Flag-like singular integrals and associated Hardy spaces on a kind of nilpotent Lie groups of step two
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| Format: | Preprint |
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2024
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| _version_ | 1866909215548768256 |
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| author | Wang, Wei Wu, Qingyan |
| author_facet | Wang, Wei Wu, Qingyan |
| contents | The Cauchy-Szegö singular integral is a fundamental tool in the study of holomorphic $H^p$ Hardy space. But for a kind of Siegel domains, the Cauchy-Szegö kernels are neither product ones nor flag ones on the Shilov boundaries, which have the structure of nilpotent Lie groups $\mathscr N $ of step two. We use the lifting method to investigate flag-like singular integrals on $\mathscr N $, which includes these Cauchy-Szegö ones as a special case. The lifting group is the product $\tilde {\mathscr N }$ of three Heisenberg groups, and naturally geometric or analytical objects on $\mathscr N $ are the projection of those on $\tilde {\mathscr N } $. As in the flag case, we introduce various notions on $\mathscr N $ adapted to geometric feature of these kernels, such as tubes, nontangential regions, tube maximal functions, Littlewood-Paley functions, tents, shards and atoms etc. They have the feature of tri-parameters, although the second step of the group $\mathscr N$ is only $2$-dimensional, i.e. there exists a hidden parameter as in the flag case. We also establish the corresponding Calderón reproducing formula, characterization of $ L ^p (\mathscr N)$ by Littlewood-Paley functions, $ L ^p $-boundedness of tube maximal functions and flag-like singular integrals and atomic decomposition of $H^1$ Hardy space on $ {\mathscr N } $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01453 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flag-like singular integrals and associated Hardy spaces on a kind of nilpotent Lie groups of step two Wang, Wei Wu, Qingyan Functional Analysis Classical Analysis and ODEs The Cauchy-Szegö singular integral is a fundamental tool in the study of holomorphic $H^p$ Hardy space. But for a kind of Siegel domains, the Cauchy-Szegö kernels are neither product ones nor flag ones on the Shilov boundaries, which have the structure of nilpotent Lie groups $\mathscr N $ of step two. We use the lifting method to investigate flag-like singular integrals on $\mathscr N $, which includes these Cauchy-Szegö ones as a special case. The lifting group is the product $\tilde {\mathscr N }$ of three Heisenberg groups, and naturally geometric or analytical objects on $\mathscr N $ are the projection of those on $\tilde {\mathscr N } $. As in the flag case, we introduce various notions on $\mathscr N $ adapted to geometric feature of these kernels, such as tubes, nontangential regions, tube maximal functions, Littlewood-Paley functions, tents, shards and atoms etc. They have the feature of tri-parameters, although the second step of the group $\mathscr N$ is only $2$-dimensional, i.e. there exists a hidden parameter as in the flag case. We also establish the corresponding Calderón reproducing formula, characterization of $ L ^p (\mathscr N)$ by Littlewood-Paley functions, $ L ^p $-boundedness of tube maximal functions and flag-like singular integrals and atomic decomposition of $H^1$ Hardy space on $ {\mathscr N } $. |
| title | Flag-like singular integrals and associated Hardy spaces on a kind of nilpotent Lie groups of step two |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2406.01453 |