Endpoint Schatten class properties of commutators

We study trace ideal properties of the commutators [(-Δ)^ϵ/2,M_f] of a power of the Laplacian with the multiplication operator by a function f on ℝ^d. For a certain range of ϵ∈ℝ, we show that this commutator belongs to the weak Schatten class ℒ_d/1-ϵ,∞ if and only if the distributional gradient of f belongs to L_d/1-ϵ. Moreover, in this case we determine the asymptotics of the singular values. Our proofs use, among other things, the tool of Double Operator Integrals.