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 $$\int_\Omega Mf(x)^{p(x)}\,dx \ \leq c_1 \int_\Omega |f(x)|^{q(x)}\,dx + c_2,$$ 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 $L^p(\Omega)$, $1<p<\infty$.