  We study special properties of solutions to the IVP associated to the
Camassa-Holm equation on the line related to the regularity and the decay of
solutions. The first aim is to show how the regularity on the initial data is
transferred to the corresponding solution in a class containing the "peakon
solutions". In particular, we shall show that the local regularity is similar
to that exhibited by the solution of the inviscid Burger's equation with the
same initial datum. The second goal is to prove that the decay results obtained
in a paper of Himonas, Misio{\l}ek, Ponce, and Zhou extend to the class of
solutions considered here.
