Remarques sur certains sous-espaces de $BMO ({\mathbb {R}}^n)$ et de $bmo({\mathbb {R}}^n)$

Bourdaud, Gérard

doi:10.5802/aif.1915

Remarques sur certains sous-espaces de

B M O (ℝ^{n})

et de

b m o (ℝ^{n})

Bourdaud, Gérard ¹

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.

Résumé
Abstract

On décrit de diverses façons les fermetures respectives, dans l’espace $B M O (ℝ^{n})$ et dans sa version locale $b m o (ℝ^{n})$ , 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.

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 $B M O (ℝ^{n})$ and in its local version $b m o (ℝ^{n})$ , 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.

MR Zbl

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

Affiliations des auteurs :

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, Tome 52 (2002) no. 4, pp. 1187-1218. doi : 10.5802/aif.1915. https://www.numdam.org/articles/10.5802/aif.1915/

Bibliographie
Cité par

[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 $B M O$ to $L^{\infty}$ , 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 $V M O$ 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 $B M O$ , 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

Bényi, Árpád; Torres, Rodolfo H. An Update on the Compactness of Bilinear Commutators, The Mathematical Heritage of Guido Weiss (2025), p. 83 | DOI:10.1007/978-3-031-76793-7_4
Guo, Qing Dong; Betancor, Jorge J.; Yang, Dong Yong Characterizations of VMO and CMO Spaces in the Bessel Setting, Acta Mathematica Sinica, English Series, Volume 40 (2024) no. 12, p. 3055 | DOI:10.1007/s10114-024-3342-4
Burchard, Almut; Dafni, Galia; Gibara, Ryan Vanishing mean oscillation and continuity of rearrangements, Advances in Mathematics, Volume 437 (2024), p. 109379 | DOI:10.1016/j.aim.2023.109379
Mair, Adam; Moen, Kabe Two weight bump conditions for compactness of commutators, Archiv der Mathematik, Volume 120 (2023) no. 1, p. 47 | DOI:10.1007/s00013-022-01783-5
Wu, Suqing The Superposition Operator and Strauss Lemma in Logarithmic Besov Spaces, Potential Analysis, Volume 59 (2023) no. 2, p. 435 | DOI:10.1007/s11118-021-09974-5
Marín, Juan; Martell, José; Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius Geometric Measure Theory, Singular Integral Operators, Quantitative Flatness, and Boundary Problems, Volume 344 (2022), p. 27 | DOI:10.1007/978-3-031-08234-4_2
Tao, Jin; Yang, Dachun; Yuan, Wen A Survey on Function Spaces of John–Nirenberg Type, Mathematics, Volume 9 (2021) no. 18, p. 2264 | DOI:10.3390/math9182264
Butaev, Almaz; Dafni, Galia Approximation and Extension of Functions of Vanishing Mean Oscillation, The Journal of Geometric Analysis, Volume 31 (2021) no. 7, p. 6892 | DOI:10.1007/s12220-020-00526-8
LIU, Liguang; YANG, Dachun; YUAN, Wen Functional Calculus on BMO-type Spaces of Bourgain, Brezis and Mironescu, Tokyo Journal of Mathematics, Volume 44 (2021) no. 2 | DOI:10.3836/tjm/1502179329
Martell, José María; Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO, Analysis PDE, Volume 12 (2019) no. 3, p. 605 | DOI:10.2140/apde.2019.12.605
Butaev, Almaz On some refinements of the embedding of critical Sobolev spaces into BMO, Pacific Journal of Mathematics, Volume 298 (2019) no. 1, p. 1 | DOI:10.2140/pjm.2019.298.1
Betancor, Jorge J.; Castro, Alejandro J.; Fariña, Juan C.; Rodríguez-Mesa, L. Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions, Journal of Mathematical Analysis and Applications, Volume 431 (2015) no. 1, p. 440 | DOI:10.1016/j.jmaa.2015.05.069
Yuan, Wen; Sickel, Winfried; Yang, DaChun Interpolation of Morrey-Campanato and related smoothness spaces, Science China Mathematics, Volume 58 (2015) no. 9, p. 1835 | DOI:10.1007/s11425-015-5047-8
Liang, Yiyu; Yang, Dachun; Yuan, Wen Vanishing mean oscillation spaces associated with operators satisfying Davies–Gaffney estimates, Kyoto Journal of Mathematics, Volume 52 (2012) no. 2 | DOI:10.1215/21562261-1550958
RUSS, EMMANUEL RACINES CARRÉES D'OPÉRATEURS ELLIPTIQUES ET ESPACES DE HARDY, Confluentes Mathematici, Volume 03 (2011) no. 01, p. 1 | DOI:10.1142/s1793744211000278
Song, Liang; Xu, Ming VMO spaces associated with divergence form elliptic operators, Mathematische Zeitschrift, Volume 269 (2011) no. 3-4, p. 927 | DOI:10.1007/s00209-010-0774-6
Jiang, Renjin; Yang, Dachun Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators, Integral Equations and Operator Theory, Volume 67 (2010) no. 1, p. 123 | DOI:10.1007/s00020-010-1774-y
Deng, Donggao; Duong, Xuan Thinh; Song, Liang; Tan, Chaoqiang; Yan, Lixin Functions of vanishing mean oscillation associated with operators and applications, Michigan Mathematical Journal, Volume 56 (2008) no. 3 | DOI:10.1307/mmj/1231770358
Marchand, Fabien Weak–strong uniqueness criteria for the critical quasi-geostrophic equation, Physica D: Nonlinear Phenomena, Volume 237 (2008) no. 10-12, p. 1346 | DOI:10.1016/j.physd.2008.03.011
Auscher, Pascal; Russ, Emmanuel Hardy spaces and divergence operators on strongly Lipschitz domains of Rn, Journal of Functional Analysis, Volume 201 (2003) no. 1, p. 148 | DOI:10.1016/s0022-1236(03)00059-4

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