On a Class of Non-Integrable Multipliers for the Jacobi Transform

TL;DR

This paper identifies conditions under which certain bounded functions induce bounded convolution operators for the Jacobi transform, extending previous results to a broader class of functions and settings.

Contribution

It introduces a new criterion involving boundary values of functions for ensuring $L^p$-boundedness of Jacobi convolution operators, generalizing prior work on symmetric spaces.

Findings

Bounded functions not integrable at infinity can produce bounded Jacobi convolution operators.

Boundary value conditions relate Jacobi multipliers to Euclidean Fourier multipliers.

Extension of previous results to more general spaces and functions.

Abstract

We show that a bounded function $m$ on $\R$ not necessarily integrable at infinity may still yield $L^p$-bounded convolution operators for the Jacobi transform if the nontangential boundary values of $\omega m$ along the edges of a certain strip in $\C$ yield Euclidean Fourier multipliers, for $\omega$ suitably defined. This partially generalizes similar results by Giulini, Mauceri, and Meda (on rank one symmetric spaces) and Astengo (on Damek--Ricci spaces).