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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.10895 |
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| _version_ | 1866910911786123264 |
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| author | Bui, The Anh |
| author_facet | Bui, The Anh |
| contents | Let $ν=(ν_1,\ldots,ν_n)\in (-1,\vc)^n$, $n\ge 1$, 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. The $j$-th partial derivative associated with $L_ν$ is given by \[ δ_{ν_j} = \frac{\partial}{\partial x_j} + x_j-\frac{1}{x_j}\Big(ν_j + \f{1}{2}\Big), \ \ \ \ j=1,\ldots, n. \] In this paper, we investigate the weighted estimates of the higher-order Riesz transforms $δ_ν^k\mathcal L^{-|k|/2}_ν, k\in \mathbb N^n$, where $δ_ν^k=δ_{ν_n}^{k_n}\ldots δ_{ν_1}^{k_1}$. This completes the description of the boundedness of the higher-order Riesz transforms with the full range $ν\in (-1,\vc)^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted norm inequalities of higher-order Riesz transforms associated with Laguerre expansions Bui, The Anh Classical Analysis and ODEs Let $ν=(ν_1,\ldots,ν_n)\in (-1,\vc)^n$, $n\ge 1$, 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. The $j$-th partial derivative associated with $L_ν$ is given by \[ δ_{ν_j} = \frac{\partial}{\partial x_j} + x_j-\frac{1}{x_j}\Big(ν_j + \f{1}{2}\Big), \ \ \ \ j=1,\ldots, n. \] In this paper, we investigate the weighted estimates of the higher-order Riesz transforms $δ_ν^k\mathcal L^{-|k|/2}_ν, k\in \mathbb N^n$, where $δ_ν^k=δ_{ν_n}^{k_n}\ldots δ_{ν_1}^{k_1}$. This completes the description of the boundedness of the higher-order Riesz transforms with the full range $ν\in (-1,\vc)^n$. |
| title | Weighted norm inequalities of higher-order Riesz transforms associated with Laguerre expansions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2504.10895 |