Paraproducts via \(H^{\infty}\)-functional calculus (Q1955678)

This paper is one of many devoted to harmonic analysis of singular ``non-integral'' operators that arise from the functional calculus of a rough differential operator \(L\). It is a technical prelude to [\textit{D. Frey} and \textit{P. C. Kunstmann}, Math. Ann. 357, No. 1, 215--278 (2013; Zbl 1280.42006)], where the \(L^2\)-boundedness of such operators is characterized. Here, the author studies mapping properties of associated paraproducts \[ \Pi(f,b)=\int_0^\infty Q_t[(Q_t b)(A_tP_t f)]dt/t, \] where \(P_t\) and \(Q_t\) are approximations and resolutions of the identity associated with \(L\), and \(A_t\) is an additional spatial averaging, needed in the absence of pointwise bounds for \(P_t\).