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arXiv · 2405.10652

Endpoint Schatten class properties of commutators

Rupert L. Frank·Fedor Sukochev·Dmitriy Zanin
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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.

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Rupert L. Frank, Fedor Sukochev, Dmitriy Zanin. 2024-05-17. Endpoint Schatten class properties of commutators. https://arxiv.org/abs/2405.10652

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