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 $\Delta_{G}$, and let $\Pi_{X}$ 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 ${\rm sym}_{X}$ the principal symbol mapping of $\Pi_{X}$. For any $S\in\Pi_{X}$, we prove that, in the framework of $C^{\ast}$-algebra, \begin{align*} \lim_{t\rightarrow\infty}t^{\frac{1}{p}}\mu(t,S(1+\Delta_G)^{-\frac{d}{2p}}) =(2\pi\sqrt[d]{d})^{-\frac{1}{p}}\Big\|{\rm sym}_{X}(S)\Big\|_{L_{p}(T^{\ast}X,e^{-q_{G}}d\lambda)}, \end{align*} where $0<p<\infty$, $e^{-q_{G}}$ is the canonical weight on $X$, and $d\lambda$ is the Liouville measure on the cotangent bundle $T^{\ast}X$.