Analysis related to all admissible type parameters in the Jacobi setting
Journal article, 2015
We derive an integral representation for the Jacobi-Poisson kernel valid for all admissible
type parameters a and b in the context of Jacobi expansions. This enables us to develop a technique
for 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 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 a and b.