Singular value asymptotics on compact smooth Riemannian manifolds

Fedor Sukochev; Fulin Yang; Dmitriy Zanin

doi:10.4171/jncg/683

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Singular value asymptotics on compact smooth Riemannian manifolds

Fedor Sukochev
UNSW, Kensington, Australia
Fulin Yang
Beijing Institute of Mathematical Sciences and Applications, P. R. China; Tsinghua University, Beijing, P. R. China
Dmitriy Zanin
Central South University, Changsha, P. R. China
- zbMATH
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Abstract

Let $(X, G)$ be a $d$ -dimensional compact smooth Riemannian manifold equipped with Laplace–Beltrami operator $Δ_{G}$ , and let $Π_{X}$ be the $C^{*}$ -algebra obtained by locally transferring the $C^{*}$ -algebra generated by multiplication operators and Riesz transforms on $R^{d}$ . Denote by $sym_{X}$ the principal symbol mapping of $Π_{X}$ . For any $S \in Π_{X}$ , we prove that, in the framework of $C^{*}$ -algebra,

t \to \infty lim t^{\frac{1}{p}} μ (t, S (1 + Δ_{G})^{- \frac{d}{2 p}}) = (2 π d d)^{- \frac{1}{p}} ∥ sym_{X} (S) ∥_{L_{p} (T^{*} X, e^{- q_{G}} d λ)},

where $0 < p < \infty$ , $e^{- q_{G}}$ is the canonical weight on $X$ , and $d λ$ is the Liouville measure on the cotangent bundle $T^{*} X$ .

Cite this article

Fedor Sukochev, Fulin Yang, Dmitriy Zanin, Singular value asymptotics on compact smooth Riemannian manifolds. J. Noncommut. Geom. (2026), published online first

DOI 10.4171/JNCG/683