Harmonic analysis operators related to symmetrized Jacobi expansions for all admissible parameters

TL;DR

This paper extends the analysis of harmonic operators related to symmetrized Jacobi expansions, removing previous parameter restrictions and providing new insights into their mapping properties and related classical expansions.

Contribution

It fully removes earlier parameter restrictions using recent results and offers shorter proofs, also exploring symmetrized Jacobi function expansions and classical cases.

Findings

Complete removal of parameter restrictions for harmonic analysis operators

Shorter, more transparent proofs of previous results

New results on classical Jacobi polynomial and function expansions

Abstract

This is an ultimate completion of our earlier paper [Acta.\ Math.\ Hungar.\ 140 (2013), 248--292] where mapping properties of several fundamental harmonic analysis operators in the setting of symmetrized Jacobi trigonometric expansions were investigated under certain restrictions on the underlying parameters of type. In the present article we take advantage of very recent results due to Nowak, Sj\"ogren and Szarek to fully release those restrictions, and also to provide shorter and more transparent proofs of the previous restricted results. Moreover, we also study mapping properties of analogous operators in the parallel context of symmetrized Jacobi function expansions. Furthermore, as a consequence of our main results we conclude some new results related to the classical non-symmetrized Jacobi polynomial and function expansions.