Boundedness of paraproducts on spaces of homogeneous type II. Issue 6 (13th April 2022) - Chang, Der-Chen; Fu, Xing; Yang, Dachun
Abstract : This article is the second part of two works by the same authors on the same topic. Let ( X, d, μ ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors introduce the notion of paraproducts on X and obtain their boundedness from H p ( X ) × H q ( X ) into H r ( X ) for any given p, q, r ∈ ( n / ( n + η ), ∞ ) satisfying 1 / r = 1 / p + 1 / q, where H p ( X ), for any given p ∈ ( n / ( n + η ), ∞ ), denotes the Hardy space on X . Moreover, the authors also establish the endpoint boundedness of these paraproducts on X when p or q is ∞ or 1 with H p ( X ) or H q ( X ) replaced by B M O ( X ), L ∞ ( X ) or L 1 ( X ), and hence give a complete picture on the boundedness of paraproducts on X . The main novelty of this article is the avoidance of dependence on the reverse doubling assumption of the considered measure μ, which is achieved by fully using the geometrical properties of X expressed via its equipped quasi-metric d, dyadic reference points and dyadic cubes.