Special John-Nirenberg-Campanato spaces via congruent cubes

Let p ∈ 1, ∞), q ∈ 1, ∞), α ∈ ?, and s be a non-negative integer. Inspired by the space JN_p introduced by John and Nirenberg (1961) and the space \({\cal B}\) introduced by Bourgain et al. (2015), we introduce a special John-Nirenberg-Campanato space \({\rm{JN}}_{{{(p,q,s)}_\alpha }}^{{\rm{con}}}\) over ?ⁿ or a given cube of ?ⁿ 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 ?ⁿ or a given cube of ?ⁿ 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.