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数学进展 - 2016, Vol. 45(2): 233-251
研究论文
各向异性的Musielak-Orlicz型帐篷空间及其应用
Anisotropic Tent Spaces of Musielak-Orlicz Type and Their Applications

范兴亚,李宝德
FAN Xingya, LI Baode

新疆大学数学与系统科学学院, 乌鲁木齐, 新疆, 830046
College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, 830046, P. R. China

收稿日期: 2014-07-10
2016, Vol. 45(2): 233-251
DOI: 10.11845/sxjz.2014109b


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摘要 设$A$是一个扩张矩阵, $\varphi : \mathbb{R}^n\times [0,\infty)\to[0,\infty)$是一个各向异性的Musielak-Orlicz函数. 本文通过各向异性的面积函数引进了各向异性Musielak-Orlicz型的帐篷空间$T^\varphi_A(\mathbb{R}^n\times\mathbb{Z})$, 并得到了它的原子分解. 此类空间包括了Coifman等人建立的经典帐篷空间、Bui等人建立的加权帐篷空间以及侯绍雄等人建立的经典Musielak-Orlicz型的帐篷空间.另外, 本文引进了各向异性Musielak-Orlicz型的BMO空间$\mathrm{BMO}^\varphi_A(\mathbb{R}^n)$,并证明了它是各向异性Musielak-Orlicz型Hardy空间的对偶空间.此类空间包括了John和Nirenberg的经典BMO空间、Bownik的各向异性的BMO空间、Muckenhoupt和Wheenden的加权BMO空间及Ky的Musielak-Orlicz型的BMO空间.作为各向异性Musielak-Orlicz型帐篷空间原子分解的应用,本文得到了$\mathrm{BMO}^\varphi_A(\mathbb{R}^n)$的各向异性$\varphi$-Carleson测度特征.
Abstract:Let $A$ be an expansive dilation and $\varphi : \mathbb{R}^n\times [0,\infty)\to[0,\infty)$ be an anisotropic Musielak-Orlicz function.This article introduces anisotropic tent spaces of Musielak-Orlicz type, $T^\varphi_A(\mathbb{R}^n\times\mathbb{Z})$, via anisotropic square-type functions and characterizes these spaces via anisotropic atomic decompositions. These spaces include classical tent spaces of Coifman et al., weighted tent spaces of Bui et al., and classical tent spaces of Musielak-Orlicz type of Hou et al. as special cases.Moreover, this article introduces anisotropic BMO spaces of Musielak-Orlicz type, $\mathrm{BMO}^\varphi_A(\mathbb{R}^n)$, which is found to be the dual spaces of anisotropic Hardy spaces of Musielak-Orlicz type. These spaces include classical $\mathrm{BMO}(\mathbb{R}^n)$ spaces of John and Nirenberg, anisotropic $\mathrm{BMO}(\mathbb{R}^n)$ spaces of Bownik, weighted $\mathrm{BMO}(\mathbb{R}^n)$ spaces of Muckenhoupt and Wheenden, and classical $\mathrm{BMO}(\mathbb{R}^n)$ spaces of Musielak-Orlicz type of Ky as special cases. Then the anisotropic $\varphi$-Carleson measure characterization of $\mathrm{BMO}^\varphi_A(\mathbb{R}^n)$ is obtained as an application of the anisotropic atomic decomposition of $T^\varphi_A(\mathbb{R}^n\times\mathbb{Z})$.
PACS:  O174.2  
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