Paraproducts via $H^{\infty}$-functional calculus

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 Calder\'on-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 Calder\'on-Zygmund theory, which will be the subject of a forthcoming paper.