A bridge connecting Lebesgue and Morrey spaces via Riesz norms

Abstract

In this article, via combining Riesz norms with Morrey norms, the authors introduce and study the so-called Riesz-Morrey space, which differs from the John-Nirenberg-Campanato space in subtracting integral means. These spaces provide a bridge connecting both Lebesgue spaces and Morrey spaces which prove to be the endpoint spaces of Riesz-Morrey spaces. Moreover, the authors introduce a block-type space which proves to be the predual space of the Riesz-Morrey space.

A Survey on Function Spaces of John–Nirenberg Type

“…which coincides with f L p (Q 0 ) due to Riesz [41]. Corresponding to the JNC space, the following triple index Riesz-type space R p,q,α (X), called the Riesz-Morrey space, was introduced and studied in [37] and, independently, by Fofana et al [87] when X = R n .…”

“…Observe that the Riesz-Morrey norm • RM p,q,α (X) is different from the JNC norm • JN (p,q,s)α (X) with s = 0, only in subtracting mean oscillations (see [37], Remark 2, for more details). It is easy to see that • R p,1,0 (Q 0 ) = • R p (Q 0 ) , and, as a generalization of the above equivalence in Riesz [41], the following proposition is just [37] (Proposition 1). Proposition 28.…”

“…with equivalent norms. Moreover, it is shown in [37] (Theorem 1 and Corollary 1) that the endpoint spaces of Riesz-Morrey spaces are Lebesgue spaces or Morrey spaces. In this sense, we regard the Riesz-Morrey space as a bridge connecting the Lebesgue space and the Morrey space.…”

“…Therefore, John-Nirenberg spaces are new spaces between Lebesgue spaces and weak Lebesgue spaces, which motivates us to study the properties of JN p . Furthermore, various John-Nirenberg-type spaces have also attracted a lot of attention in recent years (see, for instance, [31][32][33][34][35][36][37] for the Euclidean space case and [30,[38][39][40] for the metric measure space case).…”

“…In Section 4, via subtracting integral means in the JNC space, we first give the definition of the Riesz-type space appearing in [37] and then present some basic facts about this space in Section 4.1. Moreover, the predual space (namely, the block-type space) and the corresponding dual theorem of the Riesz-type space are also displayed in this subsection.…”