Remarques sur certains sous-espaces de BMO( n ) et de bmo( n )
[Remarks on some subspaces of BMO( n ) and of bmo( n )]
Annales de l'Institut Fourier, Volume 52 (2002) no. 4, pp. 1187-1218.

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( n ) and in its local version bmo( 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.

On décrit de diverses façons les fermetures respectives, dans l’espace BMO( n ) et dans sa version locale bmo( 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.

DOI: 10.5802/aif.1915
Classification: 46E30,  42B35
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)
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Bourdaud, Gérard. Remarques sur certains sous-espaces de $BMO ({\mathbb {R}}^n)$ et de $bmo({\mathbb {R}}^n)$. Annales de l'Institut Fourier, Volume 52 (2002) no. 4, pp. 1187-1218. doi : 10.5802/aif.1915. http://www.numdam.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 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

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