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

Abstract

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 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 $L$. We show various boundedness properties on $L^p(X)$ and the recently developed Hardy and BMO spaces $H_L^p(X)$ and ${\mathrm{BMO}}_L(X)$. Generalizing 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 that are beyond the reach of Calderón–Zygmund theory, which is the subject of a forthcoming paper.