Bilinear Decompositions of Products of Hardy and Lipschitz or BMO Spaces Through Wavelets

Jun Cao, Luong Dang Ky, Dachun Yang

Published 2015-10-15Version 1

The aim of this article is to give the bilinear decompositions of the products of some Hardy spaces and their duals. The authors establish the bilinear decompositions of the product spaces $H^p(\mathbb{R}^n)\times\dot\Lambda_{\alpha}(\mathbb{R}^n)$ and $H^p(\mathbb{R}^n)\times\Lambda_{\alpha}(\mathbb{R}^n)$, where, for all $p\in(\frac{n}{n+1},\,1)$ and $\alpha:=n(\frac{1}{p}-1)$, $H^p(\mathbb{R}^n)$ denotes the classical real Hardy space, and $\dot\Lambda_{\alpha}$ and $\Lambda_{\alpha}$ denote, respectively, the homogeneous and the inhomogeneous Lipschitz spaces. Sharpness of these two bilinear decompositions are considered. Moreover, the authors also give the corresponding bilinear decompositions of the associated local product spaces $h^1(\mathbb{R}^n) \times {\mathop\mathrm{bmo}}(\mathbb{R}^n)$ and $h^p(\mathbb{R}^n) \times \Lambda_{\alpha}(\mathbb{R}^n)$ with $p\in(\frac{n}{n+1},\,1)$ and $\alpha:=n(\frac{1}{p}-1)$, where, for all $p\in(\frac{n}{n+1},\,1]$, $h^p(\mathbb{R}^n)$ denotes the local Hardy space and ${\mathop\mathrm{bmo}}(\mathbb{R}^n)$ the local BMO space in the sense of D. Goldberg. As an application, the authors establish some div-curl lemmas at the endpoint case.