Nontriviality of Riesz--Morrey Spaces

In this article, the authors completely answer an open question, presented in [Banach J. Math. Anal. 15 (2021), no. 1, 20], via showing that the Riesz--Morrey space is truly a new space larger than a particular Lebesgue space with critical index. Indeed, this Lebesgue space is just the real interpolation space of the Riesz--Morrey space for suitable indices. Moreover, the authors further show the aforementioned inclusion is also proper, namely, this embedding is sharp in some sense, via constructing two nontrivial spare functions, respectively, on $\mathbb{R}^n$ and any given cube $Q_0$ of $\mathbb{R}^n$ with finite side length. The latter constructed function is inspired by the striking function constructed by Dafni et al. [J. Funct. Anal. 275 (2018), 577--603]. All the proofs of these results strongly depend on some exquisite geometrical analysis on cubes of $\mathbb{R}^n$. As an application, the relationship between Riesz--Morrey spaces and Lebesgue spaces is completely clarified on all indices.