## Remarks on some subspaces of BMO($${\mathbb R}^n$$) and of bmo($${\mathbb R}^n$$).(French)Zbl 1061.46025

On the $$n$$-dimensional Euclidean space, the author introduces the function space VMO (resp., CMO) as the BMO-closure of all infinitely differentiable functions of bounded mean oscillation (resp., of all infinitely differentiable functions with compact support). Various characterizations of these spaces are provided. For example, $$f\in \text{VMO}$$ iff $$f$$ is the limit in BMO of a sequence of uniformly continuous BMO-functions iff $$f=f_0 + R_1 f_1+\cdots+R_n f_n$$ where the $$f_j$$ are bounded uniformly continuous functions and the $$R_j$$ are the Riesz transformations iff $$\lim_{t\to 0}\sup_{| B| \leq t}| B| ^{-1}\int_B | f-f_B| =0$$ (the last condition is the usual definition of VMO; throughout, $$B$$ is a ball and $$f_B$$ is the average of $$f$$ over $$B$$), etc.
Next, $$f\in \text{CMO}$$ iff $$f$$ is in the BMO-closure of the set of continuous functions vanishing at infinity iff $$f$$ is in the intersection of VMO and the BMO-closure of the set of compactly supported continuous functions iff $$f=f_0+R_1 f_1 +\cdots+R_n f_n$$ with all $$f_j$$ continuous and vanishing at infinity, etc. The BMO-closure of the set of compactly supported BMO-functions is characterized in the same spirit. Also, a similar story is told about the closures of various natural function classes in the space bmo determined by the norm $\sup_{| B| <1} | B| ^{-1}\int_B | f-f_B| + \sup_{| B| =1}\int_B | f| .$ Various isolated facts of a similar nature can be found dispersed in the literature, but the paper under review seems to present the first systematic treatment of this range of problems. At the end, some mistakes of forerunners are indicated and the corresponding counterexamples are given.

### MSC:

 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B35 Function spaces arising in harmonic analysis

### Keywords:

bounded mean oscillation; continuous mean oscillation
Full Text:

### References:

 [1] Analysis of second order elliptic operators whitout boundary conditions and with VMO or Hölderian coefficients, Multiscale Wavelet Methods for PDEs, 495-539, (1997), Academic Press [2] Analyse fonctionnelle dans l’espace Euclidien, (1995), Pub. Math. Univ. Paris 7 · Zbl 0627.46048 [3] Functional calculus on BMO and related spaces, J. Funct. Anal., 189, 515-538, (2002) · Zbl 1007.47028 [4] The dual of Hardy spaces on a bounded domain in $$\mathbb R^n,$$ Forum Math, 6, 65-81, (1994) · Zbl 0803.42014 [5] Factorization theorems for Hardy spaces in several variables, Ann. of Math, 103, 611-635, (1976) · Zbl 0326.32011 [6] Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc, 83, 569-645, (1977) · Zbl 0358.30023 [7] $$H^p$$ spaces of several variables, Acta Math, 129, 137-193, (1972) · Zbl 0257.46078 [8] The distance in $$BMO$$ to $$L^∞,$$ Ann. of Math, 108, 373-393, (1978) · Zbl 0383.26010 [9] A local version of real Hardy space, Duke Math. J, 46, 27-42, (1979) · Zbl 0409.46060 [10] Riesz transforms and elliptic PDEs with $$VMO$$ coefficients, J. Anal. Math, 74, 183-212, (1998) · Zbl 0909.35039 [11] On functions with conditions on mean oscillation, Ark. Mat, 14, 189-196, (1976) · Zbl 0341.43005 [12] On functions of bounded mean oscillation, Comm. Pure Appl. Math, 14, 415-426, (1961) · Zbl 0102.04302 [13] Extension theorems for $$BMO,$$ Indiana Univ. Math. J, 29, 41-66, (1980) · Zbl 0432.42017 [14] Constructive decomposition of functions of finite central mean oscillation, Proc. Amer. Math. Soc, 127, 2375-2384, (1999) · Zbl 0922.42008 [15] Pseudo-differential operators with non-regular symbols, (1985) · Zbl 0695.47047 [16] Fractional integration on the space $$H^1$$ and its dual, Studia Math, 53, 175-189, (1975) · Zbl 0269.44012 [17] Analyse réelle et complexe, (1975), Masson, Paris · Zbl 0333.28001 [18] Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, (1996), De Gruyter · Zbl 0873.35001 [19] Functions of vanishing mean oscillation, Trans. Amer. Math. Soc, 207, 391-405, (1975) · Zbl 0319.42006 [20] Bounded Toeplitz operators on $$H^1$$ and applications of duality between $$H^1$$ and the functions of bounded mean oscillation, Amer. J. Math, 98, 573-589, (1976) · Zbl 0335.47018 [21] Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, (1993), Princeton University Press, Princeton · Zbl 0821.42001 [22] Real-Variable Methods in Harmonic Analysis, (1986), Academic Press · Zbl 0621.42001 [23] On the compactness of operators of Hankel type, Tôhoku Math. J, 30, 163-171, (1978) · Zbl 0384.47023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.