Singular value asymptotics on compact smooth Riemaniann manifolds

Abstract

Let $(X,G)$ be a $d$-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator $\hat{I}{”}_G$, and let ${\hat{I}}_X$ 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 ${\mathrm{sym}}_X$ the principal symbol mapping of ${\hat{I}}_X$. For any $S\in {\hat{I}}_X$, 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 $d\hat{I}»$ is the Liouville measure on the cotangent bundle $T^{\ast }X$.