Abstract Suppose that the d -dimensional unit cube Q is the union of three disjoint “simple” sets E , F and G and that the volumes of E and F are both greater than half the volume of G . Does this imply that, for some cube W contained in Q . the volumes of E ∩ W and F ∩ W both exceed s times the volume of W for some absolute positive constant s ? Here, by “simple” we mean a set which is a union of finitely many dyadic cubes. We prove that an affirmative answer to this question would have deep consequences for the important space B M O of functions of bounded mean oscillation introduced by John and Nirenberg. We recall and use the notion of a John–Stromberg pair which is closely related to the above question. The above mentioned result is obtained as a consequence of a general result about these pairs. We also present a number of additional results about these pairs.

中文翻译：

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Rn 中的立方体的并集、Zn 中的组合以及 John-Nirenberg 和 John-Strömberg 不等式

摘要 假设 d 维单位立方体 Q 是三个不相交的“简单”集合 E 、 F 和 G 的并集，并且 E 和 F 的体积都大于 G 体积的一半。这是否意味着，对于包含在 Q 中的某些立方体 W。对于某个绝对正常数 s，E ∩ W 和 F ∩ W 的体积都超过 W 体积的 s 倍？在这里，“简单”是指一个集合，它是有限多个二元立方体的并集。我们证明，对这个问题的肯定回答将对 John 和 Nirenberg 引入的有界平均振荡函数的重要空间 BMO 产生深远的影响。我们回忆并使用与上述问题密切相关的 John-Stromberg 对的概念。上述结果是作为关于这些对的一般结果的结果而获得的。