On extension of Calderón-Zygmund type singular integrals and their commutators
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916672160399360 |
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| author | Bagchi, Sayan Garg, Rahul Singh, Joydwip |
| author_facet | Bagchi, Sayan Garg, Rahul Singh, Joydwip |
| contents | Motivated by the recent works [Huan Yu, Quansen Jiu, and Dongsheng Li, 2021] and [Yanping Chen and Zihua Guo, 2021], we study the following extension of Calderón-Zygmund type singular integrals $$ T_βf (x) = p.v. \int_{\mathbb{R}^n} \frac{Ω(y)}{|y|^{n-β}} f(x-y) \, dy, $$ for $0 < β< n$, and their commutators. We establish estimates of these singular integrals on Lipschitz spaces, Hardy spaces and Muckenhoupt $A_p$-weighted $L^p$-spaces. We also establish Lebesgue and Hardy space estimates of their commutators. Our estimates are uniform in small $β$, and therefore one can pass onto the limits as $β\to 0$ to deduce analogous estimates for the classical Calderón-Zygmund type singular integrals and their commutators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_12161 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On extension of Calderón-Zygmund type singular integrals and their commutators Bagchi, Sayan Garg, Rahul Singh, Joydwip Classical Analysis and ODEs Analysis of PDEs Functional Analysis Primary 42B20, Secondary 42B35, 42B37 Motivated by the recent works [Huan Yu, Quansen Jiu, and Dongsheng Li, 2021] and [Yanping Chen and Zihua Guo, 2021], we study the following extension of Calderón-Zygmund type singular integrals $$ T_βf (x) = p.v. \int_{\mathbb{R}^n} \frac{Ω(y)}{|y|^{n-β}} f(x-y) \, dy, $$ for $0 < β< n$, and their commutators. We establish estimates of these singular integrals on Lipschitz spaces, Hardy spaces and Muckenhoupt $A_p$-weighted $L^p$-spaces. We also establish Lebesgue and Hardy space estimates of their commutators. Our estimates are uniform in small $β$, and therefore one can pass onto the limits as $β\to 0$ to deduce analogous estimates for the classical Calderón-Zygmund type singular integrals and their commutators. |
| title | On extension of Calderón-Zygmund type singular integrals and their commutators |
| topic | Classical Analysis and ODEs Analysis of PDEs Functional Analysis Primary 42B20, Secondary 42B35, 42B37 |
| url | https://arxiv.org/abs/2204.12161 |