A boundedness criterion for the maximal operator on variable Lebesgue spaces

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