Discrete Harmonic Analysis Associated with Jacobi Expansions

Résumé

In this work we consider the operator associated with the three-term recurrence relation for the Jacobi polynomials and we study some classical operators in Harmonic Analysis in this context. Particularly, we are interested in the heat and Poisson semigroups and in the maximal operators related to them, in the Riesz transforms, and in the Littlewood-Paley-Stein g_k-functions. We obtain weighted l^p-inequalities for the heat and Poisson maximal operators and for the Riesz transforms when p>1 and the parameters of the Jacobi polynomials are greater than or equal to -1/2, and weighted weak inequalities in the case p=1 and the parameters greater than or equal to -1/2. We give weighted l^p-estimates for the g_k-functions when p>1 and the parameters are greater than or equal to -1/2. The method to prove these inequalities is based on the vector-valued Calderón-Zygmund theory in spaces of homogeneous type.