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arXiv 0903.3775math.CAmath.FA
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加权各向异性积Hardy空间与次线性算子的有界性

Weighted Anisotropic Product Hardy Spaces and Boundedness of Sublinear Operators

Marcin Bownik, Baode Li, Dachun Yang, Yuan Zhou

备注：
Math. Nachr. (to appear)
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摘要：

令$A_1$和$A_2$分别为${\mathbb R}^n$和${\mathbb R}^m$上的扩张性扩张。令$\vec A\equiv(A_1, A_2)$和$\mathcal A_p(\vec A)$是$p\in(1, \infty]$的${\mathbb R}^n\times{\mathbb R}^m$上的乘积Muckenhoupt权重的类。当$p\in(1, \infty)$和$w\in{\mathcal A}_p(\vec A)$时，作者通过与$\vec A$相关的各向异性Lusin面积函数刻画了加权Lebesgue空间$L^p_w({\mathbb R}^n\times{\mathbb R}^m)$.当$p\in(0, 1]$，$w\in {\mathcal A}_\infty(\vec A)$时，通过各向异性的Lusin面积函数引入加权各向异性积Hardy空间$H^p_w({\mathbb R}^n\times{\mathbb R}^m; \vec A)$，并建立了它的原子分解.证明了$H^p_w({\mathbb R}^n\times{\mathbb R}^m;\vec A)$的稠密子空间上的有限原子范数与标准的无限原子分解范数是等价的.作为应用，证明了若$T$是次线性算子，且所有原子映射到拟Banach空间$\mathcal B $的一致有界元上，则$T$唯一地扩张为从$H^p_w({\mathbb R}^n\times{\mathbb R}^m;\vec A)$到$\mathcal B$的有界次线性算子.本文的结果改进了加权积Hardy空间中已有的结果，并且在非加权各向异性情形下也是新的。

英文摘要：

Let $A_1$ and $A_2$ be expansive dilations, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$. Let $\vec A\equiv(A_1, A_2)$ and $\mathcal A_p(\vec A)$ be the class of product Muckenhoupt weights on ${\mathbb R}^n\times{\mathbb R}^m$ for $p\in(1, \infty]$. When $p\in(1, \infty)$ and $w\in{\mathcal A}_p(\vec A)$, the authors characterize the weighted Lebesgue space $L^p_w({\mathbb R}^n\times{\mathbb R}^m)$ via the anisotropic Lusin-area function associated with $\vec A$. When $p\in(0, 1]$, $w\in {\mathcal A}_\infty(\vec A)$, the authors introduce the weighted anisotropic product Hardy space $H^p_w({\mathbb R}^n\times{\mathbb R}^m; \vec A)$ via the anisotropic Lusin-area function and establish its atomic decomposition. Moreover, the authors prove that finite atomic norm on a dense subspace of $H^p_w({\mathbb R}^n\times{\mathbb R}^m;\vec A)$ is equivalent with the standard infinite atomic decomposition norm. As an application, the authors prove that if $T$ is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space $\mathcal B $, then $T$ uniquely extends to a bounded sublinear operator from $H^p_w({\mathbb R}^n\times{\mathbb R}^m;\vec A)$ to $\mathcal B$. The results of this paper improve the existing results for weighted product Hardy spaces and are new even in the unweighted anisotropic setting.

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