Modular inequalities for the maximal operator in variable Lebesgue spaces

A now classical result in the theory of variable Lebesgue spaces due to Lerner is that a modular inequality for the Hardy-Littlewood maximal function in $L^{\pp}(\R^n)$ 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 $\R^n$. 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 $L^p(\Omega)$, $1<p<\infty$.