On the kernel conditions of operators mapping atoms to molecules in local Hardy spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866912532709507072 |
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| author | Lau, Chun Ho Vasconcelos, Claudio |
| author_facet | Lau, Chun Ho Vasconcelos, Claudio |
| contents | In this paper, we explore the relationship between the operators mapping atoms to molecules in local Hardy spaces $h^p(\mathbb{R}^n)$ and the size conditions of its kernel. In particular, we show that if the kernel of a Calderón--Zygmund-type operator satisfies an integral-type size condition and a $T^*-$type cancellation, then the operator maps $h^p(\mathbb{R}^n)$ atoms to molecules. On the other hand, assuming that $T$ is an integral type operator bounded on $L^2(\mathbb{R}^n)$ that maps atoms to molecules in $h^p(\mathbb{R}^n)$, then the kernel of such operator satisfies the same integral-type size conditions. We also provide the $L^1(\mathbb{R}^n)$ to $L^{1,\infty}(\mathbb{R}^n)$ boundedness for such operators connecting our integral-type size conditions on the kernel with others presented in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_04604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the kernel conditions of operators mapping atoms to molecules in local Hardy spaces Lau, Chun Ho Vasconcelos, Claudio Classical Analysis and ODEs Functional Analysis 42B30, 42B35, 42B20 In this paper, we explore the relationship between the operators mapping atoms to molecules in local Hardy spaces $h^p(\mathbb{R}^n)$ and the size conditions of its kernel. In particular, we show that if the kernel of a Calderón--Zygmund-type operator satisfies an integral-type size condition and a $T^*-$type cancellation, then the operator maps $h^p(\mathbb{R}^n)$ atoms to molecules. On the other hand, assuming that $T$ is an integral type operator bounded on $L^2(\mathbb{R}^n)$ that maps atoms to molecules in $h^p(\mathbb{R}^n)$, then the kernel of such operator satisfies the same integral-type size conditions. We also provide the $L^1(\mathbb{R}^n)$ to $L^{1,\infty}(\mathbb{R}^n)$ boundedness for such operators connecting our integral-type size conditions on the kernel with others presented in the literature. |
| title | On the kernel conditions of operators mapping atoms to molecules in local Hardy spaces |
| topic | Classical Analysis and ODEs Functional Analysis 42B30, 42B35, 42B20 |
| url | https://arxiv.org/abs/2503.04604 |