Endpoint Schatten class properties of commutators on quantum tori
Abstract
We study endpoint Schatten class properties of commutators [Iθε,Mx] on quantum tori of dimension d≥2, where Iθε is the Riesz potential of order ε and Mx denotes the left multiplication operator by an operator x on quantum tori 𝕋θd. For −d/2<ε<1, ε≠0, and x∈ L1(𝕋θd), we prove that [Iθε,Mx]∈ ℒd/(1−ε),∞ if and only if x∈Ẇd/(1−ε)1(𝕋θd). We also determine the singular value asymptotics of this commutator for x∈Ẇd/(1−ε)1(𝕋θd). The necessary Sobolev condition holds throughout 1−d<ε<1, ε≠0. For ε≤−d/2, we characterise Hilbert--Schmidt membership, with a logarithmically weighted Fourier condition at ε=−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).
Related
- Endpoint Schatten class properties of commutators on quantum Euclidean spaces
- Schatten--Lorentz properties of commutators with matrix weights
- Single commutators in Schatten ideals
- Schatten limits for semi-commutators in Toeplitz quantization on Fock space
- Critical Schatten exponents for the Berger-Coburn endpoint problem
BibTeX
@misc{zhang2026endpoint,
title = {{Endpoint Schatten class properties of commutators on quantum tori}},
author = {Xinyu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16747},
eprint = {2610.16747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16747},
}
RIS
TY - UNPB
TI - Endpoint Schatten class properties of commutators on quantum tori
AU - Xinyu Zhang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16747
UR - https://arxiv.org/abs/2610.16747
AB - We study endpoint Schatten class properties of commutators [I_θ^ε,M_x] on quantum tori of dimension d≥2, 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 𝕋_θ^d. For −d/2<ε<1, ε≠0, and x∈ L_1(𝕋_θ^d), we prove that [I_θ^ε,M_x]∈ ℒ_d/(1−ε),∞ if and only if x∈Ẇ_d/(1−ε)^1(𝕋_θ^d). We also determine the singular value asymptotics of this commutator for x∈Ẇ_d/(1−ε)^1(𝕋_θ^d). The necessary Sobolev condition holds throughout 1−d<ε<1, ε≠0. For ε≤−d/2, we characterise Hilbert--Schmidt membership, with a logarithmically weighted Fourier condition at ε=−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).
ER -