Some weighted estimates for discrete fractional maximal operators

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 [E. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982) 1–11]. Similar results also hold 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] (respectively, [Formula: see text]) on discrete weighted Morrey spaces.