Endpoint Schatten class properties of commutators

TL;DR

This paper characterizes when certain commutators involving fractional Laplacians and multiplication operators belong to weak Schatten classes, linking this membership to the integrability of the function's distributional gradient, and analyzes their singular value asymptotics.

Contribution

It establishes a precise criterion for the Schatten class membership of commutators with fractional Laplacians based on the gradient's integrability, using Double Operator Integrals.

Findings

Commutators belong to weak Schatten class if and only if the gradient is in L_{d/(1-ε)}.

Determines asymptotics of singular values for these commutators.

Provides a characterization linking operator ideals to function regularity.

Abstract

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