Abstract
Abstract There still exist many unsolved problems on the study related to John–Nirenberg spaces. In this article, the authors introduce two new vanishing subspaces of the John–Nirenberg space JN p ( ℝ n ) {\mathrm{JN}_{p}(\mathbb{R}^{n})} denoted, respectively, by VJN p ( ℝ n ) {\mathrm{VJN}_{p}(\mathbb{R}^{n})} and CJN p ( ℝ n ) {\mathrm{CJN}_{p}(\mathbb{R}^{n})} , and establish their equivalent characterizations which are counterparts of those characterizations for the classic spaces VMO ( ℝ n ) {\mathrm{VMO}(\mathbb{R}^{n})} and CMO ( ℝ n ) {\mathrm{CMO}(\mathbb{R}^{n})} obtained, respectively, by D. Sarason and A. Uchiyama. All these results shed some light on the mysterious space JN p ( ℝ n ) {\mathrm{JN}_{p}(\mathbb{R}^{n})} . The approach strongly depends on the fine geometrical properties of dyadic cubes, which enable the authors to subtly classify any collection of interior pairwise disjoint cubes.
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