Modular inequalities for the maximal operator in variable Lebesgue spaces

TL;DR

This paper extends classical results on modular inequalities for the Hardy-Littlewood maximal operator in variable Lebesgue spaces, providing new necessary and sufficient conditions and simplified proofs for these inequalities.

Contribution

It generalizes Lerner's result, offering a new proof and establishing necessary and sufficient conditions for modular inequalities involving variable exponents.

Findings

Modular inequality holds only if the exponent is constant.

Necessary and sufficient conditions for weaker modular inequalities are identified.

Sufficient conditions for modular inequalities for bounded operators are provided.

Abstract

A now classical result in the theory of variable Lebesgue spaces due to Lerner [A. K. Lerner, On modular inequalities in variable $L^p$ spaces, Archiv der Math. 85 (2005), no. 6, 538-543] is that a modular inequality for the Hardy-Littlewood maximal function in $L^{p(\cdot)}(\mathbb{R}^n)$ holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality \[ \int_\Omega Mf(x)^{p(x)}\,dx \ \leq c_1 \int_\Omega |f(x)|^{q(x)}\,dx + c_2, \] where $c_1,\,c_2$ are non-negative constants and $\Omega$ is any measurable subset of $\mathbb{R}^n$. As a corollary we get sufficient conditions for the modular inequality \[ \int_\Omega |Tf(x)|^{p(x)}\,dx \ \leq c_1 \int_\Omega |f(x)|^{q(x)}\,dx + c_2, \] where $T$ is any operator that is bounded on $L^p(\Omega)$, $1<p<\infty$.