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

TL;DR

This paper characterizes a class of Musielak--Orlicz BMO spaces through the boundedness of commutators of fractional integral operators, linking these spaces to Hardy spaces and extending results to general kernels.

Contribution

It introduces a new class of Musielak--Orlicz BMO spaces characterized by commutator boundedness with fractional integrals, expanding the understanding of these function spaces.

Findings

Boundedness of commutators characterizes Musielak--Orlicz BMO spaces.

Commutators map Hardy spaces to Musielak--Orlicz spaces under certain conditions.

Results extend to fractional integrals with general homogeneous kernels.

Abstract

The fractional integral operators $I_\alpha$ can be used to characterize the Musielak--Orlicz Hardy spaces. This paper shows that for $b\in \rm BMO(\mathbb R^n)$, the commutators $[b,I_\alpha]$ generated by fractional integral operators $I_\alpha$ 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 $[b,I_\alpha]$ 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.