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, p. 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 : https://doi.org/10.5802/aif.1915
Classification:  46E30,  42B35
Keywords: bounded mean oscillations, vanishing mean oscillations, continuous mean oscillations
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {4},
     year = {2002},
     pages = {1187-1218},
     doi = {10.5802/aif.1915},
     mrnumber = {1927078},
     zbl = {1061.46025},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_2002__52_4_1187_0}
}
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. https://aif.centre-mersenne.org/item/AIF_2002__52_4_1187_0/

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

[2] G. Bourdaud Analyse fonctionnelle dans l'espace Euclidien, Pub. Math. Univ. Paris 7 (1995) | Zbl 0627.46048

[3] G. Bourdaud; M. Lanza; De Cristoforis; W. Sickel Functional calculus on BMO and related spaces, J. Funct. Anal., Tome 189 (2002), pp. 515-538 | Article | MR 1892179 | Zbl 1007.47028

[4] D.C. Chang The dual of Hardy spaces on a bounded domain in n , Forum Math, Tome 6 (1994), pp. 65-81 | Article | MR 1253178 | Zbl 0803.42014

[5] R. Coifman; R. Rochberg; G. Weiss Factorization theorems for Hardy spaces in several variables, Ann. of Math, Tome 103 (1976), pp. 611-635 | Article | MR 412721 | Zbl 0326.32011

[6] R. Coifman; G. Weiss Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc, Tome 83 (1977), pp. 569-645 | Article | MR 447954 | Zbl 0358.30023

[7] C. Fefferman; E.M. Stein H p spaces of several variables, Acta Math, Tome 129 (1972), pp. 137-193 | Article | MR 447953 | Zbl 0257.46078

[8] J.B. Garnett; P.W. Jones The distance in BMO to L , Ann. of Math, Tome 108 (1978), pp. 373-393 | Article | MR 506992 | Zbl 0383.26010

[9] D. Goldberg A local version of real Hardy space, Duke Math. J, Tome 46 (1979), pp. 27-42 | Article | MR 523600 | Zbl 0409.46060

[10] T. Iwaniec; C. Sbordone Riesz transforms and elliptic PDEs with VMO coefficients, J. Anal. Math, Tome 74 (1998), pp. 183-212 | Article | MR 1631658 | Zbl 0909.35039

[11] S. Janson On functions with conditions on mean oscillation, Ark. Mat, Tome 14 (1976), pp. 189-196 | Article | MR 438030 | Zbl 0341.43005

[12] F. John; L. Nirenberg On functions of bounded mean oscillation, Comm. Pure Appl. Math, Tome 14 (1961), pp. 415-426 | Article | MR 131498 | Zbl 0102.04302

[13] P.W. Jones Extension theorems for BMO, Indiana Univ. Math. J, Tome 29 (1980), pp. 41-66 | Article | MR 554817 | Zbl 0432.42017

[14] J.D. Lakey Constructive decomposition of functions of finite central mean oscillation, Proc. Amer. Math. Soc, Tome 127 (1999), pp. 2375-2384 | Article | MR 1486741 | Zbl 0922.42008

[15] J. Marschall Pseudo-differential operators with non-regular symbols (1985) (Thèse FU Berlin) | Zbl 0695.47047

[16] U. Neri Fractional integration on the space H 1 and its dual, Studia Math, Tome 53 (1975), pp. 175-189 | MR 388074 | Zbl 0269.44012

[17] W. Rudin Analyse réelle et complexe, Masson, Paris (1975) | MR 662565 | Zbl 0333.28001

[18] T. Runst; W. Sickel Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter (1996) | MR 1419319 | Zbl 0873.35001

[19] D. Sarason Functions of vanishing mean oscillation, Trans. Amer. Math. Soc, Tome 207 (1975), pp. 391-405 | Article | MR 377518 | Zbl 0319.42006

[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, Tome 98 (1976), pp. 573-589 | Article | MR 420326 | Zbl 0335.47018

[21] E.M. Stein Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton (1993) | MR 1232192 | Zbl 0821.42001

[22] A. Torchinsky Real-Variable Methods in Harmonic Analysis, Academic Press (1986) | MR 869816 | Zbl 0621.42001

[23] A. Uchiyama On the compactness of operators of Hankel type, Tôhoku Math. J, Tome 30 (1978), pp. 163-171 | Article | MR 467384 | Zbl 0384.47023