# Algebras of Continuous Fourier Multipliers on Variable Lebesgue Spaces

## TL;DR

This paper establishes the equivalence of various definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces, providing new insights even for classical Lebesgue spaces and answering longstanding questions.

## Contribution

It demonstrates the equivalence of multiple definitions of these algebras under natural conditions and addresses open questions in the field.

## Key findings

- Several definitions are equivalent under natural assumptions.
- New results extend to standard Lebesgue spaces.
- Answers two open questions about Fourier multiplier algebras.

## Abstract

We show that several definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces over the real line are equivalent under some natural assumptions on variable exponents. Some of our results are new even in the case of standard Lebesgue spaces and give answers on two questions about algebras of continuous Fourier multipliers on Lebesgue spaces over the real line posed by H. Mascarenhas, P. Santos and M. Seidel.