This article is cited in 1 paper

Variable Lebesgue algebra on a Locally Compact group

P. Saha^a, B. Hazarika<sup>b

a</sup> Department of Mathematics, Sipajhar College, Sipajhar, Darrang-784145, Assam, India
^b Department of Mathematics, Gauhati University, Guwahati-781014, Assam, India

Abstract: For a locally compact group $H$ with a left Haar measure, we study the variable Lebesgue algebra $\mathcal{L}^{p(\cdot)}(H)$ with respect to convolution. We show that if $\mathcal{L}^{p(\cdot)}(H)$ has a bounded exponent, then it contains a left approximate identity. We also prove a necessary and sufficient condition for $\mathcal{L}^{p(\cdot)}(H)$ to have an identity. We observe that a closed linear subspace of $\mathcal{L}^{p(\cdot)}(H)$ is a left ideal if and only if it is left translation invariant.

Keywords: variable Lebesgue space, bounded exponent, approximate identity, Haar measure.

UDC: 517.986.6

MSC: 43A10, 43A15, 43A75, 43A77

Received: 17.07.2022
Revised: 26.12.2022
Accepted: 29.12.2022

DOI: 10.15393/j3.art.2023.12110

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