  We study the homogeneous Boltzmann equation with the fractional Laplacian
term. Working on the Fourier side we solve the resulting integral equation, and
improve a previous result by Y.-K. Cho. We replace the initial data space with
a certain space $\mathcal{M}^\alpha$ introduced by Morimoto, Wang, and Yang.
This space precisely captures the Fourier image of probability measures with
bounded fractional moments, providing a more natural initial condition. We show
existence of a unique global solution, in addition to the expected maximal
growth estimates and stability estimates. As a consequence we obtain a
continuous density solution of the original equation.
