Paraproducts via HH^{\infty}-functional calculus

  • Dorothee Frey

    Australian National University, Canberra, Australia

Abstract

Let XX be a space of homogeneous type and let LL be a sectorial operator with bounded holomorphic functional calculus on L2(X)L^2(X). We assume that the semigroup {etL}t>0\{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 LL. We show various boundedness properties on Lp(X)L^p(X) and the recently developed Hardy and BMO spaces HLp(X)H^p_L(X) and BMOL(X){\rm BMO}_L(X). Generalizing standard paraproducts constructed via convolution operators, we show L2(X)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 LL.

The results of this paper are fundamental for the proof of a T(1)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 HH^{\infty}-functional calculus. Rev. Mat. Iberoam. 29 (2013), no. 2, pp. 635–663

DOI 10.4171/RMI/733