Abstract
The aim of these lectures is to prove results on 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. These results are obtained by constructing a parametrix, with ultradistribution kernel, represented by means of Fourier integral operators of infinite order defined on Gevrey spaces.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aoki T., ‘Calcul exponentiel des opérateurs microdifférentiels d’ordre infini I’, Ann. Inst. Fourier, Grenoble, 33, (1983), 227–250; ‘Calcul exponentiel des opérateurs microdifferéntiels d’or-dre infini II’, to appear.
Bolley P. — Camus J. — Metivier G., ‘Regularité Gevrey et iterés pour une classe d’opérateurs hypoelliptiques’, Rend. Sem. Mat.Univ. Politecn. Torino, numero speciale (1983).
Bony J.M., ‘Propagation des singularitiés differéntiables pour une classe d’opérateurs différentiels à coefficients analytiques’, Astérisque, 34–35 (1976), 43–91.
Bony J.M. — Schapira P., ‘Propagation des singularités analytiques pour les solutions des equations aux dérivées partielles’, Ann. Inst. Fourier, Grenoble, 26 (1976), 81–140.
Boutet de Monvel L., ‘Opérateurs pseudo-différentiels analytiques et opérateurs d’ordre infini’, Ann. Inst. Fourier, Grenoble, 22 (1972), 229–268.
Boutet de Monvel L. — Krée P., ‘Pseudo-differential operators and Gevrey classes’, Ann. Inst. Fourier, Grenoble, 17 (1967), 295–323.
Cattabriga L. — Mari D., ‘Parametrix of infinite order for a Cauchy problem for certain hyperbolic operators’, to appear.
Chazarain I.J., ‘Propagation des singularités pour une classe d’opé rateurs à caractéristiques multiples et résolubilité locale’, Ann. Inst. Fourier, Grenoble, 24 (1974), 209–223.
Duistermaat J.J. — Hörmander L., ‘Fourier integral operators II’, I, Acta Math., 128 (1972), 183–269.
Hashimoto S. — Matsuzawa T. — Morimoto Y., ‘Opérateurs pseudo différentiels et classes de Gevrey’, Comm. Partial Differential Equations, 8 (1983), 1277–1289.
Hörmander L., ‘Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients’, Comm. Pure Appl. Math., 24 (1971), 671–704.
Iftimie V., ‘Opérateurs hypoelliptiques dans des espaces de Gevrey’, Bull. Soc. Sci. Math. R.S. Roumanie, 27 (1983), 317–333.
Ivrii V.Ja., ‘Condizioni di correttezza in classi di Gevrey del problema di Cauchy per operatori non strettamente iperbolici’, Sibirsk. Mat. Z., 17 (1976), 547–563 = Siberian Math. J., 17 (1976), 422–435.
Kajitani K., ‘Leray-Volevich’s system and Gevrey class’, J. Math. Kyoto Univ., 21 (1981), 547–574.
Kessab A., ‘Propagation des singularités Gevrey pour des opérateurs à caractéristiques involutives’, Thése, Université de Paris-Sud, Centre d’Orsay, 1984.
Komatsu H., ‘Ultradistributions, I. Structure theorems and a characterization’, J. Fac. Sci. Univ. Tokyo, Sect. I A Math. 20 (1973), 25–105.
Komatsu H., ‘Ultradistributions, II. The kernel theorem and ultra-distributions with support in a submanifold’, J. Fac. Sci. Univ. Tokyo, Sect. I A Math. 24 (1977), 607–628.
Komatsu H., ‘Linear hyperbolic equations with Gevrey coefficients’, J. Math, pures et appl., 59 (1980), 145–185.
Kumano-go H., Pseudodifferential operators, MIT Press, 1981.
Leray J., ‘Equations hyperboliques non strictes: contre-exemples du type De Giorgi aux théoremes d’existence et d’unicité’, Math. Annalen, 162 (1966), 228–236.
Leray J. — Ohya Y., ‘Systémes linéaires, hyperboliques non-stricts’, Centre Belg. de Rech. Math. Deuxieme Colloq. sur l’Analyse fonctionelle, Liège, 1964, 105–144.
Metivier G., ‘Analytic hypoellipticity for operators with multiple characteristics’, Comm. in Partial Differential Equations, 6 (1981), 1–90.
Mizohata S., ‘Propagation de la régularité au sens de Gevrey pour les opérateurs différentiels a multiplicité constante’, J. Vaillant, Sèminaire Equations aux dérivées partielles hyperboliques et holomorphes, Hermann Paris, 1984.
Mizohata S., ‘On perfect factorizations in Gevrey classes’, personal comunication.
Ohya Y., ‘Le problème de Cauchy pour les équations hyperboliques à caractéristique multiple’, J. Math. Soc. Japan, 16 (1964), 268–286.
Rodino L. — Zanghirati L., ‘Pseudo differential operators with multiple characteristics and Gevrey singularities’, to appear.
Sjöstrand J., ‘Propagation of singularities for operators with multiple involutive characteristics’, Ann. Inst. Fourier, Grenoble, 26 (1976), 141–155.
Talenti G., ‘Un problema di Cauchy’, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (3), 18 (1964), 165–186.
Taniguchi K., ‘Fourier integral operators in Gevrey class on Rn and the fundamental solution for a hyperbolic operator’, Publ. RIMS, Kyoto Univ., 20 (1984), 491–542.
Treves F., Introduction to pseudo differential and Fourier integral operators, vol. I, Plenum Press, 1981.
Volevic L.R., ‘Pseudo-differential operators with holomorphic symbols and Gevrey classes’, Trudy Moskov Mat. Obsc., 24 (1971), 43–68 = Trans. Moscow Math. Soc., 24 (1974), 43–72.
Zanghirati L., ‘Operatori pseudodifferenziali di ordine infinito e classi di Gevrey’, Seminario di Analisi Matematica, Dipartimento di Matematica dell’Università di Bologna, 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 D. Reidel Publishing Company
About this chapter
Cite this chapter
Cattabriga, L., Zanghirati, L. (1986). Fourier Integral Operators of Infinite Order on Gevrey Spaces. Applications to the Cauchy Problem for Hyperbolic Operators. In: Garnir, H.G. (eds) Advances in Microlocal Analysis. NATO ASI Series, vol 168. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4606-4_3
Download citation
DOI: https://doi.org/10.1007/978-94-009-4606-4_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8546-5
Online ISBN: 978-94-009-4606-4
eBook Packages: Springer Book Archive