  The study of positivity of solutions to the Boltzmann equation goes back to
Carleman (1933), and the initial argument of Carleman was developed
byPulvirenti-Wennberg (1997), the second author and Briant (2015). The
appearance of a lower bound with Gaussian decay had however remained an open
question for long-range interactions (the so-called non-cutoff collision
kernels). We answer this question and establish such Gaussian lower bound for
solutions to the Boltzmann equation without cutoff, in the case of hard and
moderately soft potentials, with spatial periodic conditions, and under the
sole assumption that hydrodynamic quantities (local mass, local energy and
local entropy density) remain bounded. The paper is mostly self-contained,
apart from the uniform upper bound on the solution established by the third
author (2016).
