Wiener algebras of Fourier integral operators

We construct a one-parameter family of algebras FIO(Xi, s), 0 <= s <= infinity, consisting of Fourier integral operators. We derive boundedness results, composition rules, and the spectral invariance of the operators in FIO(Xi, s). 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 = infinity, our 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. (C) 2012 Elsevier Masson SAS. All rights reserved.