## Functional calculus on $$BMO$$ and related spaces.(English)Zbl 1007.47028

It is considered a nonlinear autonomous superposition operator ${\mathbf T}_f [g] = f\circ g$ on a certain subspace $$X$$ of the space $$BMO({\mathbb R}^n)$$. The question is to characterise those Borel measurable functions $$f$$ for which the operator $${\mathbf T}_f$$ is continuous or differentiable on $$X$$. The following subspaces $$X$$ are considered: $$BMO({\mathbb R}^n)$$, $$VMO({\mathbb R}^n)$$ and $$CMO({\mathbb R}^n)$$ (closure in $$BMO$$-norm of $$C_c^{\infty}({\mathbb R}^n)$$ as well as their respective inhomogeneous counterparts. Namely, $$X = bmo({\mathbb R}^n)$$ (a linear subspace of those functions having finite mean on the unit cube in $${\mathbb R}^n$$), $$X = cmo({\mathbb R}^n)$$ (closure in $$bmo$$-norm of $$C_c^{\infty}({\mathbb R}^n)$$), $$X = vmo({\mathbb R}^n) = VMO({\mathbb R}^n) \cap bmo({\mathbb R}^n)$$. The results by Fominykh-Chevalier, Brezis-Nirenberg and Marcus-Mizel are extended.

### MSC:

 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 47A60 Functional calculus for linear operators
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### References:

 [1] Bourdaud, G., Fonctions qui opèrent sur LES espaces de Besov et de Triebel, Ann. inst. H. Poincaré anal. non linéaire, 10, 413-422, (1993) · Zbl 0741.46010 [2] Brezis, H.; Nirenberg, L., Degree theory and BMO. part I. compact manifolds without boundaries, Selecta math. (N.S.), 1, 197-263, (1995) · Zbl 0852.58010 [3] Chevalier, L., Quelles sont LES fonctions qui opèrent de BMO dans BMO ou de BMO dans L∞?, Bull. London math. soc., 27, 590-594, (1995) · Zbl 0843.42009 [4] Coifman, R.; Weiss, G., Extension of Hardy spaces and their use in analysis, Bull. amer. math. soc., 83, 569-645, (1977) · Zbl 0358.30023 [5] DeVore, R.A.; Lorentz, G.G., Constructive approximation, (1993), Springer-Verlag Berlin · Zbl 0797.41016 [6] Fominykh, M.A., Transformation of BMO functions, Vestnik moskov. univ. ser. I mat. mekh., 94, 20-24, (1985) · Zbl 0653.42011 [7] Janson, S., On functions with conditions on Mean oscillation, Ark. mat., 14, 189-196, (1976) · Zbl 0341.43005 [8] Katznelson, Y., An introduction to harmonic analysis, (1976), Dover New York · Zbl 0169.17902 [9] Marcus, M.; Mizel, V.J., Every superposition operator mapping one Sobolev space into another is continuous, J. funct. anal., 33, 217-229, (1979) · Zbl 0418.46024 [10] Sarason, D., Functions of vanishing Mean oscillation, Trans. amer. math. soc., 207, 391-405, (1975) · Zbl 0319.42006 [11] Stegenga, D.A., Bounded Toeplitz operators on H1 and applications of duality between H1 and the functions of bounded Mean oscillation, Amer. J. math., 98, 573-589, (1976) · Zbl 0335.47018 [12] Stein, E.M., Harmonic analysis, (1993), Princeton Univ. Press Princeton
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