Discrete harmonic analysisassociated with Jacobi expansions I: the heat semigroup

Abstract In this paper we commence the study of discrete harmonic analysis associated with Jacobi orthogonal polynomials of order ( α , β ) . Particularly, we analyse the heat semigroup W t ( α , β ) , t ≥ 0 , related to the operator J ( α , β ) − I , where J ( α , β ) is the three-term recurrence relation for the normalised Jacobi polynomials and I is the identity operator. First, we prove the positivity of the operator W t ( α , β ) under some suitable restrictions on the parameters α and β. In our main result, we investigate the mapping properties of the maximal operator defined by the heat semigroup in weighted l p -spaces using discrete vector-valued local Calderon-Zygmund theory. Moreover, we treat the Poisson semigroup by means of an appropriate subordination identity.