A compact $T1$ theorem for Calderón-Zygmund operators associated with Zygmund dilations
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| Format: | Preprint |
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2023
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| author | Cao, Mingming Chen, Jiao Li, Zhengyang Liao, Fanghui Yabuta, Kôzô Zhang, Juan |
| author_facet | Cao, Mingming Chen, Jiao Li, Zhengyang Liao, Fanghui Yabuta, Kôzô Zhang, Juan |
| contents | We develop a compact version of $T1$ theorem for singular integrals of Zygmund type on $\mathbb{R}^3$. More specifically, if a $(D_θ, δ_1, δ_{2, 3})$-Calderón-Zygmund operator $T$ associated with Zygmund dilations admits the compact full and partial kernel representations, and satisfies the weak compactness property and the cancellation condition, then $T$ can be extended to a compact operator on $L^p(w)$ whenever (i) $p \in (1, \infty)$, $w \in A_{p, \mathcal{R}}$, and $θ, δ_1, δ_{2, 3} \in (0, 1]$, or (ii) $p \in (1, \infty)$, $w \in A_{p, \mathcal{Z}}$, $θ= δ_1 = 1$, and $δ_{2, 3} \in (0, 1]$. Here $A_{p, \mathcal{R}}$ and $A_{p, \mathcal{Z}}$ respectively denote the class of of strong $A_p$ weights and the class of Zygmund $A_p$ weights. Beyond that, under similar bilinear assumptions, we prove bilinear Calderón-Zygmund operators associated with Zygmund dilations are compact from $L^{p_1}(\mathbb{R}^3) \times L^{p_2}(\mathbb{R}^3)$ to $L^p(\mathbb{R}^3)$ for all $p_1, p_2 \in (1, \infty)$, where $\frac1p = \frac{1}{p_1} + \frac{1}{p_2}$. The core of the proof is a compact dyadic representation, which asserts that under the hypotheses above, a (bilinear) Calderón-Zygmund operator associated with Zygmund dilations can be represented an average of some compact (bilinear) dyadic shifts of Zygmund nature. This further deepens our understanding of the compactness of singular integral operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_13932 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A compact $T1$ theorem for Calderón-Zygmund operators associated with Zygmund dilations Cao, Mingming Chen, Jiao Li, Zhengyang Liao, Fanghui Yabuta, Kôzô Zhang, Juan Classical Analysis and ODEs 42B20, 42B25 We develop a compact version of $T1$ theorem for singular integrals of Zygmund type on $\mathbb{R}^3$. More specifically, if a $(D_θ, δ_1, δ_{2, 3})$-Calderón-Zygmund operator $T$ associated with Zygmund dilations admits the compact full and partial kernel representations, and satisfies the weak compactness property and the cancellation condition, then $T$ can be extended to a compact operator on $L^p(w)$ whenever (i) $p \in (1, \infty)$, $w \in A_{p, \mathcal{R}}$, and $θ, δ_1, δ_{2, 3} \in (0, 1]$, or (ii) $p \in (1, \infty)$, $w \in A_{p, \mathcal{Z}}$, $θ= δ_1 = 1$, and $δ_{2, 3} \in (0, 1]$. Here $A_{p, \mathcal{R}}$ and $A_{p, \mathcal{Z}}$ respectively denote the class of of strong $A_p$ weights and the class of Zygmund $A_p$ weights. Beyond that, under similar bilinear assumptions, we prove bilinear Calderón-Zygmund operators associated with Zygmund dilations are compact from $L^{p_1}(\mathbb{R}^3) \times L^{p_2}(\mathbb{R}^3)$ to $L^p(\mathbb{R}^3)$ for all $p_1, p_2 \in (1, \infty)$, where $\frac1p = \frac{1}{p_1} + \frac{1}{p_2}$. The core of the proof is a compact dyadic representation, which asserts that under the hypotheses above, a (bilinear) Calderón-Zygmund operator associated with Zygmund dilations can be represented an average of some compact (bilinear) dyadic shifts of Zygmund nature. This further deepens our understanding of the compactness of singular integral operators. |
| title | A compact $T1$ theorem for Calderón-Zygmund operators associated with Zygmund dilations |
| topic | Classical Analysis and ODEs 42B20, 42B25 |
| url | https://arxiv.org/abs/2307.13932 |