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To shed some light on the John--Nirenberg space, the authors in this articleintroduce the John--Nirenberg-$Q$ space via congruent cubes,$JNQ^\alpha_{p,q}(\mathbb{R}^n)$, which when $p=\infty$ and $q=2$ coincideswith 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 withthe congruent John--Nirenberg space, or that the (fractional) Sobolev space iscontinuously embedded into $JNQ^\alpha_{p,q}(\mathbb{R}^n)$. Furthermore, theauthors characterize $JNQ^\alpha_{p,q}(\mathbb{R}^n)$ via mean oscillations,and then use this characterization to study the dyadic counterparts. Also, theauthors obtain some properties of composition operators on such spaces. Themain novelties of this article are twofold: establishing a general equivalenceprinciple for a kind of `almost increasing' set function introduced in thisarticle, and using the fine geometrical properties of dyadic cubes to properlyclassify any collection of cubes with pairwise disjoint interiors and equaledge length.

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