},
publisher = {Association des Annales de l'Institut Fourier},
title = {Remarques sur certains sous-espaces de $BMO (\{\mathbb \{R\}\}^n)$ et de $bmo(\{\mathbb \{R\}\}^n)$},
url = {http://eudml.org/doc/116007},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Bourdaud, Gérard
TI - Remarques sur certains sous-espaces de $BMO ({\mathbb {R}}^n)$ et de $bmo({\mathbb {R}}^n)$
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 4
SP - 1187
EP - 1218
AB - On décrit de diverses façons les fermetures respectives, dans l’espace $BMO({\mathbb {R}}^n)$ et dans sa version locale $bmo({\mathbb {R}}^n)$, de l’ensemble des fonctions à support compact et de l’ensemble des fonctions $C^\infty $ à support compact. Certains de ces résultats sont nouveaux; d’autres, considérés comme classiques, ne semblent pas avoir fait l’objet de publication. Des contre-exemples permettent de vérifier la diversité des sous-espaces considérés.
LA - fre
KW - bounded mean oscillations; vanishing mean oscillations; continuous mean oscillations
UR - http://eudml.org/doc/116007
ER -

References

top
  1. J.M. Angeletti, S. Mazet, Ph. Tchamitchian, Analysis of second order elliptic operators whitout boundary conditions and with VMO or Hölderian coefficients, Multiscale Wavelet Methods for PDEs (1997), 495-539, Academic Press 
  2. G. Bourdaud, Analyse fonctionnelle dans l'espace Euclidien, (1995), Pub. Math. Univ. Paris 7 Zbl0627.46048
  3. G. Bourdaud, M. Lanza, de Cristoforis, W. Sickel, Functional calculus on BMO and related spaces, J. Funct. Anal. 189 (2002), 515-538 Zbl1007.47028MR1892179
  4. D.C. Chang, The dual of Hardy spaces on a bounded domain in n , Forum Math 6 (1994), 65-81 Zbl0803.42014MR1253178
  5. R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math 103 (1976), 611-635 Zbl0326.32011MR412721
  6. R. Coifman, G. Weiss, Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc 83 (1977), 569-645 Zbl0358.30023MR447954
  7. C. Fefferman, E.M. Stein, H p spaces of several variables, Acta Math 129 (1972), 137-193 Zbl0257.46078MR447953
  8. J.B. Garnett, P.W. Jones, The distance in B M O to L , Ann. of Math 108 (1978), 373-393 Zbl0383.26010MR506992
  9. D. Goldberg, A local version of real Hardy space, Duke Math. J 46 (1979), 27-42 Zbl0409.46060MR523600
  10. T. Iwaniec, C. Sbordone, Riesz transforms and elliptic PDEs with V M O coefficients, J. Anal. Math 74 (1998), 183-212 Zbl0909.35039MR1631658
  11. S. Janson, On functions with conditions on mean oscillation, Ark. Mat 14 (1976), 189-196 Zbl0341.43005MR438030
  12. F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math 14 (1961), 415-426 Zbl0102.04302MR131498
  13. P.W. Jones, Extension theorems for B M O , Indiana Univ. Math. J 29 (1980), 41-66 Zbl0432.42017MR554817
  14. J.D. Lakey, Constructive decomposition of functions of finite central mean oscillation, Proc. Amer. Math. Soc 127 (1999), 2375-2384 Zbl0922.42008MR1486741
  15. J. Marschall, Pseudo-differential operators with non-regular symbols, (1985) Zbl0695.47047
  16. U. Neri, Fractional integration on the space H 1 and its dual, Studia Math 53 (1975), 175-189 Zbl0269.44012MR388074
  17. W. Rudin, Analyse réelle et complexe, (1975), Masson, Paris Zbl0333.28001MR662565
  18. T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, (1996), De Gruyter Zbl0873.35001MR1419319
  19. D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc 207 (1975), 391-405 Zbl0319.42006MR377518
  20. D.A. Stegenga, Bounded Toeplitz operators on H 1 and applications of duality between H 1 and the functions of bounded mean oscillation, Amer. J. Math 98 (1976), 573-589 Zbl0335.47018MR420326
  21. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, (1993), Princeton University Press, Princeton Zbl0821.42001MR1232192
  22. A. Torchinsky, Real-Variable Methods in Harmonic Analysis, (1986), Academic Press Zbl0621.42001MR869816
  23. A. Uchiyama, On the compactness of operators of Hankel type, Tôhoku Math. J 30 (1978), 163-171 Zbl0384.47023MR467384

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.