Discrete harmonic analysis associated 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 Wt(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 Wt(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 ℓp-spaces using discrete vector-valued local Calderón-Zygmund theory. Moreover, we treat the Poisson semigroup by means of an appropriate subordination identity.