Singular Integrals and Maximal Functions

奇异积分和极大函数

• 批准号: 9731647
• 负责人: Stephen Wainger
• 金额: $ 8.59万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-05-15 至 2002-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9731647&HistoricalAwards=false
• 关键词: Singular; Integrals; Maximal; Functions

Proposal: DMS-973647Principal Investigator: Stephen WaingerAbstract: Wainger plans to study maximal and singular averages of functions defined on n-dimensional space, where n is greater than or equal to 2. In contrast to the theory of Calderon and Zygmund, the integration in these averages is to take place over curves or surfaces of positive codimension. Wainger is interested in L-p bounds for these operators that relate to curvature properties of the curves and surfaces. He also plans to study discrete analogues of these questions, in which integration is replaced by a summation over a discrete set of points.An important problem in mathematics in the last 250 years has been that of approximating arbitrary functions by combinations of simpler functions. This problem arose in the study of vibrating strings in the 18th century and in the study of heat flow in the 19th century. More recently this problem has been the focus of considerable activity in conjunction with the electronic transmission of data. This approximation problem has led in modern times to the problem of finding estimates for the sizes of certain "averages" of functions in terms of the size of the functions. Wainger's proposed research involves the averaging of functions of n variables, with n greater than or equal to 2, now over surfaces or curves in n-dimensional space rather than all of n-dimensional space. He then seeks estimates for these averages in terms of geometric properties of the surfaces or curves, as well as of the functions themselves.

提案：DMS-973647首席研究员：Stephen Wainger摘要：Wainger计划研究定义在n维空间上的函数的最大和奇异平均值，其中n大于或等于2。与卡尔德龙和齐格蒙德的理论相反，这些平均值的积分是在正余维的曲线或曲面上进行的。Wainger感兴趣的是这些算子的L-p界，这些算子与曲线和曲面的曲率性质有关。他还计划研究这些问题的离散类似物，其中积分被一组离散点的求和所取代。在过去的250年里，数学中的一个重要问题是用简单函数的组合来近似任意函数。这个问题出现在18世纪对振动弦的研究和19世纪对热流的研究中。最近，这一问题已成为与数据电子传输有关的大量活动的焦点。这个近似问题导致在现代的问题找到估计的大小某些“平均值”的功能方面的功能。Wainger提出的研究涉及平均函数的n个变量，与n大于或等于2，现在超过表面或曲线在n维空间，而不是所有的n维空间。然后，他寻求估计这些平均值的几何性质的表面或曲线，以及功能本身。