Suppose Formula: see text, Formula: see text and Formula: see text. Let Formula: see text, Formula: see text and Formula: see text be three sequences of positive real numbers. Assume that Formula: see text satisfies the doubling condition: Formula: see text for all integral intervals Formula: see text. Then Formula: see text for all sequences Formula: see text if and only if Formula: see text for all integral intervals Formula: see text, where Formula: see text denotes the discrete Formula: see text-fractional maximal operator: Formula: see text where Formula: see text, which are the discrete variants of Sawyer’s corresponding results Studia Math., 1982, 75: 1-11. Similar results also holds for discrete dyadic Formula: see text-fractional maximal operator Formula: see text. When picking Formula: see text, we further obtain some sufficient conditions for the boundedness of Formula: see text (resp. Formula: see text) on discrete weighted Morrey spaces.
Chen et al. (Fri,) studied this question.