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

We study commutators of the Riesz potential $I_\alpha$ with functions $b$ in the capacitary space $\mathrm{BMO}^\beta(\mathbb{R}^n)$, defined through the Hausdorff content $\mathcal{H}^\beta_\infty$. We prove a Chanillo-type theorem characterising $\mathrm{BMO}^\beta(\mathbb{R}^n)$ via the boundedness of the commutator $[b,I_\alpha]$ 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 $\beta$-dimensional sharp maximal function of the commutator, together with a capacitary Fefferman-Stein inequality recently proved in [CC24].