Orlicz Space on Groupoids

TL;DR

This paper develops a framework for Orlicz spaces on groupoids, establishing Banach algebra structures and ideal characterizations, extending classical group results to the more general setting of groupoids.

Contribution

It introduces continuous fields of Orlicz spaces on groupoids and characterizes Banach algebra and ideal structures within this context.

Findings

Established Banach algebra structure for certain Orlicz spaces on groupoids.

Provided conditions for submodules to be left ideals.

Extended classical convolution characterizations to groupoids.

Abstract

Let $G$ be a locally compact second countable groupoid with a fixed Haar system $\lambda=\{\lambda^{u}\}_{u\in G^{0}}$ and $(\Phi,\Psi)$ be a complementary pair of $N$-functions satisfying $\Delta_{2}$-condition. In this article, we introduce the continuous field of Orlicz space $(L^{\Phi}_{0},\Delta_{1})$ and provide a sufficient condition for the space of continuous sections vanishing at infinity, denoted $E^{\Phi}_{0}$, to be an Banach algebra under a suitable convolution. Further, the condition for a closed $C_{b}(G^{0})$-submodule $I$ of $E^{\Phi}_{0}$ to be a left ideal is established. Moreover, we provide a groupoid analogue of the characterization of the space of convolutors of Morse-Transue space for locally compact groups.