Bilinear Decompositions of Products of Hardy and Lipschitz Spaces Through Wavelets

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 $H^p({\mathbb{R}}^n)$ and their duals in the case when $p<1$ and near to $1$, via wavelets, paraproducts and the theory of bilinear Calder'on-Zygmund operators. Precisely, the authors establish the bilinear decompositions of the product spaces $H^p({\mathbb{R}}^n)\times {\dot{\mathrm{\Lambda }}}_{\alpha }({\mathbb{R}}^n)$ and $H^p({\mathbb{R}}^n)\times {\mathrm{\Lambda }}_{\alpha }({\mathbb{R}}^n)$, 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 ${\dot{\mathrm{\Lambda }}}_{\alpha }$ and ${\mathrm{\Lambda }}_{\alpha }$ 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.