Endpoint boundedness of Orlicz-BMO commutators on Orlicz-Hardy type spaces

Abstract

Given a growth function $\phi :[0,\infty )\to [0,\infty )$, it is established that the commutators generated by sublinear operators and Orlicz-$\mathrm{BMO}$ function $b$ are bounded from $H_b^{\phi }({\mathbb{R}}^n)$ to $L^1({\mathbb{R}}^n)$, and from $H^{\phi }({\mathbb{R}}^n)$ to $L^{1,\,\infty}(\rn)$, where $H_b^{\phi }({\mathbb{R}}^n)$ is a specific subspace of Orlicz-Hardy space $H^{\phi }({\mathbb{R}}^n)$ and sublinear operators include Lusin area integral, g-function, Marcinkiewicz integral and Bochner-Riesz mean operator. Under the assumptions $T^{\ast }1=0$ and $T^{\ast }b=0$, it is shown that the Orlicz-$\mathrm{BMO}$ commutator associated with the Bochner-Riesz mean operator admits endpoint boundedness from $H_b^{\phi }({\mathbb{R}}^n)$ 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.

Commutator; Orlicz-BMO; Orlicz-Hardy; Sublinear operator; Boundedness