Endpoint boundedness of Orlicz-BMO commutators on Orlicz-Hardy type spaces
Given a growth function $\varphi:[0,\infty)\rightarrow [0,\infty)$, it is established that the commutators generated by sublinear operators and Orlicz-$\mathrm{BMO}$ function $b$ are bounded from $H_{b}^{\varphi}(\mathbb{R}^{n})$ to $L^{1}(\mathbb{R}^{n})$, and from $H^{\varphi}(\mathbb{R}^{n})$ to $L^{1,\,\infty}(\rn)$, where $H_{b}^{\varphi}(\mathbb{R}^{n})$ is a specific subspace of Orlicz-Hardy space $H^{\varphi}(\mathbb{R}^{n})$ 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 $H_{b}^{\varphi}(\mathbb{R}^{n})$ 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.