On entangled and multi-parameter commutators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916505960054784 |
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| author | Li, Kangwei Martikainen, Henri |
| author_facet | Li, Kangwei Martikainen, Henri |
| contents | We complement the recent theory of general singular integrals $T$ invariant under the Zygmund dilations $(x_1, x_2, x_3) \mapsto (s x_1, tx_2, st x_3)$ by proving necessary and sufficient conditions for the boundedness and compactness of commutators $[b,T]$ from $L^p \to L^q$. Previously, only the $p=q$ upper bound in terms of a Zygmund type little $\operatorname{BMO}$ space was known for general operators, and it appears that there has been some confusion about the corresponding lower bound in recent literature. We give complete characterizations whenever $p \le q$ for a general class of non-degenerate Zygmund type singular integrals. Some of the results are somewhat surprising in view of existing papers - for instance, compactness always forces $b$ to be constant. Even in the simpler situation of bi-parameter singular integrals it appears that this has not been observed previously. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_02497 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On entangled and multi-parameter commutators Li, Kangwei Martikainen, Henri Classical Analysis and ODEs 42B20 We complement the recent theory of general singular integrals $T$ invariant under the Zygmund dilations $(x_1, x_2, x_3) \mapsto (s x_1, tx_2, st x_3)$ by proving necessary and sufficient conditions for the boundedness and compactness of commutators $[b,T]$ from $L^p \to L^q$. Previously, only the $p=q$ upper bound in terms of a Zygmund type little $\operatorname{BMO}$ space was known for general operators, and it appears that there has been some confusion about the corresponding lower bound in recent literature. We give complete characterizations whenever $p \le q$ for a general class of non-degenerate Zygmund type singular integrals. Some of the results are somewhat surprising in view of existing papers - for instance, compactness always forces $b$ to be constant. Even in the simpler situation of bi-parameter singular integrals it appears that this has not been observed previously. |
| title | On entangled and multi-parameter commutators |
| topic | Classical Analysis and ODEs 42B20 |
| url | https://arxiv.org/abs/2412.02497 |