Bilinear Decomposition and Divergence-Curl Estimates on Products Related to Local Hardy Spaces and Their Dual Spaces

Abstract

Let $p\in (0,1)$, $\alpha :=1/p-1$ and, for any $\tau \in [0,\infty )$, ${\Phi }_p(\tau ):=\tau /(1+{\tau }^{1-p})$. Let $H^p({\mathbb{R}}^n)$, $h^p({\mathbb{R}}^n)$ and ${\Lambda }_{n\alpha }({\mathbb{R}}^n)$ be, respectively, the Hardy space, the local Hardy space and the inhomogeneous Lipschitz space on ${\mathbb{R}}^n$. In this article, applying the inhomogeneous renormalization of wavelets, the authors establish a bilinear decomposition for multiplications of elements in $h^p({\mathbb{R}}^n)$ [or $H^p({\mathbb{R}}^n)$] and ${\Lambda }_{n\alpha }({\mathbb{R}}^n)$, and prove that these bilinear decompositions are sharp in some sense. As applications, the authors also obtain some estimates of the product of elements in the local Hardy space $h^p({\mathbb{R}}^n)$ with $p\in (0,1]$ and its dual space, respectively, with zero $\lfloor n\alpha \rfloor$-inhomogeneous curl and zero divergence, where $\lfloor n\alpha \rfloor$ denotes the largest integer not greater than $n\alpha$. Moreover, the authors find new structures of $h^{{\Phi }_p}({\mathbb{R}}^n)$ and $H^{{\Phi }_p}({\mathbb{R}}^n)$ by showing that $h^{{\Phi }_p}({\mathbb{R}}^n)=h^1({\mathbb{R}}^n)+h^p({\mathbb{R}}^n)$ and $H^{{\Phi }_p}({\mathbb{R}}^n)=H^1({\mathbb{R}}^n)+H^p({\mathbb{R}}^n)$ with equivalent quasi-norms, and also prove that the dual spaces of both $h^{{\Phi }_p}({\mathbb{R}}^n)$ and $h^p({\mathbb{R}}^n)$ coincide. These results give a complete picture on the multiplication between the local Hardy space and its dual space.