Endpoint Estimates for Bergman Commutators and New Characterizations of the Bloch Space and $H^\infty$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917299807584256 |
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| author | Christopherson, Adam B. Huo, Zhenghui Wagner, Nathan A. Zeytuncu, Yunus E. |
| author_facet | Christopherson, Adam B. Huo, Zhenghui Wagner, Nathan A. Zeytuncu, Yunus E. |
| contents | We prove an $\LlogL $-type distributional inequality for the commutator of the Bergman projection with a conjugate Bloch symbol function on the unit ball. Such an inequality can be seen as a Bergman version of a result due to C. Pérez for real-variable Calderón-Zygmund operators and BMO functions. We also prove that this inequality characterizes membership of analytic functions in the Bloch space and is further equivalent to a kind of modified restricted weak-type estimate, where one only tests over characteristic functions of sets comparable to Bergman balls. We also show our estimate is sharp in the sense that there exists a Bloch function $b$ so that the commutator $[\bar{b},P]$ is not weak-type $(1,1)$, and prove $[\bar{b},P]$ with $b$ analytic is weak-type $(1,1)$ if and only if $b \in H^\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_24186 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Endpoint Estimates for Bergman Commutators and New Characterizations of the Bloch Space and $H^\infty$ Christopherson, Adam B. Huo, Zhenghui Wagner, Nathan A. Zeytuncu, Yunus E. Complex Variables Functional Analysis 32A25, 32A50, 32A18 We prove an $\LlogL $-type distributional inequality for the commutator of the Bergman projection with a conjugate Bloch symbol function on the unit ball. Such an inequality can be seen as a Bergman version of a result due to C. Pérez for real-variable Calderón-Zygmund operators and BMO functions. We also prove that this inequality characterizes membership of analytic functions in the Bloch space and is further equivalent to a kind of modified restricted weak-type estimate, where one only tests over characteristic functions of sets comparable to Bergman balls. We also show our estimate is sharp in the sense that there exists a Bloch function $b$ so that the commutator $[\bar{b},P]$ is not weak-type $(1,1)$, and prove $[\bar{b},P]$ with $b$ analytic is weak-type $(1,1)$ if and only if $b \in H^\infty$. |
| title | Endpoint Estimates for Bergman Commutators and New Characterizations of the Bloch Space and $H^\infty$ |
| topic | Complex Variables Functional Analysis 32A25, 32A50, 32A18 |
| url | https://arxiv.org/abs/2602.24186 |