Modular inequalities for the maximal operator in variable Lebesgue spaces

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 $${\int }_{\mathrm{\Omega }}Mf(x{)}^{p(x)}dx\le c_1{\int }_{\mathrm{\Omega }}\mathrm{\mid }f(x){\mathrm{\mid }}^{q(x)}dx+c_2,$$ where $c_1,c_2$ are non-negative constants and $\mathrm{\Omega }$ is any measurable subset of ${\mathbb{R}}^n$. As a corollary we get sufficient conditions for the modular inequality $${\int }_{\mathrm{\Omega }}\mathrm{\mid }Tf(x){\mathrm{\mid }}^{p(x)}dx\le c_1{\int }_{\mathrm{\Omega }}\mathrm{\mid }f(x){\mathrm{\mid }}^{q(x)}dx+c_2,$$ where $T$ is any operator that is bounded on $L^p(\mathrm{\Omega })$, $1<p<\mathrm{\infty }$.