Let be a space of homogeneous type and let be a sectorial operator with bounded holomorphic functional calculus on . We assume that the semigroup satisfies the Davies–Gaffney estimates. In this paper, we introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated with . We show various boundedness properties on and the recently developed Hardy and BMO spaces and . Generalizing standard paraproducts constructed via convolution operators, we show 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 .
The results of this paper are fundamental for the proof of a -Theorem for operators that are beyond the reach of Calderón–Zygmund theory, which is the subject of a forthcoming paper.
Cite this article
Dorothee Frey, Paraproducts via -functional calculus. Rev. Mat. Iberoam. 29 (2013), no. 2, pp. 635–663DOI 10.4171/RMI/733