  In this paper, we are interested in the $L^p$-estimates of the Boltzmann
equation in the case that the distribution function stays around a travelling
local Maxwellian. For this, we divide both sides of the Boltzmann equation by
the velocity distribution function with a fractional exponent and reformulate
the Boltzmann equation into a regularized one. This amounts to endowing
additional integrability on the collision kernel, which in turn enables us to
apply simple H\"{o}lder type inequalities. Our results cover the whole range of
Lebesgue exponents: $0<p\leq \infty$.
