Discrete harmonic analysis associated with ultraspherical expansions

TL;DR

This paper investigates discrete harmonic analysis related to ultraspherical functions, establishing boundedness of maximal operators, g-functions, and transplantation operators, supported by a new vector-valued Calderón-Zygmund theorem in the discrete setting.

Contribution

It introduces new boundedness results for operators linked to ultraspherical functions and develops a discrete Calderón-Zygmund theorem, advancing harmonic analysis techniques.

Findings

Weighted l^p-boundedness of maximal operators

Boundedness of Littlewood-Paley g-functions

Boundedness of transplantation operators

Abstract

We study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted l^p-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by certain difference operator. We also prove weighted l^p-boundedness properties of transplantation operators associated to the system of ultraspherical functions. In order to show our results we previously establish a vector-valued local Calder\'on-Zygmund theorem in our discrete setting.