Analysis related to all admissible type parameters in the Jacobi setting

We derive an integral representation for the Jacobi-Poisson kernel valid for all admissibletype parameters a and b in the context of Jacobi expansions. This enables us to develop a techniquefor proving standard estimates in the Jacobi setting, which works for all possible a and b.As a consequence, we can prove that several fundamental operators in the harmonic analysis ofJacobi expansions are (vector-valued) Calder\uf3n-Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory.The new Jacobi-Poisson kernel representation also leads to sharp estimates of thiskernel. The paper generalizes methods and results existing in the literature, but validor justified only for a restricted range of a and b