Abstract. We prove basic properties of Orlicz-Morrey spaces and give a necessary and sufficient condition for boundedness of the Hardy-Littlewood maximal operator M from one Orlicz-Morrey space to another. For example, if f ∈ L(log L)(R n ), then M f is in a (generalized) Morrey space (Example 5.1). As an application of boundedness of M , we prove the boundedness of generalized fractional integral operators, improving earlier results of the author.1. Introduction. Orlicz spaces, introduced in [29,30], are generalizations of Lebesgue spaces L p . They are useful tools in harmonic analysis and its applications. For example, the Hardy-Littlewood maximal operator is bounded on L p for 1 < p ≤ ∞, but not on L 1 . Using Orlicz spaces, we can investigate the boundedness of the operator near p = 1 precisely (see Kita [14,15] and Cianchi [4]). It is known that the fractional integral operator I α is bounded from L p (R n ) to L q (R n ) for 1 < p < q < ∞ and −n/p+α = −n/q (the Hardy-Littlewood-Sobolev theorem). Trudinger [40] investigated the boundedness of I α near q = ∞. The Hardy-Littlewood-Sobolev theorem and Trudinger's result have been generalized by several authors: [28,37,38,5,4,23,24,25], etc. For the theory of Orlicz spaces, see [18,16,33].On the other hand, Morrey spaces were introduced in [19] to estimate solutions of partial differential equations, and studied in many papers. For the boundedness of the Hardy-Littlewood maximal operator and fractional integral operators, see [31,1,3,20].
