Pseudo-Differential Operators Associated with the Jacobi Differential Operator

AbstractWe introduce two classesSmandSm0, of symbols withSm⊂Sm0. We define pseudo-differential operators associated with symbols belonging to these classes. We prove that a pseudo-differential operator associated with a symbol inSm0is a continuous linear mapping from some subspace of the Schwartz space into itself. Next we consider symbols inSmand 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 certainL1-norm inequality