A Note on the Maximal Operator on Weighted Morrey Spaces

In this paper we consider weighted Morrey spaces \({\cal M}_{\lambda ,{\cal F}}^p(w)\) adapted to a family of cubes \({\cal F}\), with the norm

$$\Vert f\Vert{_{{\cal M}_{\lambda ,{\cal F}}^p(w)}}: = \mathop {\sup }\limits_{Q \in {\cal F}} {\left( {{1 \over {|Q{|^\lambda }}}\int_Q {|f{|^p}w} } \right)^{1/p}},$$

and the question we deal with is whether a Muckenhoupt-type condition characterizes the boundedness of the Hardy–Littlewood maximal operator on \({\cal M}_{\lambda ,{\cal F}}^p(w)\).

In the case of the global Morrey spaces (when \({\cal F}\) is the family of all cubes in ℝⁿ) this question is still open. In the case of the local Morrey spaces (when \({\cal F}\) 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 \({\cal F}\) is the family of all cubes centered at a sequence of points in ℝⁿ satisfying a certain lacunary-type condition.