Singular value asymptotics on compact smooth Riemaniann manifolds

Let $(X,G)$ be a $d$-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator ${\mathrm{\Delta }}_G$, and let ${\mathrm{\Pi }}_X$ 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 ${\mathrm{s}\mathrm{y}\mathrm{m}}_X$ the principal symbol mapping of ${\mathrm{\Pi }}_X$. For any $S\in {\mathrm{\Pi }}_X$, 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 $0<p<\mathrm{\infty }$, $e^{-q_G}$ is the canonical weight on $X$, and $d\lambda$ is the Liouville measure on the cotangent bundle $T^{\ast }X$.