Singular Integrals and Maximal Functions

建议：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平均和奇异平均，其中n大于或等于2。与Calderon和Zygmund的理论不同，这些平均中的积分是在正余维的曲线或曲面上进行的。Wainger对这些算子的L-p界感兴趣，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比，其中积分被一组离散点的求和所取代。过去250年来，数学中的一个重要问题是用更简单的函数的组合来逼近任意函数。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，这一问题已成为与电子数据传输相关的大量活动的焦点。在现代，这种近似问题导致了根据函数的大小来估计函数的某些“平均”大小的问题。Wainger提出的研究涉及n个变量的函数的平均，其中n大于或等于2，现在是在n维空间的曲面或曲线上，而不是所有的n维空间。然后，他根据曲面或曲线的几何性质以及函数本身，寻求这些平均值的估计。

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.