Bounds on pseudodifferential operators and Fourier restriction for Schatten classes

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 $\mu ,$ classical Fourier restriction estimates with respect to the measure $\mu$ are equivalent to quantum restriction estimates for the Fourier-Wigner transform for Schatten classes.