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Paraproducts via \(H^{\infty}\)-functional calculus. (English) Zbl 1277.42020
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 [D. Frey and 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 \(P_t\).

42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
47A60 Functional calculus for linear operators
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