The Fourier transform in variable exponent Lebesgue spaces

TL;DR

This paper introduces a Fourier transform for variable exponent Lebesgue spaces, defining it via distributional derivatives and establishing its properties, including isometric isomorphism, inversion, and exchange theorems.

Contribution

It defines a novel Fourier transform in variable exponent Lebesgue spaces and proves key properties, expanding harmonic analysis in these variable exponent contexts.

Findings

Fourier transform is an isometric isomorphism with $L^{p(ullet)}$

Inversion formula holds in norm

Exchange theorem established for the transform

Abstract

In this work we define a Fourier transform for each $f\in L^{p(\cdot)}(\mathbb{R})$, for a large class of exponent functions $p(\cdot)$, as the distributional derivative of a H\"older continuous function. A norm is defined in the space of such Fourier transforms so that it is isometrically isomorphic to $L^{p(\cdot)}(\mathbb{R})$. We also prove several properties of this Fourier transform, such as inversion in norm and an exchange theorem.