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The Boltzmann equation without an angular cutoff in a three-dimensionalperiodic domain is considered. The global-in-time existence of solutions in afunction space $ W_k^{\zeta,p}L^\infty_TL^2_v $ with $p>1$ and$\zeta>3(1-\frac{1}{p})$ is established in the perturbation framework and thelong-time behavior of solutions is also obtained for both hard and softpotentials. The proof is based on several norm estimates.

Global solutions in $W_k^{\zeta,p}L^\infty_TL^2_v$ for the Boltzmann equation without cutoff