Commutators of Fractional Integrals with $\operatorname{BMO}^β$ Functions

We study commutators of the Riesz potential $I_α$ with functions $b$ in the capacitary space $\mathrm{BMO}^β(\mathbb{R}^n)$, defined through the Hausdorff content $\mathcal{H}^β_\infty$. We prove a Chanillo-type theorem characterising $\mathrm{BMO}^β(\mathbb{R}^n)$ via the boundedness of the commutator $[b,I_α]$ on capacitary Lebesgue spaces. In addition, we obtain the endpoint estimate in the form of a capacitary modular weak-type inequality. These results follow from a pointwise estimate for the $β$-dimensional sharp maximal function of the commutator, together with a capacitary Fefferman-Stein inequality recently proved in [CC24].