Pseudo-differential operators associated with a singular differential operator in ]0, +∞[

Using the harmonic analysis associated with a singular differential operator Delta on ]0, +infinity[, we define two classes S-m and S-0(m) of symbols with S-m subset of S-0(m) and the corresponding pseudo-differential operators. We prove that a pseudo-differential operator associated with a symbol in S-0(m), is a continuous linear mapping from some subspace of the Schwartz space into itself. Next we consider symbols in S-m, and we give an integral representation of the pseudo-differential operators associated with these symbols. Finally, we show that these pseudo-differential opeators satisfy a certain L-1-norm inequality.