  In this paper, we consider the cutoff Boltzmann equation near Maxwellian, we
proved the global existence and uniqueness for the cutoff Boltzmann equation in
polynomial weighted space for all $\gamma \in (-3, 1]$. We also proved
initially polynomial decay for the large velocity in $L^2$ space will induce
polynomial decay rate, while initially exponential decay will induce
exponential rate for the convergence. Our proof is based on newly established
inequalities for the cutoff Boltzmann equation and semigroup techniques.
Moreover, by generalizing the $L_x^\infty L^1_v \cap L^\infty_{x, v}$ approach,
we prove the global existence and uniqueness of a mild solution to the
Boltzmann equation with bounded polynomial weighted $L^\infty_{x, v}$ norm
under some small condition on the initial $L^1_x L^\infty_v$ norm and entropy
so that this initial data allows large amplitude oscillations.
