  This article presents a new approach of semigroup analysis and
pseudo-differential calculus for deriving the regularizing estimate on
non-cutoff linearized Boltzmann equation. We are able to obtain regularizing
estimate of semigroup $e^{tB}$ that is continuous from weighted Sobolev space
$H(a^{-1/2})H^m_x$ to $H(a^{1/2})H^m_x$ with a sharp large time decay. With
these properties, we prove the existence of global-in-time unique solution to
the non-cutoff Boltzmann equation for hard potential on the whole space with
weak regularity assumption on initial data. We consider the hard potential case
since $H(a^{1/2})$ can be embedded in $L^2$. This work develops the application
of pseudo-differential calculus, spectrum analysis and semigroup theory to
non-cutoff Boltzmann equation.
