Published Paper

Inserted: 30 apr 2021
Last Updated: 14 jun 2023

Journal: Nonlinear Analysis
Year: 2017
Doi: 10.1016/j.na.2018.01.007

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 $L^p(\Omega)$, $1<p<\infty$.