Nontriviality of Riesz–Morrey spaces

Abstract

In this article, the authors completely answer an open question, presented in Tao et al. [A bridge connecting Lebesgue and Morrey spaces via Riesz norms. Banach J Math Anal. 2021;15(1):20], via showing that the Riesz–Morrey space is truly a new space larger than a particular Lebesgue space with a 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 that 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. [The space $J{N}_p$: nontriviality and duality. J Funct Anal. 2018;275:577–603]. 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.

Keywords:

• Lebesgue space
• Morrey space
• Riesz norm
• Riesz–Morrey space

2020 Mathematics Subject Classifications:

• Primary: 42B35
• Secondary: 46E30
• 46E35

Acknowledgments

Zongze Zeng and Jin Tao would like to thank Yangyang Zhang for some helpful discussions on Proposition 3.3.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This project is supported by the National Natural Science Foundation of China [grant numbers 11971058 and 12071197] and the National Key Research and Development Program of China [grant number 2020YFA0712900], and Der-Chen Chang is supported by an NSF [grant DMS-1408839] and a McDevitt Endowment Fund at Georgetown University.