We consider a simplified Boltzmann equation: spatially homogeneous,
two-dimensional, radially symmetric, with Grad's angular cutoff, and
linearized around its initial condition.
We prove that for a sufficiently singular velocity cross section,
the solution may become instantaneously a function,
even if the initial condition is a singular measure.
To our knowledge, this is the first regularization result
in the case with
cutoff: all the previous results were relying on the non-integrability
of the angular cross section.
Furthermore, our result is quite surprising: the regularization
occurs for initial conditions that are not too singular,
but also not too regular.
The objective of the present work is to explain that
the singularity of the velocity cross section, which is often considered
as a (technical) obstacle to regularization, seems on the contrary to
help the regularization.