This talk concerns the global existence of low regularity solutions to the Boltzmann equation without angular cutoff. We introduce a new function space to treat the problem in cases when the spatial domain is either a torus, or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the time-velocity norm of the distribution function is in the Wiener algebra in the spatial variables.
