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

Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons