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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.

MSC:
42B25 Maximal functions, Littlewood-Paley theory
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