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We derive an integral representation for the Jacobi–Poisson kernel valid for all admissible type parameters  in the context of Jacobi expansions. This enables us to develop a technique for proving standard estimates in the Jacobi setting that works for all possible  and . As a consequence, we can prove that several fundamental operators in the harmonic analysis of Jacobi expansions are (vector-valued) Calderón–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 this kernel. The paper generalizes methods and results existing in the literature but valid or justified only for a restricted range of  and .

Analysis Related to All Admissible Type Parameters in the Jacobi Setting