Vanishing John–Nirenberg spaces

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.

A Survey on Function Spaces of John–Nirenberg Type

“…Very recently, the vanishing subspaces of John-Nirenberg spaces were also studied in [60,110]. Indeed, as a counterpart of Section 5.1, the vanishing subspaces of JNC spaces enjoy similar characterizations, which are summarized in this subsection.…”

“…namely [110] (5.2), directly from Theorems 20 and 3 because, in the statements of these dual theorems, q cannot equal 1. Indeed, (40) still holds true due to the equivalence of JN p,q (Q 0 ) with q ∈ [1, p).…”

“…The following proposition indicates that the convolution is a suitable tool when approximating functions in JN p (R n ), which is a counterpart of [6] (Lemma 1). Indeed, the approximate functions in the proofs of both Theorems 21 and 22 are constructed via the convolution (see [110] for more details).…”

“…where the supremum is taken over all collections {Q i } i of interior pairwise disjoint subcubes of R n with side lengths no more than a. Now, we recall another vanishing subspace of JN p (R n ) introduced in [110], which is of the CMO type. Definition 18.…”

“…Question 15. The proof of [110] (Theorem 3.2) indicates that (i) and (iii) of Theorem 21 are equivalent when p = 1. However, the corresponding equivalent characterization of CJN 1 (R n ) is still unclear.…”