  The Cauchy problem for the Boltzmann equation with soft potential, in the
framework of small perturbation of an equilibrium state, has been studied in
many spaces. The method of strongly continuous semigroup has been applied by
Caflisch\cite{Caflisch1980a} and Ukai-Asano\cite{Ukai1982} for the case of soft
potential, where they obtained the $L^\infty$ solution without requiring any
velocity deviation. By generalizing the estimate on linearized collision
operator $L$ to the case of very soft potential, we obtain a similar global
existence result for $\gamma\in[0,d)$. For soft potential, the spectrum
structure of the linearized Boltzmann operator couldn't give spectral gap, so
we use the method of integration by parts and consider a weighted velocity
space in order to obtain algebraic decay in time.
