  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.
