John--Nirenberg-$Q$ Spaces via Congruent Cubes

Abstract

To shed some light on the John--Nirenberg space, the authors in this article introduce the John--Nirenberg-$Q$ space via congruent cubes, $JNQ^\alpha_{p,q}(\mathbb{R}^n)$, which when $p=\infty$ and $q=2$ coincides with the space $Q_\alpha(\mathbb{R}^n)$ introduced by Ess\'en et al. in [Indiana Univ. Math. J. 49 (2000), 575-615]. Moreover, the authors show that, for some particular indices, $JNQ^\alpha_{p,q}(\mathbb{R}^n)$ coincides with the congruent John--Nirenberg space, or that the (fractional) Sobolev space is continuously embedded into $JNQ^\alpha_{p,q}(\mathbb{R}^n)$. Furthermore, the authors characterize $JNQ^\alpha_{p,q}(\mathbb{R}^n)$ via mean oscillations, and then use this characterization to study the dyadic counterparts. Also, the authors obtain some properties of composition operators on such spaces. The main novelties of this article are twofold: establishing a general equivalence principle for a kind of `almost increasing' set function introduced in this article, and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.