论文标题
双线性分解和发散性 - 折扣估计与当地耐力空间及其双空间有关的产品
Bilinear Decomposition and Divergence-Curl Estimates on Products Related to Local Hardy Spaces and Their Dual Spaces
论文作者
论文摘要
令$ p \ in（0,1）$，$α：= 1/p-1 $，对于[0，\ infty）$中的任何$τ\，$φ__{p}（τ）（τ）：=τ/（1+τ^{1-p}）$。令$ h^p（\ mathbb r^n）$，$ h^p（\ mathbb r^n）$和$λ_{nα}（\ mathbb {r}^n）$分别为hardy空间，当地的硬质空间和不菲的limschitz space on $ \ mathbb {r} n $。在本文中，应用小波的不均匀重归于，作者建立了双线性分解，以$ h^p（\ mathbb r^n）$ [或$ h^p（\ mathbb r^n）$]和$λ_{nα}（nα}（\ mathbb {\ mathbb {r}^n），并列出了这些eant bel，并列出了这些deste deste neyb，并列为有些感觉。作为应用，作者还获得了一些估计当地耐力空间中元素的产品$ h^p（\ mathbb r^n）$，分别$ p \ in（0,1] $及其双空间及其双空间，零$ \ lfloornα\ lfloornα\ rfloor $ rfloor $ -inhomogeenos $ -Inhomogeenous curl and deNlo $ n white n white n white n white n llo \ rof the最大的整数不大于$nα$。 r^n）$和$ h^{φ_p}（\ Mathbb r^n）= h^1（\ Mathbb r^n）+h^p（\ Mathbb r^n）$具有等效的quasi-norms，也证明了这两个$ h^{φ_p}（φ_p} $ MATHB r^n）的双空间重合这些结果，就当地耐力空间及其双重空间之间的乘法提供了完整的图片。
Let $p\in(0,1)$, $α:=1/p-1$ and, for any $τ\in [0,\infty)$, $Φ_{p}(τ):=τ/(1+τ^{1-p})$. Let $H^p(\mathbb R^n)$, $h^p(\mathbb R^n)$ and $Λ_{nα}(\mathbb{R}^n)$ 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 $Λ_{nα}(\mathbb{R}^n)$, 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 $\lfloor nα\rfloor$-inhomogeneous curl and zero divergence, where $\lfloor nα\rfloor$ denotes the largest integer not greater than $nα$. Moreover, the authors find new structures of $h^{Φ_p}(\mathbb R^n)$ and $H^{Φ_p}(\mathbb R^n)$ by showing that $h^{Φ_p}(\mathbb R^n)=h^1(\mathbb R^n)+h^p(\mathbb R^n)$ and $H^{Φ_p}(\mathbb R^n)=H^1(\mathbb R^n)+H^p(\mathbb R^n)$ with equivalent quasi-norms, and also prove that the dual spaces of both $h^{Φ_p}(\mathbb R^n)$ 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.
