We introduce and study the median maximal function \mathcal{ M} f, defined in the same manner as the classical Hardy-Littlewood maximal function, only replacing integral averages of f by medians throughout the definition. This change has a qualitative impact on the mapping properties of the maximal operator: in contrast with the Hardy-Littlewood operator, which is not bounded on L^1, we prove that \mathcal{ M} is bounded on L^p(w) for all 0 < p < \infty, if and only if w \in A_{ \infty}. The characterization is purely qualitative and does not give the dependence on [w]_{ A_{ \infty}}. However, the sharp bound \|\mathcal{ M}\|_{ L^1(w) \to L^1(w)} \lesssim [w]_{ A_1} is established.
Categories: math.CA