×

The FBI transforms and their use in microlocal analysis. (English) Zbl 1395.35007

In the paper under review, certain problems of microlocal analysis are studied. This topic deals with smooth microlocal singularities (the collection of directions in which the Fourier transform does not decay rapidly), the set of which is called a smooth wave front set. Using a more general class of FBI transforms, introduced by S. Berhanu and J. Hounie, the authors completely characterize regularity and microregularity in Denjoy-Carleman (non-quasi-analytic) classes, which includes the Gevrey classes and M. Christ version of the FBI transform as examples. They also exhibit a result on microlocal regularity for solutions of first order partial differential equations in these classes, that do not seem possible to prove using the classical FBI transform.

MSC:

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
35A18 Wave front sets in context of PDEs
35A21 Singularity in context of PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35N10 Overdetermined systems of PDEs with variable coefficients
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adwan, Z.; Berhanu, S., On microlocal analyticity and smoothness of solutions of first-order nonlinear PDE’s, Math. Ann., 352, 1, 239-258, (2012) · Zbl 1298.35006
[2] Adwan, Z.; Hoepfner, G., Approximate solutions and micro-regularity in the Denjoy-Carleman classes, J. Differential Equations, 249, 9, 2269-2286, (2010) · Zbl 1205.35008
[3] Adwan, Z.; Hoepfner, G., Denjoy-Carleman classes: boundary values, approximate solutions and applications, J. Geom. Anal., 25, 3, 1720-1743, (2015) · Zbl 1326.35013
[4] Adwan, Z.; Hoepfner, G.; Raich, A., Global \(L^q\)-Gevrey functions and their applications, J. Geom. Anal., 27, 3, 1874-1913, (2017) · Zbl 1376.42030
[5] Bang, T., Om quasi-analytiske funktioner, (1946), University of Copenhagen, Thesis
[6] Baouendi, M. S.; Chang, C. H.; Trèves, F., Microlocal hypo-analyticity and extension of CR functions, J. Differential Geom., 18, 3, 331-391, (1983) · Zbl 0575.32019
[7] Baouendi, M. S.; Trèves, F., A microlocal version of Bochner’s tube theorem, Indiana Univ. Math. J., 31, 6, 885-895, (1982) · Zbl 0505.32013
[8] Berhanu, S., On microlocal analyticity of solutions of first-order nonlinear PDE, Ann. Inst. Fourier, 59, 4, 1267-1290, (2009) · Zbl 1195.35011
[9] Berhanu, S., On involutive systems of first-order nonlinear partial differential equations, Complex Anal., 25-50, (2010) · Zbl 1198.35010
[10] Berhanu, S.; Cordaro, P.; Hounie, J., An introduction to involutive structures, New Math. Monogr., vol. 6, (2008), Cambridge University Press Cambridge · Zbl 1151.35011
[11] Berhanu, S.; Hailu, A., Characterization of Gevrey regularity by a class of FBI transforms, Appl. Numer. Harmon. Anal., 451-482, (2017)
[12] Berhanu, S.; Hounie, J., A class of FBI transforms, Comm. Partial Differential Equations, 37, 1, 38-57, (2012) · Zbl 1238.32002
[13] Berhanu, S.; Xiao, M., On the regularity of CR mappings between CR manifolds of hypersurface type, Trans. Amer. Math. Soc., 369, 9, 6073-6086, (2017) · Zbl 1434.32048
[14] Bony, J. M., Equivalence des diverses notions de spectre singulier analytique. Séminaire goulaouic-Schwartz, (Équations aux dérivées partielles et analyse fonctionnelle, Centre Math., vol. 3, (1977), École Polytech Palaiseau), 12 pp. · Zbl 0367.46036
[15] Borel, E., Sur la généralisation du prolongement analytique, C. R. Acad. Sci. Paris, 130, 1115-1118, (1900) · JFM 31.0411.02
[16] Borel, E., Sur LES séries de polynômes et de fractions rationnelles, Acta Math., 24, 309-387, (1902)
