Singular Integrals and Maximal Functions

摘要Wainger教授希望他的主要研究重点是连续算子族的离散类似物的l（p）估计。在连续的情况下，对于n维空间中的每一点x，给出一个正余维通过x的变量V（x）。给定一个函数f，在n维空间上考虑f的最大或奇异平均，其中x的平均是在变量V（x）上。然后我们寻求这些平均值的L（p）估计。在离散模拟中，我们考虑定义在n维空间中格点m上的函数，平均值是包含m的算术集。在上面提到的连续问题中，大部分工作假设变量V（x）与其切平面有有限阶接触。Wainger教授还计划继续研究V（x）与其切平面具有无穷阶接触的情况。Wainger教授计划研究n维空间中定义在格点上的函数的平均值。（晶格点是具有整数坐标的点。）对于每个格点m，考虑包含m的算术子集上的平均值。我们的目标是将这些平均值的分析性质与这些子集的算术性质联系起来。这种类型的结果有应用问题的遍历理论。在过去的四十年里，一个相应的连续性问题受到了极大的关注。在这个问题中，一个平均函数定义在n维空间的子流形的n维空间。我们的目标是将这些平均值的分析估计与子流形的几何性质联系起来。这项工作主要假设子流形和它的切平面之间只有有限阶的接触。Wainger教授计划继续研究子流形与其切平面具有无穷阶接触的情况。

ABSTRACTProfessor Wainger expects the primary focus of his research to be on l(p) estimates for discrete analogues of a family of continuous operators. In the continuous situation one is given for each point, x, in n-dimensional space a variety, V (x), of positive codimension passing through x. Given a function, f , on n-dimensional space one considers maximal or singular averages of f where the average of x is over the variety V (x). We then seek L(p) estimates for these averages. In the discrete analogue we consider functions defined over the lattice points, m, in n-dimensional space, and the average is over an arithmetic set containing m. In the continuous problem alluded to above, most of the work assumes the variety V (x) has finite order of contact with its tangent plane. Professor Wainger also plans to continue his investigation into the case that V (x) has infinite order of contact with its tangent plane. Professor Wainger plans to study averages of functions defined on latticepoints in n-dimensional space. (Lattice points are points with integral coordinates.) For each lattice point, m, one considers averages over an arithmetic subset containing m. The goal is to relate analytical properties of these averages to arithmetic properties of these subsets. Results of this type have applications to problems of Ergodic theory. A corresponding contin uous problem has received a great deal of attention in the last forty years. In this problem one averages functions defined on n-dimensional space over submanifolds of n-dimensional space. The goal is to relate analytical esti matesfor these averages to geometric properties of the submanifold. This work largely assumes that there is only finite order of contact between the submanifold and its tangent plane. Professor Wainger plans to continue his study of the situation in which the submanifold has infinite order of contact with its tangent plane.