×
Bony, J.-M.; Lerner, N.

Quantification asymptotique et microlocalisations d’ordre supérieur. I. (French) Zbl 0753.35005

Ann. Sci. Éc. Norm. Supér. (4) 22, No. 3, 377-433 (1989).
This paper presents a complete and deep investigation of the so called \(k\)-microlocalization. The authors develop a microlocal calculus of higher order studying \(k\)-microdifferential operators, \(k\)-microlocal regularity and other geometric and analytic topics of the microlocal analysis generalizing the usual (first order) microlocalization. The exposition is technical but the authors have the intention to build a \(k\)-microlocal calculus which can be easily applied to the analysis of different problems and in particular for the examination of the propagation of singularities for nonlinear problems. The previous second and third order microlocalizations are considered as examples and the link with other higher order microlocalizations is discussed.
Reviewer: C.M.Brauner (Bordeaux)

Cited in 3 Reviews
Cited in 45 Documents

MSC:

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35L67 Shocks and singularities for hyperbolic equations

Keywords:

\(k\)-microdifferential operators; \(k\)-microlocal regularity; propagation of singularities; nonlinear problems; higher order microlocalizations

Citations:

Zbl 0627.35065; Zbl 0669.35073
Cite Review PDF
Full Text: DOI Numdam EuDML

References:

[1] J.-M. BONY , Second Microlocalization and Propagation of Singularities for Semi-linear Hyperbolic Equations , Hyperbolic Equations and Related Topics, Mizohata éd., Kinokunya, 1986 , p. 11-49. MR 89e:35099 | Zbl 0669.35073 · Zbl 0669.35073
[2] J.-M. BONY , Singularités des solutions de problèmes de Cauchy hyperboliques non linéaires , Advances in Microlocal Analysis, M. G. Garnir éd., 15-39, 1986 by D. Reidel P. Company. MR 87m:35150 | Zbl 0627.35065 · Zbl 0627.35065
[3] A. P. CALDERON et R. VAILLANCOURT , A class of bounded pseudo-differential operators , Proc. Nat. Acad. Sci. U.S.A., 69, 1972 , p. 1185-1187. MR 45 #7532 | Zbl 0244.35074 · Zbl 0244.35074 · doi:10.1073/pnas.69.5.1185
[4] M. COTLAR , A combinatorical inequality and its application to L2 spaces , Rev. Math. Cuyana, 1, 1955 , p. 41-55. MR 18,219a | Zbl 0071.33301 · Zbl 0071.33301
[5] N. DENCKER , The Weyl calculus with locally temperate metrics and weights , Arkiv for Mat., 24, 1986 , n^\circ 1, p. 59-79. MR 87m:47111 | Zbl 0621.47045 · Zbl 0621.47045 · doi:10.1007/BF02384389
[6] L. HÖRMANDER , The analysis of linear partial differential operators III , Springer-Verlag, 1985 . Zbl 0601.35001 · Zbl 0601.35001
[7] A.-W. KNAPP et E. M. STEIN , Singular integrals and principal series , Proc. Nat. Acad. U.S.A., 63, 1969 , p. 281-284. MR 41 #8588 | Zbl 0181.12501 · Zbl 0181.12501 · doi:10.1073/pnas.63.2.281
[8] Y. LAURENT , Théorie de la deuxième microlocalisation dans le domaine complexe , Progress in Math., vol. 53, Birkhaüser, 1985 . MR 86k:58113 | Zbl 0561.32013 · Zbl 0561.32013
[9] G. LEBEAU , Deuxième microlocalisation sur les sous-variétés isotropes , Ann. Inst. Fourier, Grenoble, 35, 2, 1985 , p. 145-216. Numdam | MR 87h:58205 | Zbl 0539.58038 · Zbl 0539.58038 · doi:10.5802/aif.1014
[10] R. MELROSE et N. RITTER , Interaction of progressing waves for semi-linear wave equation II (à paraître). · Zbl 0653.35058
[11] I. SEGAL , Transforms for operators and asymptotic automorphisms over a locally compact abelian group , Math. Scand., 13, 1963 , p. 31-43. MR 29 #486 | Zbl 0208.39002 · Zbl 0208.39002
[12] J. SJÖSTRAND , Singularités analytiques microlocales , Astérisque, 95, S.M.F., 1982 . MR 84m:58151 | Zbl 0524.35007 · Zbl 0524.35007
[13] A. UNTERBERGER , Quantification de certains espaces hermitiens symétriques , Séminaire Goulaouic-Schwartz, n^\circ 16, 1979 - 1980 . Numdam | Zbl 0448.46053 · Zbl 0448.46053
[14] A. UNTERBERGER , Oscillateur harmonique et opérateurs pseudo-différentiels , Ann. Inst. Fourier, Grenoble, 29, 3, 1979 , p. 201-221. Numdam | MR 81m:58077 | Zbl 0396.47027 · Zbl 0396.47027 · doi:10.5802/aif.758
[15] H. WEYL , Gruppentheorie und Quantenmechanik , Verlag von S. Hirzel, Leipzig, 1928 . JFM 54.0954.03 · JFM 54.0954.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.