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

TL;DR

This paper introduces the John--Nirenberg-$Q$ space via congruent cubes, explores its properties, embeddings, and characterizations, and develops new principles for set functions and cube classifications.

Contribution

It defines a new class of function spaces, establishes their properties and embeddings, and introduces novel principles for set function equivalences and cube classifications.

Findings

$JNQ^eta_{p,q}(R^n)$ coincides with known spaces for specific indices

Fractional Sobolev spaces embed continuously into $JNQ^eta_{p,q}(R^n)$

New principles for set functions and cube classifications are established.

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.