Restricted version of Grand Lebesgue Spaces

TL;DR

This paper introduces a restricted, discrete version of Grand Lebesgue Spaces, explores their properties, and establishes conditions for their equivalence with classical GLS, also demonstrating their Banach algebra structure under convolution.

Contribution

It presents a new discrete variant of GLS, analyzes its properties, and proves its Banach algebra structure under convolution on certain groups.

Findings

Discrete GLS coincide with classical GLS under specific conditions

Restricted GLS form a Banach algebra with convolution

Spaces retain key properties of classical GLS

Abstract

We introduce a so-called restricted, in particular, discrete version of (Banach) Grand Lebesgue Spaces (GLS), investigate its properties and derive the conditions of coincidence with the classical ones. We show also that these spaces forms also a Banach algebra relative the convolution operation on the unimodular local compact topological group equipped with Haar's measure, alike in the complete GLS case.