  This paper aims to incorporate the Caflisch's decomposition into the
macro-micro decomposition in Boltzmann theory for allowing the microscopic
component to exhibit only the polynomial tail in large velocities. In
particular, we treat the Cauchy problem on the non-cutoff Boltzmann equation
under the compressible Euler scaling in case of three-dimensional whole space.
Up to a finite time we construct the Boltzmann solution around a local
Maxwellian corresponding to small-amplitude classical solutions of the full
compressible Euler system around constant states. We design a new energy
functional which can capture the convergence rate in the small Knudsen number
$\varepsilon$ and allow the microscopic part of solutions to decay polynomially
in large velocities. Moreover, the energy norm of perturbations can be of the
order $\varepsilon^{1/2}$ which the usual method of Hilbert expansion fails to
obtain. As a byproduct of the proof, our estimates immediately yield a
global-in-time existence result when the Euler solutions are taken to be
constant states.
