Pseudo-Differential Operators Associated with the Jacobi Differential Operator

We introduce two classes Sm and S0m, of symbols with Sm ;S0m. We define pseudo-differential operators associated with symbols belonging to these classes. We prove that a pseudo-differential operator associated with a symbol in S0m is a continuous linear mapping from some subspace of the Schwartz space into itself. Next we consider symbols in Sm and we give an integral representation of the pseudo-differential operators associated with these symbols. Finally, using the dual convolution associated with Jacobi functions, we show that these pseudo-differential operators satisfy a certain L1-norm inequality.