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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.05454 |
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| _version_ | 1866917191443546112 |
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| author | Stockdale, Cody B. Waters, Cody |
| author_facet | Stockdale, Cody B. Waters, Cody |
| contents | We establish a new $T1$ theorem for the compactness of bi-parameter Calderón-Zygmund singular integral operators. Namely, we show that if a bi-parameter CZO $T$ satisfies the product weak compactness property, the mixed weak compactness/CMO property, and $T1, T^t1,$ $T_t1, T_t^t1 \in \text{CMO}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$, then $T$ is compact on $L^2(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$. We also obtain endpoint compactness results for these operators and use them to deduce the necessity of most of our hypotheses. In particular, our conditions characterize the simultaneous $L^2(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$-compactness of a bi-parameter CZO and its partial transpose. Our assumptions improve upon previously known sufficient conditions, and our proof, which is shorter and simpler than earlier arguments, utilizes a new abstract compactness criterion for partially localized operators on tensor products of Hilbert spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05454 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the compactness of bi-parameter singular integrals Stockdale, Cody B. Waters, Cody Classical Analysis and ODEs 42B20, 47B07 We establish a new $T1$ theorem for the compactness of bi-parameter Calderón-Zygmund singular integral operators. Namely, we show that if a bi-parameter CZO $T$ satisfies the product weak compactness property, the mixed weak compactness/CMO property, and $T1, T^t1,$ $T_t1, T_t^t1 \in \text{CMO}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$, then $T$ is compact on $L^2(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$. We also obtain endpoint compactness results for these operators and use them to deduce the necessity of most of our hypotheses. In particular, our conditions characterize the simultaneous $L^2(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$-compactness of a bi-parameter CZO and its partial transpose. Our assumptions improve upon previously known sufficient conditions, and our proof, which is shorter and simpler than earlier arguments, utilizes a new abstract compactness criterion for partially localized operators on tensor products of Hilbert spaces. |
| title | On the compactness of bi-parameter singular integrals |
| topic | Classical Analysis and ODEs 42B20, 47B07 |
| url | https://arxiv.org/abs/2601.05454 |