Restricted boundedness of translation operators on variable Lebesgue spaces

TL;DR

This paper studies the boundedness of translation operators on variable Lebesgue spaces, establishing conditions under which translations are bounded, which advances understanding of these spaces' structure and operator behavior.

Contribution

It provides new conditions for the boundedness of translation operators on variable Lebesgue spaces, extending previous results in the field.

Findings

Established boundedness conditions for translation operators

Identified specific assumptions on functions and exponents

Contributed to the theory of variable Lebesgue spaces

Abstract

In this paper, we investigate the inequality \begin{equation*} \left\Vert f(\cdot +h)\right\Vert_{p\left( \cdot \right) }\leq A\left\Vert f\right\Vert_{p\left( \cdot \right) },\quad h\in \mathbb{R}^{n}, A>0 \end{equation*} under some suitable assumptions on the function $f$ and the variable exponent $p$.