Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators. (English) Zbl 1504.35664
Summary: We generalize the results for Banach algebras of pseudodifferential operators obtained by K. Gröchenig and Z. Rzeszotnik [Ann. Inst. Fourier 58, No. 7, 2279–2314 (2008; Zbl 1168.35050)] to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators [E. Cordero et al., J. Math. Pures Appl. (9) 99, No. 2, 219–233 (2013; Zbl 1306.42045)], which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrödinger equation with bounded perturbations, cf. [E. Cordero, G. Giacchi and L. Rodino, “Wigner analysis of operators. II: Schrödinger equations”, Preprint, arXiv:2208.00505].
MSC:
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 35S30 | Fourier integral operators applied to PDEs |
| 47G30 | Pseudodifferential operators |
| 42C15 | General harmonic expansions, frames |
Keywords:
quasi-Banach algebras; metaplectic operators; modulation spaces; short-time Fourier transform; Wiener algebraReferences:
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