Endpoint Schatten class properties of commutators
Abstract
We study the trace ideal properties of the commutators [(−Δ)^(𝜖/2),𝑀բ] 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.
Copyright and License
© 2024 The Author(s). Published by Elsevier Under a Creative Commons license.
Acknowledgement
Partial support through US National Science Foundation grant DMS-1954995 (RLF), the Deutsche Forschungsgemeinschaft (German Research Foundation) through Germany's Excellence Strategy EXC-2111-390814868 (RLF), and through Australian Research Council, Laureate Fellowship FL170100052 (FS) and DP230100434 (DZ) is acknowledged.
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Additional details
- National Science Foundation
- DMS-1954995
- Deutsche Forschungsgemeinschaft
- EXC-2111-390814868
- Australian Research Council
- FL170100052
- Australian Research Council
- DP230100434