On the compactness of the bi-commutator
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929337972817920 |
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| author | Martikainen, Henri Oikari, Tuomas |
| author_facet | Martikainen, Henri Oikari, Tuomas |
| contents | We prove compactness results and characterizations for the bi-commutator $[T_1,[b, T_2]]$ of a symbol $b$ and two non-degenerate Calderón-Zygmund singular integral operators $T_1, T_2$. Our strategy for proving sufficient conditions for compactness is to first establish them in the mixed-norm $L^{p_1}L^{p_2}\to L^{q_1}L^{q_2}$ off-diagonal case with $p_i < q_i$, and then extend these to other exponents, including the diagonal $p_i = q_i$, with a new extrapolation argument. In particular, the natural product $\operatorname{VMO}$ condition is obtained as a sufficient condition in the diagonal. A full characterization is obtained, both in terms of a vanishing mean oscillation type condition and in terms of the approximability of the symbol, whenever the inequality $p_i \le q_i$ is strict for at least one index. The extrapolation scheme for proving sufficiency requires us to prove new approximation results in relevant bi-parameter function spaces that are of independent interest. The necessity results are obtained by carefully combining recent rectangular approximate weak factorization methods with a classical idea of Uchiyama. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05460 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the compactness of the bi-commutator Martikainen, Henri Oikari, Tuomas Classical Analysis and ODEs 42B20, 42B35, 41A29, 26B35, We prove compactness results and characterizations for the bi-commutator $[T_1,[b, T_2]]$ of a symbol $b$ and two non-degenerate Calderón-Zygmund singular integral operators $T_1, T_2$. Our strategy for proving sufficient conditions for compactness is to first establish them in the mixed-norm $L^{p_1}L^{p_2}\to L^{q_1}L^{q_2}$ off-diagonal case with $p_i < q_i$, and then extend these to other exponents, including the diagonal $p_i = q_i$, with a new extrapolation argument. In particular, the natural product $\operatorname{VMO}$ condition is obtained as a sufficient condition in the diagonal. A full characterization is obtained, both in terms of a vanishing mean oscillation type condition and in terms of the approximability of the symbol, whenever the inequality $p_i \le q_i$ is strict for at least one index. The extrapolation scheme for proving sufficiency requires us to prove new approximation results in relevant bi-parameter function spaces that are of independent interest. The necessity results are obtained by carefully combining recent rectangular approximate weak factorization methods with a classical idea of Uchiyama. |
| title | On the compactness of the bi-commutator |
| topic | Classical Analysis and ODEs 42B20, 42B35, 41A29, 26B35, |
| url | https://arxiv.org/abs/2405.05460 |