Some mudular inequalities in Lebesgue spaces with variable exponent

TL;DR

This paper investigates modular inequalities for operators like the Bergman projection in Lebesgue spaces with variable exponents, showing they hold only when the exponent is essentially constant.

Contribution

It establishes a necessary and sufficient condition for modular inequalities to hold in variable exponent Lebesgue spaces, linking them to the constancy of the exponent.

Findings

Modular inequalities hold only if the variable exponent is almost everywhere constant.

A key lemma provides a lower pointwise bound for operators acting on characteristic functions.

The results characterize when these inequalities are valid in variable exponent spaces.

Abstract

Our aim is to study the modular inequalities for some operators, for example the Bergman projection acting on, in Lebesgue spaces with variable exponent. Under proper assumptions on the variable exponent, we prove that the modular inequalities are hold if and only if the exponent almost everywhere equals to a constant. In order to get the main results, we prove a lemma for a lower pointwise bound for these operators of a characteristic function.