×

On Gevrey singularities of microhyperbolic operators. (English) Zbl 1293.35375

Summary: We study the Gevrey singularities of solutions of microhyperbolic equations using exponential weighted estimates in the phase space. In particular, we recover some known results on the propagation of Gevrey regularity in an elementary way, using microlocal exponential estimates.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
Full Text: DOI

References:

[1] E. Bernardi and A. Bove, Propagation of Gevrey singularities for a class of operators with triple characteristics. I, II, Duke Math. J. 60 (1990), 187–205, 207–220. · Zbl 0721.35035 · doi:10.1215/S0012-7094-90-06006-5
[2] J. M. Bony, Equivalence des diverses notions de spectre singulier analytique, Séminaire Goulaouic-Schwartz (1976/1977), Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 3, École Polytech., Palaiseau, 1977.
[3] M. D. Bronshtein, The Cauchy problem for hyperbolic operators with variable multiple characteristics, Trudy Moskov. Mat. Obschch. 41 (1980), 83–99.
[4] M. Christ, Intermediate optimal Gevrey exponents occur, Comm. Partial Differential Equations 22 (1997), 359–379. · Zbl 0893.35021 · doi:10.1080/03605309708821267
[5] V. Ja. Ivrii, Wave fronts of solutions of certain pseudodifferential equations, Trudy Moskov. Mat. Obshch. 39 (1979), 49–82, 235.
[6] V. Ja. Ivrii, Wave fronts of solutions of certain hyperbolic pseudodifferential equations, Trudy Moskov. Mat. Obshch. 39 (1979), 83–112, 235.
[7] K. Jung, Phase space tunneling for operators with symbols in a Gevrey class, J. Math. Phys. 41 (2000), 4478–4496. · Zbl 0974.35136 · doi:10.1063/1.533355
[8] K. Kajitani and S. Wakabayashi, Microhyperbolic operators in Gevrey classes, Publ. Res. Inst. Mat. Sci. 25 (1989), 169–221. · Zbl 0705.35158 · doi:10.2977/prims/1195173608
[9] K. Kajitani and S. Wakabayashi, Propagation of singularities for several classes of pseudodifferential operators, Bull. Sci. Math. 115 (1991), 397–449. · Zbl 0758.35097
[10] T. Kawai and M. Kashiwara, Microhyperbolic pseudodifferential operators I, J. Math. Soc. Japan 27 (1975), 359–404. · Zbl 0305.35066 · doi:10.2969/jmsj/02730359
[11] B. Lascar, Propagation des singularités Gevrey pour des opérateurs hyberboliques, Amer. J. Math. 110 (1988), 413–449. · Zbl 0653.35085 · doi:10.2307/2374618
[12] B. Lascar and R. Lascar, Propagation des singularités pour des opérateurs pseudodiff érentiels à symboles réels, Duke Math. J. 53 (1986), 945–981. · Zbl 0625.35079 · doi:10.1215/S0012-7094-86-05348-2
[13] B. Lascar and J. Sjöstrand, Équation de Schrödinger et propagation des singularités pour des opérateurs pseudo-différentiels à caractéristiques réelles de multiplicité variable, Astérisque 95, Soc. Math. France, Paris, 1982, pp. 167–207.
[14] R. Lascar, Propagation des singularités des solutions d’équations pseudo-différentielles à Caractéristiques de Multiplicités variables, Springer Lecture Notes in Math. 856, 1981.
[15] O. Liess, The Fourier-Bros-Iagolnitzer transform for inhomogeneous Gevrey classes. I, Simon Stevin 60 (1986), 105–121. · Zbl 0616.46042
[16] O. Liess and L. Rodino, Inhomogeneous Gevrey classes and related pseudodifferential operators, Boll. Un. Mat. Ital. C (6) (1984), 233–323. · Zbl 0557.35131
[17] A. Martinez, Estimates on complex interactions in phase space, Math. Nachr. 167 (1994), 203–254. · Zbl 0836.35135 · doi:10.1002/mana.19941670109
[18] A. Martinez, Precise exponential estimates in adiabatic theory, J. Math. Phys. 35 (1994), 3889–3915. · Zbl 0808.47053 · doi:10.1063/1.530832
[19] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Springer-Verlag, New York, 2002. · Zbl 0994.35003
[20] A. Martinez, S. Nakamura, and V. Sordoni, Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math. 59 (2006), 1330–1351. · Zbl 1122.35027 · doi:10.1002/cpa.20112
[21] A. Martinez, S. Nakamura, and V. Sordoni, Analytic wave front set for solutions to Schrödinger equations, Adv. Math. 222 (2009), 1277–1307. · Zbl 1180.35016 · doi:10.1016/j.aim.2009.06.002
[22] A. Martinez and V. Sordoni, Microlocal WKB expansions, J. Funct. Anal. 168 (1999), 380–402. · Zbl 0941.35136 · doi:10.1006/jfan.1999.3460
[23] A. Melin and J. Sjöstrand, Fourier integral operators with complex valued phase functions, in Fourier Integral Operators and Partial Differential Equations, Springer Lecture Notes in Math. 459, 1975, pp. 120–223. · Zbl 0306.42007 · doi:10.1007/BFb0074195
[24] R. Mizuhara, Microlocal smoothing effect for the Schrödinger evolution equation in a Gevrey class, J. Math. Pures Appl. (9) 9 (2009), 115–136. · Zbl 1173.35042 · doi:10.1016/j.matpur.2008.09.005
[25] M. Petrini and V. Sordoni, Propagation of singularities for a class of hyperbolic operators with triple characteristics, Comm. Partial Differential Equations 16 (1991), 683–703. · Zbl 0735.35087 · doi:10.1080/03605309108820774
[26] M. Petrini and V. Sordoni, Propagation of singularities for hyperbolic operators with multiple involutive characteristics, Osaka J. Math. 28 (1991), 911–933. · Zbl 0760.58043
[27] M. Petrini and V. Sordoni, Propagation of singularities for hyperbolic operators with double characteristics, Ann. Mat. Pura Appl. (4) 163 (1993), 199–222. · Zbl 0799.35007 · doi:10.1007/BF01759022
[28] L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific Publishing Co., River Edge, NJ, 1993. · Zbl 0869.35005
[29] J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95, Soc. Math. France, Paris, 1982, pp. 1–166.
[30] V. Sordoni, Gaussian decay for the eigenfunctions of a Schrödinger operator with magnetic field constant at infinity, Comm. Partial Differential Equations 23 (1998), 223–242. · Zbl 0898.34077 · doi:10.1080/03605309808821344
[31] K. Uchikoshi, Construction of the solutions of microhyperbolic pseudodifferential equations, J. Math. Soc. Japan 40 (1988), 289–318. · Zbl 0689.35101 · doi:10.2969/jmsj/04020289
[32] K. Uchikoshi, Irregularities of microhyperbolic operators, J. Math. Soc. Japan 58 (2006), 453–484. · Zbl 1106.35156 · doi:10.2969/jmsj/1149166784
[33] S. Wakabayashi, The Cauchy problem for operators with constant coefficient hyperbolic principal part and propagation of singularities, Japan J. Math. 6 (1980), 179–228. · Zbl 0453.35061
[34] S. Wakabayashi, Generalized Hamilton flows and singularities of the hyperbolic Cauchy problem, in Hyperbolic Equations and Related Topics, Academic Press, Boston, MA, 1986, pp. 415–423. · Zbl 0697.35083
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.