A geometric proof of the strong maximal theorem

Abstract

limdiam(R)-O1 f(y)dy = f(x) for a.e. xc 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.