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Fourier integral operators of infinite order on Gevrey spaces. Applications to the Cauchy problem for hyperbolic operators. (English) Zbl 0601.35065

Advances in microlocal analysis, Proc. NATO Adv. Study Inst., Castelvecchio-Pascoli (Lucca)/Italy 1985, NATO ASI Ser., Ser. C 168, 41-71 (1986).
[For the entire collection see Zbl 0583.00014.]
The authors study the Gevrey wave front set of the solution of the Cauchy problem with data in spaces of Gevrey ultradistributions for hyperbolic operators with characteristics of constant multiplicity. The main tools in the analysis are Fourier integral operators with amplitudes of infinite order defined on Gevrey spaces.
They begin with the calculus of pseudo-differential operators of infinite order treated in [the second author, ”Operatori pseudodifferenziali di ordine infinito e classi di Gevrey,” Semin. Anal. Dip. Mat. Univ. Bologna (1984)] and develop some properties and calculus rules for infinite order Fourier integral operators. A parametrix with ultradistribution kernel for the Cauchy problem is constructed. Results on the existence and propagation of singularities in Gevrey spaces follow.
Reviewer: P.McCoy

MSC:

35L30 Initial value problems for higher-order hyperbolic equations
58J45 Hyperbolic equations on manifolds
35S05 Pseudodifferential operators as generalizations of partial differential operators
35L67 Shocks and singularities for hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A20 Analyticity in context of PDEs

Citations:

Zbl 0583.00014