We present a version of the John–Nirenberg inequality for a sub-class of BMO by estimating the corresponding mean oscillating distribution function via dyadic decomposition. The dominating functions are of the form of decreasing step functions which are finer than the classical exponential functions and might be much more efficient for some sophisticated analysis. We also prove that the modified BMO-norm is equivalent to the classical BMO-norm under the convexity assumption.

中文翻译：

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关于约翰·尼伦贝格不等式

通过估计二元分解的相应平均振荡分布函数，我们给出了BMO子类的John–Nirenberg不等式的一个形式。主导函数采用递减步阶函数的形式，它比经典的指数函数要精细，并且对于某些复杂的分析可能更为有效。我们还证明，在凸性假设下，修改后的BMO-范数与经典BMO-范数等效。