Boundedness of paraproduct operators on RD-spaces

Abstract

Let (X, d,µ) be an RD-space with “dimension” n, namely, a space of homogeneous type in the sense of Coifman and Weiss satisfying a certain reverse doubling condition. Using the Calderon reproducing formula, the authors hereby establish boundedness for paraproduct operators from the product of Hardy spaces H p (X) × H q (X) to the Hardy space H r (X), where p, q, r ∈ (n/(n + 1),∞) satisfy 1/p + 1/q = 1/r. Certain endpoint estimates are also obtained. In view of the lack of the Fourier transform in this setting, the proofs are based on the derivation of appropriate kernel estimates.