no. 4
Remarques sur certains sous-espaces de BMO(ℝ n ) et de bmo(ℝ n )
Bourdaud, Gérard1
1 Université Paris VII, UFR de Mathématiques, 2 place Jussieu, 75251 Paris Cedex 05 (France)
Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1187-1218.

On décrit de diverses façons les fermetures respectives, dans l’espace BMO(ℝ n ) et dans sa version locale bmo(ℝ n ), de l’ensemble des fonctions à support compact et de l’ensemble des fonctions C ∞ 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.

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 BMO(ℝ n ) and in its local version bmo(ℝ 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.

DOI : 10.5802/aif.1915
Classification : 46E30, 42B35
Mot clés : oscillations moyennes bornées, oscillations moyennes continues
Keywords: bounded mean oscillations, vanishing mean oscillations, continuous mean oscillations

Bourdaud, Gérard 1

1 Université Paris VII, UFR de Mathématiques, 2 place Jussieu, 75251 Paris Cedex 05 (France) 
@article{AIF_2002__52_4_1187_0,
     author = {Bourdaud, G\'erard},
     title = {Remarques sur certains sous-espaces de $BMO ({\mathbb {R}}^n)$ et de $bmo({\mathbb {R}}^n)$},
     journal = {Annales de l'Institut Fourier},
     pages = {1187--1218},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {4},
     year = {2002},
     doi = {10.5802/aif.1915},
     zbl = {1061.46025},
     mrnumber = {1927078},
     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1915/}
}
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
SP  - 1187
EP  - 1218
VL  - 52
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1915/
DO  - 10.5802/aif.1915
LA  - fr
ID  - AIF_2002__52_4_1187_0
ER  - 
%0 Journal Article
%A Bourdaud, Gérard
%T Remarques sur certains sous-espaces de $BMO ({\mathbb {R}}^n)$ et de $bmo({\mathbb {R}}^n)$
%J Annales de l'Institut Fourier
%D 2002
%P 1187-1218
%V 52
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1915/
%R 10.5802/aif.1915
%G fr
%F AIF_2002__52_4_1187_0
Bourdaud, Gérard. Remarques sur certains sous-espaces de $BMO ({\mathbb {R}}^n)$ et de $bmo({\mathbb {R}}^n)$. Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1187-1218. doi : 10.5802/aif.1915. https://aif.centre-mersenne.org/articles/10.5802/aif.1915/

[1] J.M. Angeletti; S. Mazet; Ph. Tchamitchian; W. Dahmen, A.J. Kurdila Analysis of second order elliptic operators whitout boundary conditions and with VMO or Hölderian coefficients, Multiscale Wavelet Methods for PDEs (1997), pp. 495-539

[2] G. Bourdaud Analyse fonctionnelle dans l'espace Euclidien, Pub. Math. Univ. Paris 7, 1995 | Zbl

[3] G. Bourdaud; M. Lanza; de Cristoforis; W. Sickel Functional calculus on BMO and related spaces, J. Funct. Anal., Volume 189 (2002), pp. 515-538 | DOI | MR | Zbl

[4] D.C. Chang The dual of Hardy spaces on a bounded domain in ℝ n , Forum Math, Volume 6 (1994), pp. 65-81 | DOI | MR | Zbl

[5] R. Coifman; R. Rochberg; G. Weiss Factorization theorems for Hardy spaces in several variables, Ann. of Math, Volume 103 (1976), pp. 611-635 | DOI | MR | Zbl

[6] R. Coifman; G. Weiss Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc, Volume 83 (1977), pp. 569-645 | DOI | MR | Zbl

[7] C. Fefferman; E.M. Stein H p spaces of several variables, Acta Math, Volume 129 (1972), pp. 137-193 | DOI | MR | Zbl

[8] J.B. Garnett; P.W. Jones The distance in BMO to L ∞ , Ann. of Math, Volume 108 (1978), pp. 373-393 | DOI | MR | Zbl

[9] D. Goldberg A local version of real Hardy space, Duke Math. J, Volume 46 (1979), pp. 27-42 | DOI | MR | Zbl

[10] T. Iwaniec; C. Sbordone Riesz transforms and elliptic PDEs with VMO coefficients, J. Anal. Math, Volume 74 (1998), pp. 183-212 | DOI | MR | Zbl

[11] S. Janson On functions with conditions on mean oscillation, Ark. Mat, Volume 14 (1976), pp. 189-196 | DOI | MR | Zbl

[12] F. John; L. Nirenberg On functions of bounded mean oscillation, Comm. Pure Appl. Math, Volume 14 (1961), pp. 415-426 | DOI | MR | Zbl

[13] P.W. Jones Extension theorems for BMO, Indiana Univ. Math. J, Volume 29 (1980), pp. 41-66 | DOI | MR | Zbl

[14] J.D. Lakey Constructive decomposition of functions of finite central mean oscillation, Proc. Amer. Math. Soc, Volume 127 (1999), pp. 2375-2384 | DOI | MR | Zbl

[15] J. Marschall Pseudo-differential operators with non-regular symbols (1985) (Thèse FU Berlin) | Zbl

[16] U. Neri Fractional integration on the space H 1 and its dual, Studia Math, Volume 53 (1975), pp. 175-189 | MR | Zbl

[17] W. Rudin Analyse réelle et complexe, Masson, Paris, 1975 | MR | Zbl

[18] T. Runst; W. Sickel Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter, 1996 | MR | Zbl

[19] D. Sarason Functions of vanishing mean oscillation, Trans. Amer. Math. Soc, Volume 207 (1975), pp. 391-405 | DOI | MR | Zbl

[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, Volume 98 (1976), pp. 573-589 | DOI | MR | Zbl

[21] E.M. Stein Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, 1993 | MR | Zbl

[22] A. Torchinsky Real-Variable Methods in Harmonic Analysis, Academic Press, 1986 | MR | Zbl

[23] A. Uchiyama On the compactness of operators of Hankel type, Tôhoku Math. J, Volume 30 (1978), pp. 163-171 | DOI | MR | Zbl

Cité par Sources :