Abstract
In this systematic review, the authors give a survey on the recent developments of both the John–Nirenberg space JNp and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO, VJNp, and CJNp on Rn or a given cube Q0⊂Rn with finite side length. In addition, some related open questions are also presented.
Highlights
In this article, a cube Q means that it has finite side length and all its sides parallel to the coordinate axes, but Q is not necessarily open or closed
We further review the compactness theorems of the Calderón–Zygmund commutators [b, T], where b belongs to the vanishing subspaces CMO (Rn) as well as XMO (Rn), and propose an open question on [b, T] with b ∈ XMO (Rn)
It is well known that the space BMO has played an important role in harmonic analysis, partial differential equations, and other mathematical fields since it was introduced by John and Nirenberg in their celebrated article [1]
Summary
A cube Q means that it has finite side length and all its sides parallel to the coordinate axes, but Q is not necessarily open or closed. We first recall their definitions and review their (except MMO (Rn)) mean oscillation characterizations, respectively, in Theorems 11–13 below.
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