Endpoint Schatten class properties of commutators

We study trace ideal properties of the commutators [(-Δ)ε/2,Mf] of a power of the Laplacian with the multiplication operator by a function f on Rᵈ. For a certain range of ε∈ R, we show that this commutator belongs to the weak Schatten class L_ d1-ε,∞ if and only if the distributional gradient of f belongs to L_ d1-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.