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

The Boltzmann equation without an angular cutoff in a three-dimensional periodic domain is considered. The global-in-time existence of solutions in a function 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 the long-time behavior of solutions is also obtained for both hard and soft potentials. The proof is based on several norm estimates.