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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.15758 |
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| _version_ | 1866917638752436224 |
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| author | Hu, Guoen Lai, Xudong Tao, Xiangxing Xue, Qingying |
| author_facet | Hu, Guoen Lai, Xudong Tao, Xiangxing Xue, Qingying |
| contents | In this paper, the authors consider the endpoint estimates for the maximal Calderón commutator defined by $$T_{Ω,\,a}^*f(x)=\sup_{ε>0}\Big|\int_{|x-y|>ε}\frac{Ω(x-y)}{|x-y|^{d+1}} \big(a(x)-a(y)\big)f(y)dy\Big|,$$ where $Ω$ is homogeneous of degree zero, integrable on $S^{d-1}$ and has vanishing moment of order one, $a$ be a function on $\mathbb{R}^d$ such that $\nabla a\in L^{\infty}(\mathbb{R}^d)$. The authors prove that if $Ω\in L\log L(S^{d-1})$, then $T^*_{Ω,\,a}$ satisfies an endpoint estimate of $L\log\log L$ type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An endpoint estimate for the maximal Calderón commutator with rough kernel Hu, Guoen Lai, Xudong Tao, Xiangxing Xue, Qingying Classical Analysis and ODEs 42B20 In this paper, the authors consider the endpoint estimates for the maximal Calderón commutator defined by $$T_{Ω,\,a}^*f(x)=\sup_{ε>0}\Big|\int_{|x-y|>ε}\frac{Ω(x-y)}{|x-y|^{d+1}} \big(a(x)-a(y)\big)f(y)dy\Big|,$$ where $Ω$ is homogeneous of degree zero, integrable on $S^{d-1}$ and has vanishing moment of order one, $a$ be a function on $\mathbb{R}^d$ such that $\nabla a\in L^{\infty}(\mathbb{R}^d)$. The authors prove that if $Ω\in L\log L(S^{d-1})$, then $T^*_{Ω,\,a}$ satisfies an endpoint estimate of $L\log\log L$ type. |
| title | An endpoint estimate for the maximal Calderón commutator with rough kernel |
| topic | Classical Analysis and ODEs 42B20 |
| url | https://arxiv.org/abs/2403.15758 |