Spectral Asymptotics for Quantized Derivatives on Quantum Euclidean Spaces

We obtain spectral asymptotics for the quantized derivatives of elements from the first-order homogeneous Sobolev space on the quantum Euclidean space, extending an earlier result of McDonald, Sukochev and Xiong (Commun. Math. Phys. 2020). Our approach is based on a noncommutative Wiener-Ikehara Tauberian theorem and a recently developed $C^{\ast }$-algebraic version of pseudo-differential operator theory.