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BMO spaces, Carleson measures, generalized Littlewood-Paley \(g\) functions, and annulation conditions. (Espace BMO, mesures de Carleson, fonctions \(g\) de Littlewood-Paley généralisées et conditions d’annulation.) (French) Zbl 0788.42007
We prove that the Littlewood-Paley \((g_ r)^ 2\) operator associated with a kernel \(r\) (satisfying the standard decay and Lipschitz conditions) maps \(L^ \infty\) into BMO if and only if a suitable cancellation condition on \(r\) holds. A similar result is obtained for BMO. As a consequence, we obtain the following extrapolation result: the Littlewood-Paley \(L^ p\) estimates \(\| g_ r(f)\|_ p\leq C_ p\| f\|_ p\) hold if and only if the \((g_ r)^ 2\) operator maps \(L^ \infty\) into BMO.

42B25 Maximal functions, Littlewood-Paley theory
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[1] Ba?uelos, R., Brossard, J.: The area integral and its density for BMO and VMO functions. Ark. Mat. (? para?tre) · Zbl 0803.42008
[2] David, G., Journ?, J.L., Semmes, S.: Op?rateurs de Calder?n-Zygmund, fonctions para-accr?tives et interpolation. Rev. Mat. Iberoan.1, 1-56 (1985) · Zbl 0604.42014
[3] Jones, P.: Square functions, Cauchy integrals, analytic capacity and harmonic measure. In: Garcia Cuerva, J. (ed.) Harmonic analysis and partial differential equations (Lect. Notes Math., vol. 1384, pp. 24-68) Berlin Heidelberg New York: Springer 1989
[4] Journ?, J.L.: Calder?n-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderon (Lect. Notes Math., vol. 994) Berlin Heidelberg New York: Springer 1983
[5] Quian, T.: On BMO boundedness of a class of operators. J. Math. Res. Expos.7, 331-334 (1987)
[6] Semmes, S.: Square function estimates and theT(b) theorem. Proc. Am. Math. Soc.110, 721-726 (1990) · Zbl 0719.42023
[7] Stein, E.M.: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans. Am. Math. Soc.88, 430-466 (1958) · Zbl 0105.05104
[8] Torschinsky, A.: Real-variable methods in harmonic analysis. London New York: Academic Press 1986
[9] Wang, S.: Some properties of Littlewood-Paley’sg-function. Sci. Sinica, Ser. A28, 252-262 (1985)
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