Characterizations of a class of Musielak--Orlicz BMO spaces via commutators of Riesz potential operators

TL;DR

This work characterizes the boundedness of commutators $[b,I_\alpha]$ on Musielak--Orlicz Hardy spaces and identifies the exact BMO-type subspaces $\mathcal{BMO}_{\varphi_1,u}(\mathbb{R}^n)$ governing these bounds. Using atomic decompositions and careful kernel estimates, it proves that $[b,I_\alpha]$ is bounded from $H^{\varphi_1}(\mathbb{R}^n)$ to $L^{\varphi_2}(\mathbb{R}^n)$ iff $b\in\mathcal{BMO}_{\varphi_1}(\mathbb{R}^n)$, and that boundedness to $H^{\varphi_2}(\mathbb{R}^n)$ holds under a broader BMO-type condition, with endpoint behavior clarified. The results extend to commutators with general homogeneous kernels $T_{\Omega,\alpha}$, broadening the Musielak--Orlicz framework for Hardy spaces and enriching the understanding of operator boundedness in these variable-exponent settings.

Abstract

The fractional integral operators $I_α$ can be used to characterize the Musielak--Orlicz Hardy spaces. This paper shows that for $b\in \rm BMO(\mathbb R^n)$, the commutators $[b,I_α]$ generated by fractional integral operators $I_α$ with $b$ are bounded from the Musielak--Orlicz Hardy spaces $H^{\varphi_1}(\mathbb R^n)$ to the Musielak--Orlicz spaces $L^{\varphi_2}(\mathbb R^n)$ (where $1<u<\infty$ and $\varphi_1$, $\varphi_2$ are growth functions) if and only if $b\in \mathcal {BMO}_{\varphi_1,u}(\mathbb R^n)$, which are a class of non-trivial subspaces of $\rm BMO(\mathbb R^n)$. Additionally, we obtain the boundedness of the commutator $[b,I_α]$ from $H^{\varphi_1}(\mathbb R^n)$ to $H^{\varphi_2}(\mathbb R^n)$. The corresponding results are also provided for commutators of fractional integrals associated with general homogeneous kernels.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Remark 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5