Spectral projection operators of the Sub-Laplacian and Laguerre calculus on non-degenerate nilpotent Lie groups of step two
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2024
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| _version_ | 1866914740968620032 |
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| author | Kang, Qianqian Chang, Der-Chen Wang, Wei |
| author_facet | Kang, Qianqian Chang, Der-Chen Wang, Wei |
| contents | In this paper, we introduce the spectral projection operators $\mathbb{P}_m$ on non-degenerate nilpotent Lie groups $\mathcal{N}$ of step two, associated to the joint spectrum of sub-Laplacian and derivatives in step two. We construct their kernels $P_m(\mathbf{y},\mathbf{t})$ by using Laguerre calculus and find a simple integral representation formula for $\mathbf{y}\neq 0$. Then we show the kernels are Lipschitzian homogeneous functions on $\mathcal{N}\setminus \{\mathbf{0}\}$ by analytic continuation. Moreover, they are shown to be Calderón-Zygmund kernels, so that the spectral projection operator $\mathbb{P}_m$ can be extended to a bounded operator from $L^p(\mathcal{N})$ to itself. We also prove a convergence theorem of the Abel sum $\lim _{R \rightarrow 1^-} \sum_{m=0}^{\infty} R^{m}\mathbb{P}_{m}ϕ=ϕ$ by estimating the $L^p(\mathcal{N})$-norms of $\mathbb{P}_m$. Furthermore, $\mathbb{P}_m$ are mutually orthogonal projection operators and $\sum_{m=0}^{\infty} \mathbb{P}_{m}ϕ=ϕ$ for $ϕ\in L^2(\mathcal{N})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_03378 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral projection operators of the Sub-Laplacian and Laguerre calculus on non-degenerate nilpotent Lie groups of step two Kang, Qianqian Chang, Der-Chen Wang, Wei Functional Analysis In this paper, we introduce the spectral projection operators $\mathbb{P}_m$ on non-degenerate nilpotent Lie groups $\mathcal{N}$ of step two, associated to the joint spectrum of sub-Laplacian and derivatives in step two. We construct their kernels $P_m(\mathbf{y},\mathbf{t})$ by using Laguerre calculus and find a simple integral representation formula for $\mathbf{y}\neq 0$. Then we show the kernels are Lipschitzian homogeneous functions on $\mathcal{N}\setminus \{\mathbf{0}\}$ by analytic continuation. Moreover, they are shown to be Calderón-Zygmund kernels, so that the spectral projection operator $\mathbb{P}_m$ can be extended to a bounded operator from $L^p(\mathcal{N})$ to itself. We also prove a convergence theorem of the Abel sum $\lim _{R \rightarrow 1^-} \sum_{m=0}^{\infty} R^{m}\mathbb{P}_{m}ϕ=ϕ$ by estimating the $L^p(\mathcal{N})$-norms of $\mathbb{P}_m$. Furthermore, $\mathbb{P}_m$ are mutually orthogonal projection operators and $\sum_{m=0}^{\infty} \mathbb{P}_{m}ϕ=ϕ$ for $ϕ\in L^2(\mathcal{N})$. |
| title | Spectral projection operators of the Sub-Laplacian and Laguerre calculus on non-degenerate nilpotent Lie groups of step two |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2404.03378 |