Special John-Nirenberg-Campanato spaces via congruent cubes

Let p ∈ [1, ∞), q ∈ [1, ∞), α ∈ ℝ, and s be a non-negative integer. Inspired by the space JNp introduced by John and Nirenberg (1961) and the space ℬ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal B}$$\end{document} introduced by Bourgain et al. (2015), we introduce a special John-Nirenberg-Campanato space JN(p,q,s)αcon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{JN}}_{{{(p,q,s)}_\alpha }}^{{\rm{con}}}$$\end{document} over ℝn or a given cube of ℝn with finite side length via congruent subcubes, which are of some amalgam features. The limit space of such spaces as p → ∞ is just the Campanato space which coincides with the space BMO (the space of functions with bounded mean oscillations) when α = 0. Moreover, a vanishing subspace of this new space is introduced, and its equivalent characterization is established as well, which is a counterpart of the known characterization for the classical space VMO (the space of functions with vanishing mean oscillations) over ℝn or a given cube of ℝn with finite side length. Furthermore, some VMO-H1-BMO-type results for this new space are also obtained, which are based on the aforementioned vanishing subspaces and the Hardy-type space defined via congruent cubes in this article. The geometrical properties of both the Euclidean space via its dyadic system and congruent cubes play a key role in the proofs of all these results.