Wiener algebras of Fourier integral operators (Q1935828)

The authors construct a one-parameter family of algebras \(FIO(\Xi,s), 0\leq s\leq \infty\), consisting of Fourier integral operators. Boundedness results, composition rules, and the spectral invariance of the operators in \(FIO(\Xi,s)\) are obtained. The operator algebra is defined by the decay properties of an associated Gabor matrix around the graph of the canonical transformation. In particular, for the limit case \(s=\infty\), the Gabor technique provides a new approach to the analysis of \(S^0_{0,0}\)-type Fourier integral operators, for which the global calculus represents a still open relevant problem.