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

The fractional integral operators $I_{\alpha }$ can be used to characterize the Musielak–Orlicz Hardy spaces. This paper shows that for $b\in \mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^{\mathrm{n}})$, the commutators $[b,I_{\alpha }]$ generated by fractional integral operators $I_{\alpha }$ with $b$ are bounded from the Musielak–Orlicz Hardy spaces $H^{{\phi }_1}({\mathbb{R}}^n)$ to the Musielak–Orlicz spaces $L^{{\phi }_2}({\mathbb{R}}^n)$ (where $1<u<\mathrm{\infty }$ and ${\phi }_1$, ${\phi }_2$ are growth functions) if and only if $b\in {\mathcal{B}\mathcal{M}\mathcal{O}}_{{\phi }_1,u}({\mathbb{R}}^n)$, which are a class of non-trivial subspaces of $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^{\mathrm{n}})$. Additionally, we obtain the boundedness of the commutator $[b,I_{\alpha }]$ from $H^{{\phi }_1}({\mathbb{R}}^n)$ to $H^{{\phi }_2}({\mathbb{R}}^n)$. The corresponding results are also provided for commutators of fractional integrals associated with general homogeneous kernels.