AI 中文总结
本文研究次线性算子与Orlicz-BMO函数生成的交换子在Orlicz-Hardy型空间上的端点有界性，证明了两类有界性结果，指出仅Bochner-Riesz平均算子对应的交换子满足特定端点有界性并给出反例。
AI 中文摘要
给定增长函数$\varphi:[0,\infty)\rightarrow [0,\infty)$，本文证明了由次线性算子与Orlicz-$\mathrm{BMO}$函数$b$生成的交换子从$H_{b}^\varphi(\mathbb{R}^{n})$到$L^{1}(\mathbb{R}^{n})$、以及从$H^\varphi(\mathbb{R}^{n})$到$L^{1,\\,\infty}(\mathbb{R}^{n})$是有界的，其中$H_{b}^\varphi(\mathbb{R}^{n})$是Orlicz-Hardy空间$H^\varphi(\mathbb{R}^{n})$的一个特定子空间，次线性算子包括Lusin面积积分、g函数、Marcinkiewicz积分与Bochner-Riesz平均算子。在$T^*1=0$和$T^*b=0$的假设下，本文证明与Bochner-Riesz平均算子相关的Orlicz-$\mathrm{BMO}$交换子具有从$H_{b}^\varphi(\mathbb{R}^{n})$到$H^{1}(\mathbb{R}^{n})$的端点有界性。但本文讨论的其他算子对应的交换子不具备上述端点有界性，并给出反例说明这一点。
英文摘要
Given a growth function $φ:[0,\infty)\rightarrow [0,\infty)$, it is established that the commutators generated by sublinear operators and Orlicz-$\mathrm{BMO}$ function $b$ are bounded from $H_{b}^φ(\mathbb{R}^{n})$ to $L^{1}(\mathbb{R}^{n})$, and from $H^φ(\mathbb{R}^{n})$ to $L^{1,\,\infty}(\rn)$, where $H_{b}^φ(\mathbb{R}^{n})$ is a specific subspace of Orlicz-Hardy space $H^φ(\mathbb{R}^{n})$ and sublinear operators include Lusin area integral, g-function, Marcinkiewicz integral and Bochner-Riesz mean operator. Under the assumptions $T^*1=0$ and $T^*b=0$, it is shown that the Orlicz-$\mathrm{BMO}$ commutator associated with the Bochner-Riesz mean operator admits endpoint boundedness from $H_{b}^φ(\mathbb{R}^{n})$ to $H^{1}(\mathbb{R}^{n})$. However, the commutators corresponding to other operators discussed in this paper do not possess the aforementioned endpoint boundedness, and a counterexample is provided to illustrate this point.
CommentsCommutator; Orlicz-BMO; Orlicz-Hardy; Sublinear operator; Boundedness
