# Discrete Harmonic Analysis associated with Jacobi expansions III: the Littlewood-Paley-Stein $g_{k}$-functions and the Laplace type multipliers

## TL;DR

This paper extends harmonic analysis related to Jacobi expansions by studying Littlewood-Paley-Stein $g_k$-functions and Laplace multipliers, establishing norm equivalences and multiplier results.

## Contribution

It introduces and analyzes Littlewood-Paley-Stein $g_k$-functions for Jacobi operators, proving norm equivalences with weights and deriving Laplace multiplier results.

## Key findings

- Established norm equivalence for $g_k^{(eta,eta)}$-functions with weights.
- Proved boundedness of Laplace type multipliers on Jacobi expansions.
- Extended harmonic analysis framework for Jacobi operators.

## Abstract

The research about Harmonic Analysis associated with Jacobi expansions carried out in \cite{ACL-JacI} and \cite{ACL-JacII} is continued in this paper. Given the operator $\mathcal{J}^{(\alpha,\beta)}=J^{(\alpha,\beta)}-I$, where $J^{(\alpha,\beta)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we define the corresponding Littlewood-Paley-Stein $g_k^{(\alpha,\beta)}$-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.