  We study the Boltzmann equation near a global Maxwellian in the case of
bounded domains. We consider the boundary conditions to be either specular
reflections or Maxwellian diffusion. Starting from the reference work of Guo in
$L^\infty_{x,v}\left(\left(1+|v|\right)^\beta e^{|v|^2/4}\right)$, we prove
existence, uniqueness, continuity and positivity of solutions for less
restrictive weights in the velocity variable; namely, polynomials and stretch
exponentials. The methods developed here are constructive.
