# Revista Matemática Iberoamericana

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**Volume 29, Issue 2, 2013, pp. 635–663**

**DOI: 10.4171/RMI/733**

Published online: 2013-04-22

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

Dorothee Frey^{[1]}(1) Australian National University, Canberra, Australia

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 $H^p_L(X)$ and ${\rm BMO}_L(X)$. 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.

*Keywords: *Paraproducts, Davies–Gaffney estimates, Hardy spaces, $H^{\infty}$-functional calculus, Carleson measures

Frey Dorothee: Paraproducts via $H^{\infty}$-functional calculus. *Rev. Mat. Iberoamericana* 29 (2013), 635-663. doi: 10.4171/RMI/733