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

### Dorothee Frey

Australian National University, Canberra, Australia

## 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 $H^p_L(X)$ 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.

## Cite this article

Dorothee Frey, Paraproducts via $H^{\infty}$-functional calculus. Rev. Mat. Iberoam. 29 (2013), no. 2, pp. 635–663

DOI 10.4171/RMI/733