Calderón Commutators on Quantum Euclidean Spaces and Quantum Tori
Abstract
In the classical setting, Calderón's first commutator theorem states that Lipschitz regularity of the symbol characterizes the L2-boundedness of the first commutators. We show that the formally analogous assertion fails on every quantum Euclidean space Rdθ with θ≠ 0. The same simultaneous failure occurs on Tθd whenever θ has an irrational entry. For rational θ, we obtain the optimal logarithmic bound for the corresponding Calderón first commutator. We also discuss the connection between Calderón's first commutator and the T1 theorem, and construct an operator-valued counterexample to Calderón's first commutator theorem, which shows that there is no direct analogue of the classical T1 theorem in the operator-valued setting. The counterexamples and the optimality argument rely on the construction of sequences of matrices Dn and An such that |[Dn,An]| is uniformly bounded, whereas |[|Dn|,An]| tends to infinity. This phenomenon reflects the failure of the absolute value function to be operator Lipschitz with respect to the operator norm. The logarithmic estimate, on the other hand, is obtained by combining matrix-valued T1 theory with a transference argument.
Related
- Endpoint Schatten class properties of commutators on quantum Euclidean spaces
- Endpoint Schatten class properties of commutators on quantum tori
- The Calderón problem for near-Euclidean metrics
- Local rigidity of conformally Euclidean metrics for the anisotropic Calderón problem
- The anisotropic Calderón problem: rigidity near the Euclidean metric
BibTeX
@misc{chen2026calderón,
title = {{Calderón Commutators on Quantum Euclidean Spaces and Quantum Tori}},
author = {Deyu Chen and Guixiang Hong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24881},
eprint = {2610.24881},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24881},
}
RIS
TY - UNPB
TI - Calderón Commutators on Quantum Euclidean Spaces and Quantum Tori
AU - Deyu Chen
AU - Guixiang Hong
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24881
UR - https://arxiv.org/abs/2610.24881
AB - In the classical setting, Calderón's first commutator theorem states that Lipschitz regularity of the symbol characterizes the L_2-boundedness of the first commutators. We show that the formally analogous assertion fails on every quantum Euclidean space R^d_θ with θ≠ 0. The same simultaneous failure occurs on T_θ^d whenever θ has an irrational entry. For rational θ, we obtain the optimal logarithmic bound for the corresponding Calderón first commutator. We also discuss the connection between Calderón's first commutator and the T1 theorem, and construct an operator-valued counterexample to Calderón's first commutator theorem, which shows that there is no direct analogue of the classical T1 theorem in the operator-valued setting. The counterexamples and the optimality argument rely on the construction of sequences of matrices D_n and A_n such that |[D_n,A_n]| is uniformly bounded, whereas |[|D_n|,A_n]| tends to infinity. This phenomenon reflects the failure of the absolute value function to be operator Lipschitz with respect to the operator norm. The logarithmic estimate, on the other hand, is obtained by combining matrix-valued T1 theory with a transference argument.
ER -