A boundedness criterion for the maximal operator on variable Lebesgue spaces

We obtain a necessary and sufficient condition on an exponent $p(\cdot )$ for which the Hardy–Littlewood maximal operator is bounded on the variable $L^{p(\cdot )}$ space. It is formulated in terms of the Muckenhoupt-type condition $A_{p(\cdot )}$, responsible for a local control of $p(\cdot )$, and a certain integral condition on $p(\cdot )$, responsible for the behaviour of $p(\cdot )$ at infinity. Our approach is based on an earlier characterization established by L. Diening and on non-increasing rearrangements.