DISSIPATION AND SEMIGROUP ON H^k_n: NON-CUTOFF LINEARIZED BOLTZMANN OPERATOR WITH SOFT POTENTIAL

In this paper, we find that the linearized collision operator L of the non-cutoff Boltzmann equation with soft potential generates a strongly continuous semigroup on H^k_n, with k, n ∈ R. In the theory of the Boltzmann equation without angular cutoff, the weighted Sobolev space plays a fundamental role. The proof is based on pseudodifferential calculus, and, in general, for a specific class of Weyl quantization, the L² dissipation implies H^k_n dissipation. This kind of estimate is also known as Gårding's inequality.