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Orlicz Space on Groupoids

K. N. Sridharan, N. Shravan Kumar

TL;DR

This paper develops a groupoid analogue of Orlicz-space convolution theory by formulating a continuous field of Orlicz spaces over the unit space $G^{0}$ for a locally compact groupoid with a fixed left Haar system. It proves that under the $Δ_{2}$ condition on the complementary pair of $N$-functions $(Φ,Ψ)$, the space of continuous sections vanishing at infinity $E_0^{Φ}$ becomes a Banach algebra under convolution and provides a criterion for when closed $C_b(G^{0})$-submodules are left ideals, via invariance under the left-regular representation. It then identifies the space of convolutors $C_{V_{Φ}}(G)$ with the dual of a groupoid Fourier-type algebra $reve{A}_{Φ}(G)$ through a canonical map, extending the group-based duality to the groupoid context. Overall, the results generalize known group and hypergroup theories to the groupoid setting, furnishing foundational tools for harmonic analysis on groupoids using Orlicz spaces.

Abstract

Let $G$ be a locally compact second countable groupoid with a fixed Haar system $λ=\{λ^{u}\}_{u\in G^{0}}$ and $(Φ,Ψ)$ be a complementary pair of $N$-functions satisfying $Δ_{2}$-condition. In this article, we introduce the continuous field of Orlicz space $(L^Φ_{0},Δ_{1})$ and provide a sufficient condition for the space of continuous sections vanishing at infinity, denoted $E^Φ_{0}$, to be an Banach algebra under a suitable convolution. Further, the condition for a closed $C_{b}(G^{0})$-submodule $I$ of $E^Φ_{0}$ 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.

Orlicz Space on Groupoids

TL;DR

This paper develops a groupoid analogue of Orlicz-space convolution theory by formulating a continuous field of Orlicz spaces over the unit space for a locally compact groupoid with a fixed left Haar system. It proves that under the condition on the complementary pair of -functions , the space of continuous sections vanishing at infinity becomes a Banach algebra under convolution and provides a criterion for when closed -submodules are left ideals, via invariance under the left-regular representation. It then identifies the space of convolutors with the dual of a groupoid Fourier-type algebra through a canonical map, extending the group-based duality to the groupoid context. Overall, the results generalize known group and hypergroup theories to the groupoid setting, furnishing foundational tools for harmonic analysis on groupoids using Orlicz spaces.

Abstract

Let be a locally compact second countable groupoid with a fixed Haar system and be a complementary pair of -functions satisfying -condition. In this article, we introduce the continuous field of Orlicz space and provide a sufficient condition for the space of continuous sections vanishing at infinity, denoted , to be an Banach algebra under a suitable convolution. Further, the condition for a closed -submodule of 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.
Paper Structure (4 sections, 18 theorems, 62 equations)

This paper contains 4 sections, 18 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Continuous Field of Orlicz Space
  4. The space of Convolutors of $E_{0}^{\Phi}$

Key Result

Theorem 3.1

For $f\in C_{c}(G)$, the function $g : G^{0}\to \mathbb{R}$ defined as $g(u) = \|f^{u}\|_{\Phi}^{0}$, where $f^{u} = f_{|_{G^{u}}}$, is continuous over $G^{0}$ with compact support.

Theorems & Definitions (32)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • Definition 3.7
  • ...and 22 more