Riesz transforms, Hardy spaces and Campanato spaces associated with Laguerre expansions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910738854969344 |
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| author | Bui, The Anh |
| author_facet | Bui, The Anh |
| contents | Let $ν\in [-1/2,\infty)^n$, $n\ge 1$, and let $\mathcal{L}_ν$ be a self-adjoint extension of the differential operator \[ L_ν:= \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(ν_i^2 - \frac{1}{4})\right] \] on $C_c^\infty(\mathbb{R}_+^n)$ as the natural domain. In this paper, we first prove that the Riesz transform associated with $\mathcal L_ν$ is a Calderón-Zygmund operator, answering the open problem in [JFA, 244 (2007), 399-443]. In addition, we develop the theory of Hardy spaces and Campanato spaces associated with $\mathcal{L}_ν$. As applications, we prove that the Riesz transform related to $\mathcal{L}_ν$ is bounded on these Hardy spaces and Campanato spaces, completing the description of the boundedness of the Riesz transform in the Laguerre expansion setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19403 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Riesz transforms, Hardy spaces and Campanato spaces associated with Laguerre expansions Bui, The Anh Classical Analysis and ODEs 42B30, 42B35 Let $ν\in [-1/2,\infty)^n$, $n\ge 1$, and let $\mathcal{L}_ν$ be a self-adjoint extension of the differential operator \[ L_ν:= \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(ν_i^2 - \frac{1}{4})\right] \] on $C_c^\infty(\mathbb{R}_+^n)$ as the natural domain. In this paper, we first prove that the Riesz transform associated with $\mathcal L_ν$ is a Calderón-Zygmund operator, answering the open problem in [JFA, 244 (2007), 399-443]. In addition, we develop the theory of Hardy spaces and Campanato spaces associated with $\mathcal{L}_ν$. As applications, we prove that the Riesz transform related to $\mathcal{L}_ν$ is bounded on these Hardy spaces and Campanato spaces, completing the description of the boundedness of the Riesz transform in the Laguerre expansion setting. |
| title | Riesz transforms, Hardy spaces and Campanato spaces associated with Laguerre expansions |
| topic | Classical Analysis and ODEs 42B30, 42B35 |
| url | https://arxiv.org/abs/2411.19403 |