Discrete harmonic analysis associated with Jacobi expansions I: the heat semigroup

TL;DR

This paper develops discrete harmonic analysis tools related to Jacobi polynomials, solving the heat equation for associated operators, establishing positivity, and analyzing maximal operators in weighted spaces.

Contribution

It introduces solutions to the heat equation for Jacobi operators, proves positivity under certain conditions, and studies maximal operators using discrete Calderón-Zygmund theory.

Findings

Explicit solution to the heat equation for Jacobi operators

Positivity of the heat semigroup operator under parameter restrictions

Boundedness of maximal operators in weighted b-spaces

Abstract

In this paper we commence the study of discrete harmonic analysis associated with Jacobi orthogonal polynomials of order $(\alpha,\beta)$. Particularly, we give the solution $W^{(\alpha,\beta)}_t$, $t\ge 0$, and some properties of the heat equation related to the operator $J^{(\alpha,\beta)}-I$, where $J^{(\alpha,\beta)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator. These results will be a consequence of a much more general theorem concerning the solution of the heat equation for Jacobi matrices. In addition, we also prove the positivity of the operator $W^{(\alpha,\beta)}_t$ under some suitable restrictions on the parameters $\alpha$ and $\beta$. Finally, we investigate mapping properties of the maximal operators defined by the heat and Poisson semigroups in weighted $\ell^{p}$-spaces using discrete vector-valued local Calder\'{o}n-Zygmund theory. For the Poisson semigroup, these properties follows readily from the control in terms of the heat one.