Singular value asymptotics on compact smooth Riemaniann manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915655634124800 |
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| author | Sukochev, Fedor Yang, Fulin Zanin, Dmitriy |
| author_facet | Sukochev, Fedor Yang, Fulin Zanin, Dmitriy |
| contents | Let $(X,G)$ be a $d$-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator $Δ_{G}$, and let $Π_{X}$ be the $C^{\ast}$-algebra obtained by locally transferring the $C^{\ast}$-algebra generated by multiplication operators and Riesz transforms on $\mathbb{R}^{d}$. Denote ${\rm sym}_{X}$ the principal symbol mapping of $Π_{X}$. For any $S\inΠ_{X}$, we prove that, in the framework of $C^{\ast}$-algebra, \begin{align*} \lim_{t\rightarrow\infty}t^{\frac{1}{p}}μ(t,S(1+Δ_G)^{-\frac{d}{2p}}) =(2π\sqrt[d]{d})^{-\frac{1}{p}}\Big\|{\rm sym}_{X}(S)\Big\|_{L_{p}(T^{\ast}X,e^{-q_{G}}dλ)}, \end{align*} where $0<p<\infty$, $e^{-q_{G}}$ is the canonical weight on $X$, and $dλ$ is the Liouville measure on the cotangent bundle $T^{\ast}X$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_02365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular value asymptotics on compact smooth Riemaniann manifolds Sukochev, Fedor Yang, Fulin Zanin, Dmitriy Functional Analysis 58C40, 46L51, 47G30 Let $(X,G)$ be a $d$-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator $Δ_{G}$, and let $Π_{X}$ be the $C^{\ast}$-algebra obtained by locally transferring the $C^{\ast}$-algebra generated by multiplication operators and Riesz transforms on $\mathbb{R}^{d}$. Denote ${\rm sym}_{X}$ the principal symbol mapping of $Π_{X}$. For any $S\inΠ_{X}$, we prove that, in the framework of $C^{\ast}$-algebra, \begin{align*} \lim_{t\rightarrow\infty}t^{\frac{1}{p}}μ(t,S(1+Δ_G)^{-\frac{d}{2p}}) =(2π\sqrt[d]{d})^{-\frac{1}{p}}\Big\|{\rm sym}_{X}(S)\Big\|_{L_{p}(T^{\ast}X,e^{-q_{G}}dλ)}, \end{align*} where $0<p<\infty$, $e^{-q_{G}}$ is the canonical weight on $X$, and $dλ$ is the Liouville measure on the cotangent bundle $T^{\ast}X$. |
| title | Singular value asymptotics on compact smooth Riemaniann manifolds |
| topic | Functional Analysis 58C40, 46L51, 47G30 |
| url | https://arxiv.org/abs/2512.02365 |