Optimal Sparse Bounds and Commutator Characterizations Without Doubling
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917050405879808 |
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| author | D'Emilio, Francesco Lin, Yongxi Wagner, Nathan A. Wick, Brett D. |
| author_facet | D'Emilio, Francesco Lin, Yongxi Wagner, Nathan A. Wick, Brett D. |
| contents | We examine dyadic paraproducts and commutators in the non-homogeneous setting, where the underlying Borel measure $μ$ is not assumed to be doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related operators with symbols $b \in \textrm{BMO}(μ)$, improving upon an earlier result of Lacey, where the symbol $b$ was assumed to satisfy a stronger Carleson-type condition, that coincides with $\textrm{BMO}$ only in the doubling setting. As an application of this result, we obtain sharpened weighted inequalities for the commutator of a dyadic Hilbert transform $\mathcal{H}$ previously studied by Borges, Conde Alonso, Pipher, and the third author. We also characterize the symbols for which the commutator $[\mathcal{H},b]$ is bounded on $L^p(μ)$ for $1<p<\infty$ and provide some interesting examples to prove that this class of symbols strictly depends on $p$ and is nested between symbols satisfying the $p$-Carleson packing condition and symbols belonging to martingale BMO (even in the case of absolutely continuous measures). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Sparse Bounds and Commutator Characterizations Without Doubling D'Emilio, Francesco Lin, Yongxi Wagner, Nathan A. Wick, Brett D. Classical Analysis and ODEs 42B20, 47B90 (secondary), 60G46 (secondary) We examine dyadic paraproducts and commutators in the non-homogeneous setting, where the underlying Borel measure $μ$ is not assumed to be doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related operators with symbols $b \in \textrm{BMO}(μ)$, improving upon an earlier result of Lacey, where the symbol $b$ was assumed to satisfy a stronger Carleson-type condition, that coincides with $\textrm{BMO}$ only in the doubling setting. As an application of this result, we obtain sharpened weighted inequalities for the commutator of a dyadic Hilbert transform $\mathcal{H}$ previously studied by Borges, Conde Alonso, Pipher, and the third author. We also characterize the symbols for which the commutator $[\mathcal{H},b]$ is bounded on $L^p(μ)$ for $1<p<\infty$ and provide some interesting examples to prove that this class of symbols strictly depends on $p$ and is nested between symbols satisfying the $p$-Carleson packing condition and symbols belonging to martingale BMO (even in the case of absolutely continuous measures). |
| title | Optimal Sparse Bounds and Commutator Characterizations Without Doubling |
| topic | Classical Analysis and ODEs 42B20, 47B90 (secondary), 60G46 (secondary) |
| url | https://arxiv.org/abs/2510.26505 |