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 $$\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathcal{M}}_{\lambda ,\mathcal{F}}^p(w)}:=\underset{Q\in \mathcal{F}}{\sup }{\left(\frac{1}{\mathrm{\mid }Q{\mathrm{\mid }}^{\lambda }}{\int }_Q\mathrm{\mid }f{\mathrm{\mid }}^{p}w\right)}^{1\mathrm{/}p},$$ 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.