Commutators of Fractional Integrals with $\operatorname{BMO}^β$ Functions
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arXiv
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| Format: | Preprint |
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2026
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| author | Chen, You-Wei Benson Claros, Alejandro |
| author_facet | Chen, You-Wei Benson Claros, Alejandro |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09742 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Commutators of Fractional Integrals with $\operatorname{BMO}^β$ Functions Chen, You-Wei Benson Claros, Alejandro Classical Analysis and ODEs Functional Analysis 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]. |
| title | Commutators of Fractional Integrals with $\operatorname{BMO}^β$ Functions |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2602.09742 |