Shifted wave equation on noncompact symmetric spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911671305371648 |
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| author | Kuznetsova, Yulia Song, Zhipeng |
| author_facet | Kuznetsova, Yulia Song, Zhipeng |
| contents | Let $G$ be a semisimple, connected, and noncompact Lie group with a finite center. We carry out a detailed analysis of oscillating integrals involving the Harish-Chandra $c$-function, in the case of real rank $l\ge 2$. This allows to obtain two main applications. Consider the Laplace-Beltrami operator $Δ$ on the homogeneous space $G/K=S$ by a maximal compact subgroup $K$. We obtain pointwise estimates for the kernel of an oscillating function $\exp( it\sqrt{|x|}) ψ(\sqrt{|x|}) $ applied to the shifted Laplacian $Δ+|ρ|^2$. We obtain a polynomial decay in time of the kernel, and of the $L^p-L^q$ norms of the operator, for $1\le p<2<q\le \infty$. For the related distinguished Laplacian, we obtain bounds for the $L^p-L^p$ norms, $1\le p\le\infty$, with a slower growth in time than predicted by earlier results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21479 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shifted wave equation on noncompact symmetric spaces Kuznetsova, Yulia Song, Zhipeng Analysis of PDEs Functional Analysis Representation Theory 43A85, 22E30, 42B15, 35L05, 43A90 Let $G$ be a semisimple, connected, and noncompact Lie group with a finite center. We carry out a detailed analysis of oscillating integrals involving the Harish-Chandra $c$-function, in the case of real rank $l\ge 2$. This allows to obtain two main applications. Consider the Laplace-Beltrami operator $Δ$ on the homogeneous space $G/K=S$ by a maximal compact subgroup $K$. We obtain pointwise estimates for the kernel of an oscillating function $\exp( it\sqrt{|x|}) ψ(\sqrt{|x|}) $ applied to the shifted Laplacian $Δ+|ρ|^2$. We obtain a polynomial decay in time of the kernel, and of the $L^p-L^q$ norms of the operator, for $1\le p<2<q\le \infty$. For the related distinguished Laplacian, we obtain bounds for the $L^p-L^p$ norms, $1\le p\le\infty$, with a slower growth in time than predicted by earlier results. |
| title | Shifted wave equation on noncompact symmetric spaces |
| topic | Analysis of PDEs Functional Analysis Representation Theory 43A85, 22E30, 42B15, 35L05, 43A90 |
| url | https://arxiv.org/abs/2504.21479 |