Endpoint Schatten class properties of commutators on quantum Euclidean spaces

We study commutators [(-Δ_θ)^ε2,Mₓ] involving fractional Laplacians on quantum Euclidean spaces associated with arbitrary real antisymmetric matrices $θ$. For d≥2 and - d2<ε≠ 0<1, we characterize critical weak Schatten class membership of [(-Δ_θ)^ε2,Mₓ] for any x as a tempered distribution by Sobolev spaces, with equivalent corresponding norms. We also establish precise asymptotics of singular values for [(-Δ_θ)^ε2,Mₓ]. For this purpose, we extend the Cwikel estimates and the required asymptotics of singular values to arbitrary real antisymmetric matrices, and we also establish a systematic approximation theory of tempered distributions on quantum Euclidean spaces.