Abstract:The explicit upper Bellman function is found for the natural dyadic maximal operator acting from into . As a consequence, it is shown that the norm of the natural operator equals for all , and so does the norm of the classical dyadic maximal operator. The main result is a partial consequence of a theorem for the so-called -trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasiperiodic structure depending on , but also allows a majorant independent of , hence a dimension-free norm constant. Also, the decay of the norm is described with respect to the growth of the difference between the average of a function on a cube and the infimum of its maximal function on that cube. An explicit norm-optimizing sequence is constructed.

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中文翻译：

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operator树上最大算子的→𝐵𝐿𝑂作用

摘要：为从到的自然二进极大算子找到了明确的上Bellman函数。结果表明，自然算子的范数对所有都相等，经典二进最大值算子的范数也相等。主要结果是定理对所谓的-tree的局部结果，该定理推广了二叉晶格。在这种情况下，Bellman函数显示出一个有趣的拟周期结构，取决于，但也允许一个主要变量独立于 ，因此是无量纲的规范常数。同样，关于立方体上函数平均值与该立方体上最大函数最小值之间的差的增长描述了范数的衰减。构造了一个明确的规范优化序列。

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