Paraproducts via$H^∞$ -functional calculus - Frey, Dorothee (2011)
Let$X$be a space of homogeneous type and let$L$be a sectorial operator with bounded holomorphic functional calculus on$L^2(X)$ . We assume that the semigroup$\{e^{-tL}\}_{t>0}$satisfies Davies-Gaffney estimates. In this paper, we introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated to$L$ . We show various boundedness properties on$L^p(X)$and the recently developed Hardy and BMO spaces$H^p_L(X)$and$BMO_L(X)$ . In generalization of standard paraproducts constructed via convolution operators, we show$L^2(X)$off-diagonal estimates as a substitute for Calderón-Zygmund kernel estimates. As an application, we study differentiability properties of paraproducts in terms of fractional powers of the operator$L$ . The results of this paper are fundamental for the proof of a T(1)-Theorem for operators beyond Calderón-Zygmund theory, which will be the subject of a forthcoming paper.