Maximal functions on the unit n-sphere
Maximal functions on the unit n-sphere
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DOI：
10.2140/pjm.1987.129.77
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发表时间：
1987-09
影响因子：
0.6
通讯作者：
P. Knopf
中科院分区：
文献类型：
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作者：
P. Knopf
导论. E. M. Stein和J.O. Strumberg [6]证明了R中的Hardy-Littlewood极大函数是弱型的（1，1），具有一个弱型常数en，其中c与n无关.他们的方法是通过R* 上的某个热扩散半群的成员的平均值的上确界来逐点约束极大函数。然后，他们应用霍普夫抽象极大遍历定理，以获得他们的结果。我们计划使用这种方法的类似版本来证明单位w-球面上的极大函数是弱型的（1，1），具有弱型常数en。在此之前，最好的弱型常数是cnyfn，参见[4]，使用完全不同的方法。本文中的许多观点已经在C. Herz [3].为了得到弱型常数en，需要比Herz论文中所指出的更大的估计。此外，还有一个主要是技术性的疏忽，导致提交人进行了一些扭曲以纠正。应该指出的是，赫兹的整体方法不仅适用于R中的单位球面和R本身，而且适用于更一般的空间。作者对N. Stanton，E. M.斯坦，还有裁判。
Introduction. E. M. Stein and J. O. Strόmberg [6] have shown that the Hardy-Littlewood maximal function in R is weak-type (1,1) with a weak-type constant en with c independent of n. Their approach is to pointwise bound the maximal function by a supremum of averages of members of a certain heat-diffusion semi-group on R*. They then apply the Hopf abstract maximal ergodic theorem to obtain their result. We plan to use an analogous version of this approach to show that the maximal function on the unit w-sphere is weak-type (1,1) with a weak-type constant en. The best weak-type constant prior to this was cnyfn, see [4], using an entirely different approach. Many of the ideas in this paper have already been presented in a paper by C. Herz [3]. In order to obtain the weak-type constant en, shaφer estimates are required than are indicated in Herz's paper. Furthermore, there is an oversight of a primarily technical nature which led this author to perform some contortions to rectify. It should be pointed out that Herz's overall approach applies not only to the unit sphere in R and R itself, but to more general spaces as well. The author is appreciative of the informative comments and helpful suggestions of N. Stanton, E. M. Stein, and the referee.