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 $L_2$-boundedness of the first commutators. We show that the formally analogous assertion fails on every quantum Euclidean space $\R^d_θ$ with $θ\neq 0$. 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 $\mathsf D_n$ and $\mathsf A_n$ such that $\|[\mathsf D_n,\mathsf A_n]\|$ is uniformly bounded, whereas $\|[|\mathsf D_n|,\mathsf A_n]\|$ 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.