Weighted norm inequalities of some singular integrals 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_ | 1866929608588263424 |
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| author | Bui, The Anh |
| author_facet | Bui, The Anh |
| contents | Let $ν=(ν_1,\ldots,ν_n)\in (-1,\infty)^n$, $n\ge 1$, 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 investigate the weighted estimates of singular integrals in the Laguerre setting including the maximal function, the Riesz transform and the square functions associated to the Laguerre operator $\mathcal L_ν$. In the special case of the Riesz transform, the paper completes the description of the Riesz transform for the full range of $ν\in (-1,\infty)^n$ which significant improves the result in [J. Funct. Anal. 244 (2007), 399--443] for $ν_i\ge -1/2, ν_i\notin (-1/2,1/2)$ for $i=1,\ldots, n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19404 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted norm inequalities of some singular integrals associated with Laguerre expansions Bui, The Anh Classical Analysis and ODEs 42B30, 42B35 Let $ν=(ν_1,\ldots,ν_n)\in (-1,\infty)^n$, $n\ge 1$, 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 investigate the weighted estimates of singular integrals in the Laguerre setting including the maximal function, the Riesz transform and the square functions associated to the Laguerre operator $\mathcal L_ν$. In the special case of the Riesz transform, the paper completes the description of the Riesz transform for the full range of $ν\in (-1,\infty)^n$ which significant improves the result in [J. Funct. Anal. 244 (2007), 399--443] for $ν_i\ge -1/2, ν_i\notin (-1/2,1/2)$ for $i=1,\ldots, n$. |
| title | Weighted norm inequalities of some singular integrals associated with Laguerre expansions |
| topic | Classical Analysis and ODEs 42B30, 42B35 |
| url | https://arxiv.org/abs/2411.19404 |