[17] Braun, R. W., An extension of Komatsu’s second structure theorem for ultradistributions, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., 40, 2, 411-417, (1993) · Zbl 0811.46031
[18] Bros, J.; Iagolnitzer, D., Causality and local analyticity: mathematical study, Ann. Inst. H. Poincaré Sect. A (N.S.), 18, 147-184, (1973) · Zbl 0286.42016
[19] Bros, J.; Iagolnitzer, D., Tuboïdes et structure analytiques des distribution. II. support essentiel et structure analytique des distributions, (Sém. Goulaouic-Lions-Schwartz, Exposé 18, (1975)), 1974-1975 · Zbl 0333.46029
[20] Bruna, J., An extension theorem of Whitney type for non-quasi-analytic classes of functions, J. Lond. Math. Soc. (2), 22, 3, 495-505, (1980) · Zbl 0419.26010
[21] Carleman, T., LES fonctions quasi-analytiques, (1926), Gauthier-Villars · JFM 52.0255.02
[22] Christ, M., Intermediate optimal Gevrey exponents occur, Comm. Partial Differential Equations, 22, 3-4, 359-379, (1997) · Zbl 0893.35021
[23] Constantine, C. M.; Savits, T. H., A multivariate faà di bruno formula with applications, Trans. Amer. Math. Soc., 348, 2, 503-520, (1996) · Zbl 0846.05003
[24] Denjoy, A., Sur LES fonctions quasi-analytiques de variable réelle, C. R. Acad. Sci. Paris, 173, 1329-1331, (1921) · JFM 48.0295.01
[25] Eastwood, M. G.; Graham, C. R., Edge of the wedge theory in hypo-analytic manifolds, Comm. Partial Differential Equations, 28, 2003-2028, (2003) · Zbl 1063.32009
[26] Fürdös, S., Ultradifferentiable CR manifolds, (2017), University of Vienne, Ph.D. Thesis
[27] Hoepfner, G.; Raich, A., Global \(L^q\) Gevrey functions, Paley-weiner theorems, and the FBI transform, Indiana Univ. Math. J., (2018), in press
[28] G. Hoepfner, A. Raich, Microglobal Regularity and the Global Wavefront Set, submitted for publication.; G. Hoepfner, A. Raich, Microglobal Regularity and the Global Wavefront Set, submitted for publication.
[29] Hörmander, L., Linear differential operators, (Actes du Congrès International des Mathématiciens, Tome 1, Nice, 1970, (1971), Gauthier-Villars Paris), 121-133
[30] Hörmander, L., Fourier integral operators I, Acta Math., 127, 79-183, (1971) · Zbl 0212.46601
[31] Hörmander, L., Uniqueness theorems and wave front sets for solutions of linear partial differential equations with analytic coefficients, Comm. Pure Appl. Math., 24, 671-704, (1971) · Zbl 0226.35019
[32] Hörmander, L., The analysis of linear partial differential operators, vol. 1, (1983), Springer-Verlag Berlin
[33] Komatsu, H., Ultradistributions. I. structure theorems and a characterization, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., 20, 25-105, (1973) · Zbl 0258.46039
[34] Lions, J. L.; Magenes, E., Problèmes aux limites non homogènes, vol. 3, (1970), Dunod Paris · Zbl 0197.06701
[35] Mandelbrojt, S., Séries adhérentes, régularisation des suites, applications, (1952), Gauthier-Villars Paris · Zbl 0048.05203
[36] Petzsche, H. J.; Vogt, D., Almost analytic extension of ultradifferentiable functions and the boundary values of holomorphic functions, Math. Ann., 267, 1, 17-35, (1984) · Zbl 0517.30028
[37] Rodino, L., Linear partial differential operators in Gevrey spaces, (1993), World Scientific · Zbl 0869.35005
[38] Sato, M., Hyperfunctions and partial differential equations, Proc. Int. Conf. on Funct. Anal. and Rel. Topics, 91-94, (1969), Tokyo University Press Tokyo
[39] Sato, M., Regularity of hyperfunction solutions of partial differential equations, Actes Congr. Int. Math. Nice, 2, 785-794, (1970)
[40] Sjöstrand, J., Singularites analytiques microlocales, Astérisque, vol. 95, (1982), Soc. Math. France Paris · Zbl 0524.35007
[41] Trèves, F., Hypo-analytic strutures: local theory, (1992), Princenton University Press · Zbl 0787.35003
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.