A geometric proof of the strong maximal theorem
A geometric proof of the strong maximal theorem
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DOI：
10.1090/s0002-9904-1975-13899-7
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发表时间：
1975-07
影响因子：
1.3
通讯作者：
A. Córdoba;R. Fefferman
中科院分区：
文献类型：
--
作者：
A. Córdoba;R. Fefferman
Limdiam(R)-O1 f(Y)dy=f(X)对于A.E.XC R“XeReB.I R R只要f在L(LOG+L)的局部”。这个极大值定理是由Jessen，Marcinkiewicz和Zygmund[2]得到的，他们证明的基本思想是通过合成MxMx2来控制算子M。其中Mx是沿第i个坐标轴方向的一维Hardy-Littlewood极大算子。这个证明的简洁性和高雅是显而易见的。另一方面，强极大值定理的几何证明似乎是可取的。主要原因是在许多情况下，要得到与强极大函数密切相关的算子的结果，唯一的方法是通过对矩形几何的深入理解。正是这种理解，我们在本文中尽了最大努力来实现。现在，文献[1]表明，在非常一般的假设下，研究关于一族有界可测集族的极大算子的性质等价于研究该族的覆盖性质。
limdiam(R)-O1 f(y)dy = f(x) for a.e. xc R" xeReB. I RI R so long as f is locally in L(log+ L)"'. This maximal theorem is due to Jessen, Marcinkiewicz, and Zygmund [2], and the basic idea of their proof is to dominate the operator M, by the composition MxMx2 .. Mxn where Mx, is the one dimensional Hardy-Littlewood maximal operator in the direction of the ith coordinate axis. The simplicity and elegance of this proof are obvious. On the other hand, it seemed desirable to have a geometric proof of the strong maximal theorem. The main reason is that in many cases, the only way to obtain results for operators intimately connected with the strong maximal function will be through a deep understanding of the geometry of rectangles. It is this understanding which we have done our best to achieve in this article. Now, it was shown in [1] that under very general hypotheses, to study the properties of a maximal operator with respect to a family of bounded measurable sets is equivalent to studying the covering properties of that family.