Journal of the Iranian Mathematical Society

Journal of the Iranian Mathematical Society

Gevrey regularity on maximally real submanifolds

Document Type : Research Article

Author
Department of Mathematics, College of Natural and Computational Science, Samara University, Ethiopia.
Abstract
The Fourier‎ -‎Br'os‎ -‎Iagolnitzer (FBI) transform is the right tool to characterize microlocal analyticity‎, ‎microlocal smoothness‎, ‎and Gevrey regularity‎. ‎In this paper‎, ‎we characterize microlocal Gevrey regularity of a distribution on a maximally real submanifold of $\mathbb{C}^m$ using the FBI transform‎.
Keywords
Subjects

[1] Z. Adwan, G. Hoepfner, A generalization of Borel’s theorem and microlocal Gevrey regularity in involutive structures, J. Differential Equations 245 (2008), no. 10, 2846–2870.
[2] M. S. Baouendi, C. H. Chang and F. Treves, Microlocal hypo-analyticity and extension of CR functions, J. Differential. Geom. 18 (1983), no. 3, 331–391.
[3] R. F. Barostichi and G. Petronilho, Gevrey micro-regularity for solutions to first order nonlinear PDE, J. Differential Equations 247 (2009), no. 6, 1899–1914.
[4] S. Berhanu, P. D. Cordaro, and J. Hounie, An introduction to involutive structures, Cambridge University Press, Cambridge, 2008.
[5] S. Berhanu and Abraham Hailu, Characterization of Gevrey regularity by a class of FBI transforms, Springer –Verlag (Birkhauser), Novel Methods in Harmonic Analysis, 2 (2017) 451–482.
[6] S. Berhanu and J. Hounie, A class of FBI transforms, Comm. Partial Differential Equations 37 (2012), no. 1, 38–57.
[7] M. Christ, Intermediate optimal Gevrey exponents occur, Comm. Partial Differential Equations, 22 (1997), no. 3-4, 359–379.
[8] P. D. Cordaro and F. Treves,Hyperfunctions on hypo-analytic manifolds, Annals of Mathematics Studies, 136, Princeton University Press, Princeton, 1994.
[9] H. Komatsu, Microlocal analysis in Gevrey classes and complex domains, Microlocal Analysis and Applications, Lect. Notes in Math, 1495, Springer, Berlin, 1989.
[10] F. Treves, Hypo-analytic structures, Local theory, Princeton University Press, Princeton, 1992.