Modular inequalities for the maximal operator in variable Lebesgue spaces

A now classical result in the theory of variable Lebesgue spaces due to Lerner (2005) is that a modular inequality for the Hardy Littlewood maximal function in L-p(.) (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