A Note on the Maximal Operator on Weighted Morrey Spaces

In this paper we consider weighted Morrey spaces ℳλ,ℱp(w) adapted to a family of cubes ℱ , with the norm ‖f‖ℳλ,ℱp(w):=supQ∈ℱ(1∣Q∣λ∫Q∣f∣pw)1/p, and the question we deal with is whether a Muckenhoupt-type condition characterizes the boundedness of the Hardy–Littlewood maximal operator on ℳλ,ℱp(w) . In the case of the global Morrey spaces (when ℱ is the family of all cubes in ℝⁿ) this question is still open. In the case of the local Morrey spaces (when ℱ is the family of all cubes centered at the origin) this question was answered positively in a recent work of Duoandikoetxea and Rosenthal [2]. We obtain an extension of [2] by showing that the answer is positive when ℱ is the family of all cubes centered at a sequence of points in ℝⁿ satisfying a certain lacunary-type condition.