Bilinear Decompositions of Products of Hardy and Lipschitz Spaces Through Wavelets

TL;DR

This paper provides a comprehensive solution for bilinear decompositions of products of Hardy and Lipschitz spaces near the critical index, using wavelets, paraproducts, and bilinear Calderón-Zygmund operators.

Contribution

It introduces new bilinear decompositions of Hardy and Lipschitz space products for p<1, extending previous results with sharpness proofs and applications to div-curl lemmas.

Findings

Established bilinear decompositions for Hardy and Lipschitz space products.

Proved sharpness of the decompositions.

Applied results to endpoint div-curl lemmas.

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 $H^p(\mathbb{R}^n)$ 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 $H^p(\mathbb{R}^n)\times\dot\Lambda_{\alpha} (\mathbb{R}^n)$ 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 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.