Bounds on pseudodifferential operators and Fourier restriction for Schatten classes

2024/12/08 by Detlef Müller, Müller, Detlef
Mathematics · #42B10 #43A30 #47B10 #47G30 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2412.05879

openalex publication_date 2024/12/08 · openalex created_date 2024/12/12 · openalex updated_date 2026/07/28

Abstract

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.

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