Schatten \(p\)-class property of pseudodifferential operators with symbols in modulation spaces (Q2880838)

The authors consider pseudodifferential operators of the kind NEWLINE\[NEWLINE \sigma_t(X,D) f(x) = \int \int_{\mathbb{R}^2d} \widehat{\sigma}(\xi,\eta) e^{2 \pi i (x+t\eta) \xi} f(\eta+x)\, d\xi\, d\eta,NEWLINE\]NEWLINE where the symbol \(\sigma\) is a suitable function or distribution. The \(\sigma_t\)-pseudodifferential calculus covers both the Kohn-Nirenberg (\(t=0\)) and the Weyl (\(t=1/2\)) correspondences. The authors study sufficient criteria on \(\sigma\) to guarantee that \(\sigma_t(X,D)\) is in the \(p\)-Schatten class for \(0 < p \leq 2\). For this purpose, they employ the modulation spaces \( M_m^{r,r}(\mathbb{R}^{2d})\), where \(m\) is a weight function and \(r>0\) a suitable exponent.NEWLINENEWLINEThe chief results show sufficiency of the following conditions: \(\sigma \in M^{2,2}_m\), for a suitably chosen weight \(m\) depending on \(p\) and \(t\), or \(\sigma \in M^{p,p}(\mathbb{R}^{2d})\).NEWLINENEWLINEThe results generalize and extend previously known results in this domain. In addition, the paper introduces a new approach to the estimation of Schatten class norms via frames, by adapting a technique using orthonormal bases due to \textit{C. A. McCarthy} [Isr. J. Math. 5, 249--271 (1967; Zbl 0156.37902)].