Orlicz–Morrey spaces and the Hardy–Littlewood maximal function

Abstract

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].

On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

“…The Orlicz-Morrey space M r,Φ (R n ) with 1 ≤ r < ∞ and a Young function Φ was originally introduced in Nakai [24] as a generalization of the Morrey space M r,1 (R n ), see Section 5 for its precise definition. Among others, we refer to Nakai [26] and Sawano-Sobukawa-Tanaka [36] concerning Orlicz-Morrey spaces and related further properties.…”

Inclusion properties of generalized Morrey spaces

“…We do not only give the sufficient condition for the inclusion properties of the generalized (weak) Morrey spaces, but also the necessary condition. The inclusion of generalized Orlicz–Morrey spaces can be found, for instance, in [, Theorem ] and [, Remark 1]. See also , for related results.…”

Generalized fractional integral operators over non-doubling metric measure spaces

“…We remark that there exist the generalized Orlicz-Morrey space of the first kind [12] and the generalized Orlicz-Morrey space of the third kind [13] (see [14,Section 9.1] for more details). Accordingly, starting from generalized Morrey spaces and generalized OrliczMorrey spaces, we can consider the weak space WL p,φ;κ (X, μ).…”