A note on the maximal operator on weighted Morrey spaces

In this paper we consider weighted Morrey spaces ${\mathcal M}_{\lambda, {\mathcal F}}^p(w)$ adapted to a family of cubes ${\mathcal F}$, with norm $$\|f\|_{{\mathcal M}_{\lambda, {\mathcal F}}^p(w)}:=\sup_{Q\in {\mathcal F}}\left(\frac{1}{|Q|^{\lambda}}\int_Q|f|^pw\right)^{1/p},$$ and the question we deal with is whether a Muckenhoupt-type condition characterizes the boundedness of the Hardy--Littlewood maximal operator on ${\mathcal M}_{\lambda, {\mathcal F}}^p(w)$. In the case of the global Morrey spaces (when ${\mathcal F}$ is the family of all cubes in ${\mathbb R}^n$) this question is still open. In the case of the local Morrey spaces (when ${\mathcal F}$ is the family of all cubes centered at the origin) this question was answered positively in a recent work of Duoandikoetxea--Rosenthal \cite{DR21}. We obtain an extension of \cite{DR21} by showing that the answer is positive when ${\mathcal F}$ is the family of all cubes centered at a sequence of points in ${\mathbb R}^n$ satisfying a certain lacunary-type condition.