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

TL;DR

This paper establishes a bilinear decomposition for products of elements in local Hardy and Lipschitz spaces, providing sharp estimates and revealing new structural relationships within these function spaces.

Contribution

It introduces a novel bilinear decomposition for products involving local Hardy and Lipschitz spaces using wavelet renormalization, and uncovers new structural identities and duality results.

Findings

Bilinear decompositions are sharp in some sense.

New structural identities for $h^{\

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.