BMO-Type characterizations and endpoint estimates for commutators of intrinsic square functions on Orlicz-Hardy spaces

The commutators of intrinsic square functions associated with the intrinsic Littlewood–Paley $g$-function, the intrinsic $g_{\lambda }^{\ast }$-function and the intrinsic Lusin area function can be used to characterize BMO-type function spaces through their boundedness properties. In this paper, we show that, for $b\in \mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$, the boundedness of each of these commutators from the Musielak–Orlicz Hardy space $H^{\phi }({\mathbb{R}}^n)$ to $L^{\phi }({\mathbb{R}}^n)$ is equivalent to $b\in {\mathrm{B}\mathrm{M}\mathrm{O}}_{\phi }({\mathbb{R}}^n)$ (a nontrivial subspace of $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$), under suitable assumptions on the growth function $\phi$. This extends the weighted characterization of Han and Wu [Proc. Amer. Math. Soc. 152(1) (2024), 281–293] to the Musielak–Orlicz setting. In addition, under suitable assumptions on a growth function $\mathrm{\Phi }$, we prove that, for $b\in {\mathrm{B}\mathrm{M}\mathrm{O}}_{\mathrm{\Phi }}({\mathbb{R}}^n)$, these commutators are bounded from the $b$-adapted Orlicz–Hardy space $H_b^{\mathrm{\Phi }}({\mathbb{R}}^n)$ to $L^1({\mathbb{R}}^n)$ and from Orlicz Hardy space $H^{\mathrm{\Phi }}({\mathbb{R}}^n)$ to $L^{1,\mathrm{\infty }}({\mathbb{R}}^n)$.