R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal, vol.152, issue.4, pp.327-355, 2000.

R. Alexandre, Y. Morimoto, S. Ukai, C. Xu, and T. Yang, Uncertainty principle and kinetic equations, J. Funct. Anal, vol.255, issue.8, pp.2013-2066, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00435676

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math, vol.55, issue.1, pp.30-70, 2002.

R. Alexandre and M. E. Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci, vol.15, issue.6, pp.907-920, 2005.

R. Alexandre and M. Elsafadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules, Discrete Contin. Dyn. Syst, vol.24, issue.1, pp.1-11, 2009.

Y. Radjesvarane-alexandre, S. Morimoto, C. Ukai, T. Xu, and . Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal, vol.198, issue.1, pp.39-123, 2010.

R. Alonso, J. A. Cañizo, I. Gamba, and C. Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Comm. Partial Differential Equations, vol.38, issue.1, pp.155-169, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00677951

R. Alonso, M. Irene, M. Gamba, and . Taskovi´ctaskovi´c, Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation, 2017.

L. Arkeryd, L ? estimates for the space-homogeneous Boltzmann equation, J. Statist. Phys, vol.31, issue.2, pp.347-361, 1983.

A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Statist. Phys, vol.88, pp.1183-1214, 1997.

A. V. Bobylev, I. M. Gamba, and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, J. Statist. Phys, vol.116, issue.5-6, pp.1651-1682, 2004.

A. V. Bobylev and I. M. Gamba, Upper Maxwellian bounds for the Boltzmann equation with pseudoMaxwell molecules, Kinet. Relat. Models, vol.10, issue.3, pp.573-585, 2017.

M. Briant, Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains, Arch. Ration. Mech. Anal, vol.218, issue.2, pp.985-1041, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01492021

M. Briant, Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions, Kinet. Relat. Models, vol.8, issue.2, pp.281-308, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01492036

M. Briant and A. Einav, On the Cauchy problem for the homogeneous Boltzmann-Nordheim equation for bosons: local existence, uniqueness and creation of moments, J. Stat. Phys, vol.163, issue.5, pp.1108-1156, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01492026

S. Cameron, L. Silvestre, and S. Snelson, Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials, vol.18, 2017.

T. Carleman, Probì emes mathématiques dans la théorie cinétique des gaz, Publ. Sci. Inst. Mittag-Leffler. 2. Almqvist & Wiksells Boktryckeri Ab, 1957.

T. Carleman, Sur la théorie de l'´ equation intégrodifférentielle de Boltzmann, Acta Math, vol.60, issue.1, pp.91-146, 1933.

C. Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol.67, 1988.

P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal, vol.22, issue.5, pp.1289-1321, 2012.

L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Arch. Rational Mech. Anal, vol.123, issue.4, pp.387-404, 1993.

L. Desvillettes and C. Mouhot, Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials, Asymptot. Anal, vol.54, issue.3-4, pp.235-245, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00079949

L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Ration. Mech. Anal, vol.193, issue.2, pp.227-253, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00079713

L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Partial Differential Equations, vol.29, issue.1-2, pp.133-155, 2004.

R. J. Diperna and P. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math, vol.130, issue.2, pp.321-366, 1989.

T. Elmroth, Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range, Arch. Rational Mech. Anal, vol.82, issue.1, pp.1-12, 1983.

F. Filbet and C. Mouhot, Analysis of spectral methods for the homogeneous Boltzmann equation, Trans. Amer. Math. Soc, vol.363, issue.4, pp.1947-1980, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00659629

I. M. Gamba, V. Panferov, and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal, vol.194, issue.1, pp.253-282, 2009.

M. Irene, N. Gamba, M. Pavlovi´cpavlovi´c, and . Taskovi´ctaskovi´c, On pointwise exponentially weighted estimates for the Boltzmann equation, 2017.

C. Francois¸golsefrancois¸francois¸golse, C. Imbert, A. F. Mouhot, and . Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Classe di Scienze, vol.19, issue.1, pp.253-295, 2019.

M. P. Gualdani, S. Mischler, and C. Mouhot, Factorization of non-symmetric operators and exponential H-theorem, p.137, 2017.

T. Gustafsson, L p -estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal, vol.92, issue.1, pp.23-57, 1986.

T. Gustafsson, Global L p -properties for the spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal, vol.103, issue.1, pp.1-38, 1988.

C. Henderson and S. Snelson, C ? smoothing for weak solutions of the inhomogeneous Landau equation, 2017.

C. Henderson, S. Snelson, and A. Tarfulea, Local existence, lower mass bounds, and smoothing for the Landau equation, 2017.

Z. Huo, Y. Morimoto, S. Ukai, and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinet. Relat. Models, vol.1, issue.3, pp.453-489, 2008.

E. Ikenberry and C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, I. J. Rational Mech. Anal, vol.5, pp.1-54, 1956.

C. Imbert and L. Silvestre, Weak Harnack inequality for the Boltzmann equation without cut-off, Journal of the European Mathematical Society, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01357047

C. Imbert and L. Silvestre, The Schauder estimate for kinetic integral equations, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01979425

X. Lu, Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation, J. Statist. Phys, vol.96, issue.3-4, pp.765-796, 1999.

X. Lu and C. Mouhot, On measure solutions of the Boltzmann equation, part I: moment production and stability estimates, J. Differential Equations, vol.252, issue.4, pp.3305-3363, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00561793

J. C. Maxwell, On the dynamical theory of gases, J. Philosophical Transactions of the Royal Society of London, vol.157, pp.49-88, 1867.

S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres. II. Self-similar solutions and tail behavior, J. Stat. Phys, vol.124, issue.2-4, pp.703-746, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00087232

S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.16, issue.4, pp.467-501, 1999.

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, vol.24, issue.1, pp.187-212, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00434249

Y. Morimoto and T. Yang, Local existence of polynomial decay solutions to the Boltzmann equation for soft potentials, Anal. Appl. (Singap.), vol.13, issue.6, pp.663-683, 2015.

C. Mouhot, Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions, Comm. Partial Differential Equations, vol.30, issue.4-6, pp.881-917, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00087249

A. J. Povzner, On the Boltzmann equation in the kinetic theory of gases, Mat. Sb. (N.S.), vol.58, issue.100, pp.65-86, 1962.

L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Communications in Mathematical Physics, vol.348, issue.1, pp.69-100, 2016.

L. Silvestre, Upper bounds for parabolic equations and the Landau equation, J. Differential Equations, vol.262, issue.3, pp.3034-3055, 2017.

C. , On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, II. J. Rational Mech. Anal, vol.5, pp.55-128, 1956.

C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, vol.I, pp.71-305, 2002.

B. Wennberg, The Povzner inequality and moments in the Boltzmann equation, Proceedings of the VIII International Conference on Waves and Stability in Continuous Media, Part II, vol.45, pp.673-681, 1995.

, Imbert) CNRS & Department of Mathematics and Applications, p.45

, address: C.Mouhot@dpmms.cam.ac.uk (L. Silvestre) Mathematics Department, vol.60637