On the convergence of some products of Fourier integral operators

Article type: Research Article

Authors: Le Rousseau, Jérôme

Affiliations: Laboratoire d'Analyse Topologie Probabilités, CNRS UMR 6632, Université d'Aix-Marseille I, France E-mail: jlerous@cmi.univ-mrs.fr

Abstract: An approximation Ansatz for the operator solution, U(z′,z), of a hyperbolic first-order pseudodifferential equation, ∂z+a(z,x,Dx) with Re(a)≥0, is constructed as the composition of global Fourier integral operators with complex phases. The symbol a(z,·) is assumed to have a regularity as low as Hölder, 𝒞0,α, with respect to the evolution parameter z. We prove a convergence result for the Ansatz to U(z′,z) in some Sobolev space as the number of operators in the composition goes to ∞, with a convergence of order α. We also study the consequences of some truncation approximations of the symbol a(z,·) in the construction of the Ansatz.

Journal: Asymptotic Analysis, vol. 51, no. 3-4, pp. 189-207, 2007