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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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##### References:
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