Dyadic analysis of compactness on product spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909541620252672 |
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| author | Cao, Mingming Yabuta, Kôzô |
| author_facet | Cao, Mingming Yabuta, Kôzô |
| contents | We develop the compactness theory of multilinear singular integrals on product spaces using a modern point of view. The first main result is a compact $T1$ theorem for multilinear Calderón--Zygmund operators on product spaces. More specifically, we prove that a multilinear singular integral operator $T$ on product spaces can be extended to a compact multilinear operator from $L^{p_1}(w_1^{p_1}) \times \cdots \times L^{p_m}(w_m^{p_m})$ to $L^p(w^p)$ for all exponents $\frac1p = \sum_{j=1}^m \frac{1}{p_j}>0$ with $p_1, \ldots, p_m \in (1, \infty]$ and for all weights $\vec{w} \in A_{\vec{p}}(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$ if the following hypotheses are satisfied: (H1) $T$ admits a compact full kernel representation, (H2) $T$ admits a compact partial kernel representation, (H3) $T$ satisfies the weak compactness property, (H4) $T$ satisfies the diagonal $\mathrm{CMO}$ condition, and (H5) $T$ satisfies the product $\mathrm{CMO}$ condition. This is a multilinear compact extension of Journé's $T1$ theorem on product spaces. The second main result establishes the mean continuity of commutators $[\boldsymbol{b}, T]_{\boldsymbolα}$ on weighted Lebesgue spaces as above, which can be viewed as a substitution of compactness because the compactness of $[\boldsymbol{b}, T]_{\boldsymbolα}$ is equivalent to $\boldsymbol{b} \equiv \text{constant}$ when $T$ is a non-degenerate bi-parameter singular integral. Our main tools include multilinear bi-parameter dyadic representation, multilinear extrapolation, multilinear interpolation, and Kolmogorov--Riesz compactness criterion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10304 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dyadic analysis of compactness on product spaces Cao, Mingming Yabuta, Kôzô Classical Analysis and ODEs 42B20, 42B35 We develop the compactness theory of multilinear singular integrals on product spaces using a modern point of view. The first main result is a compact $T1$ theorem for multilinear Calderón--Zygmund operators on product spaces. More specifically, we prove that a multilinear singular integral operator $T$ on product spaces can be extended to a compact multilinear operator from $L^{p_1}(w_1^{p_1}) \times \cdots \times L^{p_m}(w_m^{p_m})$ to $L^p(w^p)$ for all exponents $\frac1p = \sum_{j=1}^m \frac{1}{p_j}>0$ with $p_1, \ldots, p_m \in (1, \infty]$ and for all weights $\vec{w} \in A_{\vec{p}}(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$ if the following hypotheses are satisfied: (H1) $T$ admits a compact full kernel representation, (H2) $T$ admits a compact partial kernel representation, (H3) $T$ satisfies the weak compactness property, (H4) $T$ satisfies the diagonal $\mathrm{CMO}$ condition, and (H5) $T$ satisfies the product $\mathrm{CMO}$ condition. This is a multilinear compact extension of Journé's $T1$ theorem on product spaces. The second main result establishes the mean continuity of commutators $[\boldsymbol{b}, T]_{\boldsymbolα}$ on weighted Lebesgue spaces as above, which can be viewed as a substitution of compactness because the compactness of $[\boldsymbol{b}, T]_{\boldsymbolα}$ is equivalent to $\boldsymbol{b} \equiv \text{constant}$ when $T$ is a non-degenerate bi-parameter singular integral. Our main tools include multilinear bi-parameter dyadic representation, multilinear extrapolation, multilinear interpolation, and Kolmogorov--Riesz compactness criterion. |
| title | Dyadic analysis of compactness on product spaces |
| topic | Classical Analysis and ODEs 42B20, 42B35 |
| url | https://arxiv.org/abs/2410.10304 |