The Boltzmann equation, Besov spaces, and optimal time decay rates in the whole space

We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\mathbb R}^{n}_x$ with $n \ge 3$, converge in large-time to the global Maxwellian with the optimal decay rate of $O(t^{-1/2(k+\varrho+\frac{n}{2}-\frac{n}{r})})$ in the $L^r_x(L^2_{v})$-norm for any $2\leq r\leq \infty$. These results hold for any $\varrho \in [0, n/2]$ as long as initially $\| f_0|_{\dot{B}^{-\varrho,\infty}_2 L^2_{v}} < \infty$. In the hard potential case, we prove faster decay results in the sense that if $|\mathbf{P} f_0\|_{\dot{B}^{-\varrho,\infty}_2 L^2_{v}} < \infty$ and $|({\mathbf{I} - \mathbf{P}}) f_0|_{\dot{B}^{-\varrho+1,\infty}_2 L^2_{v}} < \infty$ for $\varrho \in (n/2, (n+2)/2]$ then the solution decays to zero in $L^2_v(L^2_x)$ with the optimal large time decay rate of $O(t^{-1/2\varrho})$.