Endpoint Schatten class properties of commutators on quantum tori

We study endpoint Schatten class properties of commutators $[I_{\theta }^{\epsilon },M_x]$ on quantum tori of dimension $d\ge 2$, where $I_{\theta }^{\epsilon }$ is the Riesz potential of order $\epsilon$ and $M_x$ denotes the left multiplication operator by an operator $x$ on quantum tori ${\mathbb{T}}_{\theta }^d$. For $-\frac{d}{2}<\epsilon <1$, $\epsilon \mathrm{\ne }0$, and $x\in L_1({\mathbb{T}}_{\theta }^d)$, we prove that $[I_{\theta }^{\epsilon },M_x]\in {\mathcal{L}}_{\frac{d}{1-\epsilon },\mathrm{\infty }}$ if and only if $x\in {\dot{W}}_{\frac{d}{1-\epsilon }}^1({\mathbb{T}}_{\theta }^d)$. We also determine the singular value asymptotics of this commutator for $x\in {\dot{W}}_{\frac{d}{1-\epsilon }}^1({\mathbb{T}}_{\theta }^d)$. The necessary Sobolev condition holds throughout $1-d<\epsilon <1$, $\epsilon \mathrm{\ne }0$. For $\epsilon \le -\frac{d}{2}$, we characterise Hilbert–Schmidt membership, with a logarithmically weighted Fourier condition at $\epsilon =-\frac{d}{2}$. In order to get the singular value asymptotics of commutators, we prove singular value asymptotics for pseudodifferential operators of arbitrary negative order on quantum tori using $C^{\ast }$-algebraic principal symbols. This asymptotic result confirms Conjecture~7.1 of McDonald and Ponge (\emph{Adv. Math.} 412 (2023), 108815).