Spectral asymptotics for quantized derivatives on quantum Euclidean spaces

Abstract

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–Xiong (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.