Pseudo-differential operators associated with the Jacobi differential operator (Q1269654)

The authors consider pseudo-differential operators on \((0,+\infty)\), defined in terms of the Fourier-Jacobi transform: \[ (Ff)(\xi)=\widehat f(\xi)=\int^\infty_0\varphi_\xi(x)f(x)dm(x) \] where \(\varphi_\xi(x)\) is the Jacobi function and \(dm(x)\) the associated measure. Precisely, for a suitable class of symbols \(p(x,\xi)\), one sets \[ p(x,D)f(x)=F^{-1}_{\xi\to x}\bigl(p(x,\xi)\widehat f(\xi)\bigr). \] The authors study the boundedness of the operators \(p(x,D)\) in different function spaces, giving for them an integral representation.