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

Abstract

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 ${\mathcal{H}}_{\infty }^{\beta }$. 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].

31 pages