Schatten p-class property of pseudodifferential operators with symbols in modulation spaces

Abstract

It is proved that the pseudodifferential operators ${\sigma }_t(X,D)$ belong to the Schatten $p$-class $C_p$, $0<p\le 2$, if the symbol $\sigma (x,\omega )$ is in certain modulation spaces on ${\mathbf{R}}_x^d\times {\mathbf{R}}_{\omega }^d$.