Paraproducts via $H^\infty$-functional calculus

TL;DR

This paper introduces a new class of paraproduct operators linked to sectorial operators with functional calculus on spaces of homogeneous type, establishing their boundedness and differentiability properties.

Contribution

It develops a novel approach to paraproducts via $H^$-functional calculus, extending classical theory to more general operators and spaces.

Findings

Established boundedness of new paraproducts on $L^p$ and Hardy/BMO spaces.

Proved off-diagonal estimates as a substitute for Calderf3n-Zygmund kernel estimates.

Analyzed differentiability properties of paraproducts using fractional powers of $L$.

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 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.