Endpoint Schatten class properties of commutators on quantum tori

Abstract

We study endpoint Schatten class properties of commutators $[I_θ^ε,M_x]$ on quantum tori of dimension $d\geq2$, where $I_θ^ε$ is the Riesz potential of order $ε$ and $M_x$ denotes the left multiplication operator by an operator $x$ on quantum tori $\mathbb T_θ^d$. For $-\frac{d}{2}<ε<1$, $ε\neq0$, and $x\in L_1(\mathbb T_θ^d)$, we prove that $[I_θ^ε,M_x]\in \mathcal L_{\frac{d}{1-ε},\infty}$ if and only if $x\in\dot W_{\frac{d}{1-ε}}^1(\mathbb T_θ^d)$. We also determine the singular value asymptotics of this commutator for $x\in\dot W_{\frac{d}{1-ε}}^1(\mathbb T_θ^d)$. The necessary Sobolev condition holds throughout $1-d<ε<1$, $ε\neq0$. For $ε\leq-\frac{d}{2}$, we characterise Hilbert--Schmidt membership, with a logarithmically weighted Fourier condition at $ε=-\frac{d}{2}$. 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 $C^*$-algebraic principal symbols. This asymptotic result confirms Conjecture~7.1 of McDonald and Ponge (\emph{Adv. Math.} 412 (2023), 108815).