BMO spaces, Carleson measures, generalized Littlewood-Paley \(g\) functions, and annulation conditions (Q1318164)

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.