Paraproducts via H^∞-functional calculus

2011/07/21 by Dorothee Frey, Frey, Dorothee
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #math.CA #math.FA

paper · pdf · doi:10.48550/arxiv.1107.4348

26 pages

arxiv created 2011/07/21 · arxiv updated 2011/07/22

Abstract

Let X be a space of homogeneous type and let L be a sectorial operator with bounded holomorphic functional calculus on L²(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 L²(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 beyond Calderón-Zygmund theory, which will be the subject of a forthcoming paper.

Related

• A T(1)-Theorem for non-integral operators
• A T(1)-Theorem in relation to a semigroup of operators and applications to new paraproducts
• Multilinear paraproducts on Sobolev spaces
• Operator-valued dyadic BMO spaces
• Uniform estimates for paraproducts and related multilinear multipliers