Endpoint Schatten class properties of commutators

Abstract

We study trace ideal properties of the commutators $[(-\hat{I}”{)}^{\frac{\hat{I}}{\mu }2},M_f]$ of a power of the Laplacian with the multiplication operator by a function $f$ on ${\mathbb{R}}^d$. For a certain range of $\hat{I}\mu \in \mathbb{R}$, we show that this commutator belongs to the weak Schatten class ${\mathcal{L}}_{\frac{d}{1-\hat{I}\mu },\infty }$ if and only if the distributional gradient of $f$ belongs to $L_{\frac{d}{1-\hat{I}\mu }}$. Moreover, in this case we determine the asymptotics of the singular values. Our proofs use, among other things, the tool of Double Operator Integrals.

41 pages