Bounds on pseudodifferential operators and Fourier restriction for Schatten classes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913616937091072 |
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| author | Müller, Detlef |
| author_facet | Müller, Detlef |
| contents | As main result, we show that a pseudodifferential operator in the Weyl calculus, whose symbol has compact Fourier support, lies in the Schatten class $\mathcal S^p$ if and only if its symbol lies in the Lebesgue space $L^p$ on phase space.
As an immediate consequence, this gives an alternative and very lucid proof of a recent result by Luef and Samuelsen, who had discovered that for compactly supported measures $μ,$ classical Fourier restriction estimates with respect to the measure $μ$ are equivalent to quantum restriction estimates for the Fourier-Wigner transform for Schatten classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05879 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on pseudodifferential operators and Fourier restriction for Schatten classes Müller, Detlef Classical Analysis and ODEs 42B10, 47G30, 47B10, 43A30 As main result, we show that a pseudodifferential operator in the Weyl calculus, whose symbol has compact Fourier support, lies in the Schatten class $\mathcal S^p$ if and only if its symbol lies in the Lebesgue space $L^p$ on phase space. As an immediate consequence, this gives an alternative and very lucid proof of a recent result by Luef and Samuelsen, who had discovered that for compactly supported measures $μ,$ classical Fourier restriction estimates with respect to the measure $μ$ are equivalent to quantum restriction estimates for the Fourier-Wigner transform for Schatten classes. |
| title | Bounds on pseudodifferential operators and Fourier restriction for Schatten classes |
| topic | Classical Analysis and ODEs 42B10, 47G30, 47B10, 43A30 |
| url | https://arxiv.org/abs/2412.05879 |