Compactness properties for modulation spaces (Q2279013)

Compactness criteria for ``classical'' modulation spaces are investigated in e.g. [\textit{M. Dörfler} et al., Colloq. Math. 94, No. 1, 37--50 (2002; Zbl 1017.46014); \textit{P. Boggiatto} and \textit{J. Toft}, Appl. Anal. 84, No. 3, 269--282 (2005; Zbl 1074.42010)] and, in a certain sense, in [\textit{M. A. Shubin}, Pseudodifferential operators and spectral theory. (Psevdodifferentsial'nye operatory i spektral'naya teoriya) (Russian). Moskva: Izdatel'stvo ''Nauka'' (1978; Zbl 0451.47064)]. The main results of the paper involve a broad class of modulation spaces thus extending the previous contributions in different ways. Firstly, the authors extend the class of weights by considering moderate weights (instead of usually observed weights of at most polynomial growth at infinity). Another extension is the replacement of the usual Lebesgue integrability condition on the short-time Fourier transform by imposing certain types of translation invariant solid BF-space norms instead. Finally, the usual global condition on the involved weights is relaxed into a suitable local moderate condition. As an application, the authors show how their result can be used to deduce index results and lifting properties for certain pseudo-differential operators (see [\textit{A. Abdeljawad} et al., Anal. Appl., Singap. 18, No. 4, 523--583 (2020; Zbl 1442.35573)]).