Bilinear Decompositions of Products of Hardy and Lipschitz Spaces Through Wavelets

2015/10/15 by Cao, Jun, Ky, Luong Dang, Yang, Dachun
#42C40 #46E30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 42B30 #Secondary: 42B35

paper · doi:10.48550/arxiv.1510.04384

Abstract

The aim of this article is to give a complete solution to the problem of the bilinear decompositions of the products of some Hardy spaces H^p(ℝⁿ) and their duals in the case when p<1 and near to 1, via wavelets, paraproducts and the theory of bilinear Calderón-Zygmund operators. Precisely, the authors establish the bilinear decompositions of the product spaces H^p(ℝⁿ)×Λ_α (ℝⁿ) and H^p(ℝⁿ)×Λ_α(ℝⁿ), where, for all p∈((n)/(n+1), 1) and α:=n((1)/(p)-1), H^p(ℝⁿ) denotes the classical real Hardy space, and Λ_α and Λ_α denote the homogeneous, respectively, the inhomogeneous Lipschitz spaces. Sharpness of these two bilinear decompositions are also proved. As an application, the authors establish some div-curl lemmas at the endpoint case.

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