We study the connection between the Muckenhoupt Ap weights and bounded mean oscillation (BMO) for general bases for ℝd. New classes of bases are introduced that allow for several deep results on the Muckenhoupt weights–BMO connection to hold in a very general form. The John–Nirenberg type inequality and its consequences are valid for the new class of Calderón–Zygmund bases which includes cubes in ℝd, but also the basis of rectangles in ℝd. Of particular interest to us is the Garnett–Jones theorem on the BMO distance, which is valid for cubes. We prove that the theorem is equivalent to the newly introduced A2-decomposition property of bases. Several sufficient conditions for the theorem to hold are analysed as well. However, the question whether the theorem fully holds for rectangles remains open.

中文翻译：

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Muckenhoupt 类权重分解和 BMO 到有界函数的距离

我们研究 Muckenhoupt 之间的联系一种pℝ 的一般基的权重和有界平均振荡 (BMO)d. 引入了新的基类，允许在 Muckenhoupt 权重-BMO 连接上以非常一般的形式保持几个深入的结果。John-Nirenberg 类型不等式及其结果适用于新的 Calderón-Zygmund 基类，其中包括 ℝ 中的立方体d，也是ℝ中矩形的基础d. 我们特别感兴趣的是关于 BMO 距离的 Garnett-Jones 定理，它对立方体有效。我们证明该定理等价于新引入的一种2-碱基的分解特性。还分析了定理成立的几个充分条件。然而，该定理是否完全适用于矩形的问题仍然悬而未决。