Boundedness of Intrinsic Littlewood-Paley Functions on Musielak-Orlicz Morrey and Campanato Spaces
Boundedness of Intrinsic Littlewood-Paley Functions on Musielak-Orlicz Morrey and Campanato Spaces
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DOI：
10.15352/bjma/1381782098
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发表时间：
2013-09
期刊：
影响因子：
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通讯作者：
Yiyu Liang;E. Nakai;Dachun Yang;Junqiang Zhang
中科院分区：
文献类型：
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作者：
Yiyu Liang;E. Nakai;Dachun Yang;Junqiang Zhang
令 $\varphi: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ 使得 $\vz(x,\cdot)$ 不减，$\varphi(x,0)=0$, $\varphi(x,t)>0$ 当 $t>0$ 时，$\lim_{t\to\infty}\varphi(x,t)=\infty$ $\vz(\cdot,t)$ 是 $t$ 中统一的 Muckenhoupt $A_\infty({\mathbb R^n})$ 权重。令 $\phi: [0,\infty)\to[0,\infty)$ 不减。在本文中，作者介绍了 Musielak-Orlicz Morrey 空间 $\mathcal M^{\varphi,\phi}(\mathbb R^n)$ 并获得了本征 Lusin 面积函数 $S_{\alpha}$（本征 $g$-函数 $g_{\alpha}$）在 $\mathcal M^{\varphi,\phi}(\mathbb R^n)$ 上的有界性， $g_{\lambda}^*$-函数 $g^\ast_{\lambda, \alpha}$ 及其带有 ${\rm BMO}(\rn)$ 函数的换向器，其中 $\alpha\in(0,1]$, $\lambda\in(\min\{\max\{3,\,p_1\},3+2\az/n\},\infty)$ 和 $p_1$ 表示$\vz$。令 $\Phi: [0,\infty)\to[0,\infty)$ 不减，$\Phi(0)=0$，$t>0$ 时 $\Phi(t)>0$，$\lim_{t\to\infty}\Phi(t)=\infty$，$w\in A_\infty(\mathbb R^n)$ 和 $\phi： (0,\infty)\to(0,\infty)$ 是非增的。作者还引入了加权 Orlicz-Morrey 空间 $M_w^{\Phi,\phi}(\mathbb R^n)$ 并获得了上述内在 Littlewood-Paley 函数及其与 ${\rm BMO}(\rn)$ 函数的交换子在 $M_w^{\Phi,\phi}(\mathbb R^n)$ 上的有界性。最后，对于 $q\in[1,\fz)$，上述内禀 Littlewood-Paley 函数在 Musielak Orlicz Campanato 空间 $\mathcal L^{\varphi,q}(\mathbb R^n)$ 上的有界性也成立。
Let $\varphi: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $\vz(x,\cdot)$ is nondecreasing, $\varphi(x,0)=0$, $\varphi(x,t)>0$ when $t>0$, $\lim_{t\to\infty}\varphi(x,t)=\infty$ and $\vz(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight uniformly in $t$. Let $\phi: [0,\infty)\to[0,\infty)$ be nondecreasing. In this article, the authors introduce the Musielak-Orlicz Morrey space $\mathcal M^{\varphi,\phi}(\mathbb R^n)$ and obtain the boundedness on $\mathcal M^{\varphi,\phi}(\mathbb R^n)$ of the intrinsic Lusin area function $S_{\alpha}$, the intrinsic $g$-function $g_{\alpha}$, the intrinsic $g_{\lambda}^*$-function $g^\ast_{\lambda, \alpha}$ and their commutators with ${\rm BMO}(\rn)$ functions, where $\alpha\in(0,1]$, $\lambda\in(\min\{\max\{3,\,p_1\},3+2\az/n\},\infty)$ and $p_1$ denotes the uniformly upper type index of $\vz$. Let $\Phi: [0,\infty)\to[0,\infty)$ be nondecreasing, $\Phi(0)=0$, $\Phi(t)>0$ when $t>0$, and $\lim_{t\to\infty}\Phi(t)=\infty$, $w\in A_\infty(\mathbb R^n)$ and $\phi: (0,\infty)\to(0,\infty)$ be nonincreasing. The authors also introduce the weighted Orlicz-Morrey space $M_w^{\Phi,\phi}(\mathbb R^n)$ and obtain the boundedness on $M_w^{\Phi,\phi}(\mathbb R^n)$ of the aforementioned intrinsic Littlewood-Paley functions and their commutators with ${\rm BMO}(\rn)$ functions. Finally, for $q\in[1,\fz)$, the boundedness of the aforementioned intrinsic Littlewood-Paley functions on the Musielak Orlicz Campanato space $\mathcal L^{\varphi,q}(\mathbb R^n)$ is also established.