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1997 | 123 | 3 | 249-274
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Espaces BMO, inégalités de Paley et multiplicateurs idempotents

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Generalizing the classical BMO spaces defined on the unit circle 𝕋 with vector or scalar values, we define the spaces $BMO_{ψ_{q}}(𝕋)$ and $BMO_{ψ_{q}}(𝕋,B)$, where $ψ_{q}(x) = e^{x^q} -1$ for x ≥ 0 and q ∈ [1,∞[, and where B is a Banach space. Note that $BMO_{ψ_{1}}(𝕋) = BMO(𝕋)$ and $BMO_{ψ_{1}}(𝕋,B) = BMO(𝕋,B)$ by the John-Nirenberg theorem. Firstly, we study a generalization of the classical Paley inequality and improve the Blasco-Pełczyński theorem in the vector case. Secondly, we compute the idempotent multipliers of $BMO_{ψ_{q}}(𝕋)$. Pisier conjectured that the supports of idempotent multipliers of $L^{ψ_{q}}(𝕋)$ form a Boolean algebra generated by the periodic parts and the finite parts for 2 < q < ∞. We confirm this conjecture with $L^{ψ_{q}}(𝕋)$ replaced by $BMO_{ψ_{q}}(𝕋)$.
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  • Equipe d'analyse, URA 754, Université Pierre et Marie Curie-Paris 6, Tour 46-0, Boîte 186, 4, place Jussieu 75252 Paris Cedex 05, France
Bibliografia
  • [BP] O. Blasco and A. Pełczyński, Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces, Trans. Amer. Math. Soc. 323 (1991), 335-367.
  • [CW] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
  • [GM] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, New York, 1979.
  • [H] H. Helson, Note on harmonic functions, Proc. Amer. Math. Soc. 4 (1953), 686-691.
  • [Kl] I. Klemes, Idempotent multipliers of $H^1(T)$, Canad. J. Math. 39 (1987), 1223-1234.
  • [L] H. Lelièvre, Espaces BMO et multiplicateurs idempotents, thèse de doctorat de l'Université Paris 6, 1995.
  • [Pe] A. Pełczyński, Commensurate sequences of characters, Proc. Amer. Math. Soc. 104 (1988), 525-531.
  • [Pi1] G. Pisier, Les inégalités de Khintchine-Kahane d'après C. Borel, Séminaire sur la géométrie des espaces de Banach 1977-1978, exposé VII, Ecole Polytechnique, Centre de Mathematiques, 1978.
  • [Pi2] G. Pisier, De nouvelles caractérisations des ensembles de Sidon, in: Mathematical Analysis and Applications, Part B, Adv. in Math. Suppl. Stud. 7B, Academic Press, 1981, 685-726.
  • [Pi3] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.
  • [Pi4] G. Pisier, Probabilistic methods in the geometry of Banach spaces, in: Probability and Analysis 1985, Lecture Notes in Math. 1206, Springer, 1996, 167-241.
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bwmeta1.element.bwnjournal-article-smv123i3p249bwm
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