Special John-Nirenberg-Campanato spaces via congruent cubes
Let \(p\in [1,\infty)\), \(q \in [1,\infty)\), \(\alpha \in {\mathbb R}\), and \(s\) be a non-negative integer. Inspired by the space \(JN_p\) introduced by \textit{F. John} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 14, 415--426 (1961; Zbl 0102.04302)] and the space \(B\) introduced by \textit{J. Bourgain} et al. [J. Eur. Math. Soc. (JEMS) 17, No. 9, 2083--2101 (2015; Zbl 1339.46028)], we introduce a special John-Nirenberg-Campanato space \(JN^{\mathrm{con}}_ {(p,q,s)_{\alpha}}\) over \({\mathbb R}^n\) or a given cube of \({\mathbb R}^n\) with finite side length via congruent subcubes, which are of some amalgam features. The limit space of such spaces as \(p\to \infty\) is just the Campanato space which coincides with the space BMO (the space of functions with bounded mean oscillations) when \(\alpha = 0\). 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 \({\mathbb R}^n\) or a given cube of \({\mathbb R}^n\) with finite side length. Furthermore, some VMO-\(H_{1}\)-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 proof of all these results.
scientific article; zbMATH DE number 7484068
scientific article