A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Soviet Sci. Rev. Sect. C Math. Phys, vol.7, pp.111-233, 1988.

C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, vol.67, 1988.
DOI : 10.1007/978-1-4612-1039-9

L. Desvillettes, G. Furioli, and E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules, Transactions of the American Mathematical Society, vol.361, issue.04, pp.361-1731, 2009.
DOI : 10.1090/S0002-9947-08-04574-1

L. Desvillettes and B. Wennberg, Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff, Communications in Partial Differential Equations, vol.45, issue.1-2, pp.133-155, 2004.
DOI : 10.1080/03605309408821082

E. Dolera, On the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules, Boll. UMI, vol.46, pp.67-105, 2010.

L. Glangetas and M. Najeme, Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation, Kinetic and Related Models, vol.6, issue.2, pp.407-427, 2013.
DOI : 10.3934/krm.2013.6.407
URL : https://hal.archives-ouvertes.fr/hal-00701857

T. Gramchev, S. Pilipovi´cpilipovi´c, and L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces. New Developments in Pseudo-Differential Operators, Birkhäuser Basel, pp.15-31, 2009.

G. E. Andrews, R. Askey, and R. Roy, Special functions, encyclopedia of mathematics and its applications, 1999.

M. N. Jones, Spherical harmonics and tensors for classical field theory, 1985.

N. Lekrine and C. Xu, Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation, Kinetic and Related Models, vol.2, issue.4, pp.647-666, 2009.
DOI : 10.3934/krm.2009.2.647

N. Lerner, Y. Morimoto, K. Pravda-starov, and C. Xu, Spectral and phase space analysis of the linearized non-cutoff Kac collision operator, Journal de Math??matiques Pures et Appliqu??es, vol.100, issue.6, pp.832-867, 2013.
DOI : 10.1016/j.matpur.2013.03.005
URL : https://hal.archives-ouvertes.fr/hal-01116721

N. Lerner, Y. Morimoto, K. Pravda-starov, and C. Xu, Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators, Kinetic and Related Models, vol.6, issue.3, pp.625-648, 2013.
DOI : 10.3934/krm.2013.6.625
URL : https://hal.archives-ouvertes.fr/hal-01116722

N. Lerner, Y. Morimoto, K. Pravda-starov, and C. Xu, Gelfand???Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff, Journal of Differential Equations, vol.256, issue.2, pp.797-831, 2014.
DOI : 10.1016/j.jde.2013.10.001
URL : https://hal.archives-ouvertes.fr/hal-01116715

H. Li, Cauchy problem for linearized non-cutoff boltzmann equation with distribution initial datum, Acta Mathematica Scientia, vol.35, issue.2, pp.459-476, 2015.
DOI : 10.1016/S0252-9602(15)60015-7

Y. Morimoto and S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff, Journal of Pseudo-Differential Operators and Applications, vol.143, issue.1, pp.139-159, 2010.
DOI : 10.1007/s11868-010-0008-z

Y. Morimoto, S. Ukai, C. Xu, and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin, Dyn. Syst., A special issue on Boltzmann Equations and Applications, pp.24-187, 2009.

C. Müller, Analysis of spherical symmetries in Euclidean spaces, 1998.
DOI : 10.1007/978-1-4612-0581-4

S. Ukai, Local solutions in gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan Journal of Applied Mathematics, vol.84, issue.1, pp.141-156, 1984.
DOI : 10.1007/BF03167864

C. Villani, A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, pp.71-74, 2002.

C. S. Wang-chang and G. E. Uhlenbeck, On the propagation of sound in monoatomic gases, Univ. of Michigan Press, Studies in Statistical Mechanics, 1970.

G. N. Watson, A Treatise on the Theory of Bessel Functions, The Mathematical Gazette, vol.18, issue.231, 1995.
DOI : 10.2307/3605513

E. T. Whittaker and G. N. Watson, A course of modern analysis, 1927.

T. Zhang, Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff, Journal of Differential Equations, vol.253, issue.4, pp.1172-1190, 2012.
DOI : 10.1016/j.jde.2012.04.023

L. Glangetas, C. Université-de-rouen, and . Umr, Mathématiques 76801 Saint-Etienne du Rouvray, France E-mail address: leo.glangetas@univ-rouen.fr Hao-Guang Li, School of Mathematics and statistics, Wuhan University 430072, Wuhan, P. R. China School of mathematics and statistics, South-Central University for Nationalities 430074, R. China E-mail address: lihaoguang@mail.scuec.edu.cn Chao-Jiang Xu, School of Mathematics and statistics