Singular Integrals and Maximal Functions

提案0098757I计划研究在大于等于2维欧几里得空间上的函数上定义的某些积分算子的解析估计。在这些算子中，积分是在正余维曲面上进行的，我们寻求反映曲面曲率性质的估计。例如，假设在欧几里德空间中，对于每一个点P，我们有一条从P发出的一维曲线。从一个给定的函数f，我们形成一个新的函数Mf，称为极大函数，它在点P处的值是f在从P发出的曲线上的平均值的最大值。这就定义了一个从欧几里德空间上的函数到欧几里德空间上的函数的转换。我们想知道对于什么曲线和p的什么值这个变换在勒贝格空间中是有界的它的函数的p次幂是可积的。这里的正结果意味着关于积分微分的勒贝格定理的变体。也就是说，如果从函数f到Mf的变换在一个勒贝格空间上是有界的，那么对于该勒贝格空间中的每个函数f和几乎每个点P， f(P)的值可以被恢复为f在曲线上通过P的一小部分上的平均值的极限。我也对这些算子的离散类似物感兴趣，其中积分被离散点集上的和所取代。古典分析这一数学分支存在了一百多年的一个基本问题是从函数的平均值中恢复函数。这个问题与用简单函数的组合近似任意函数的问题密切相关，而简单函数的组合又成为数学应用于现实世界问题的主要方法之一。我计划研究在至少二维的欧几里得空间中，从一小块一维曲线的平均值中恢复函数的问题。对于连续函数，这是一个简单的问题，但对于不连续的函数，这是一个微妙的问题，取决于曲线的曲率特性。我还计划研究相关的变换和这些变换的离散类似物，其中平均过程是在离散的点集上。

Abstract for Proposal 0098757I plan to study analytical estimates for certain integral operators defined on functions on the Euclidean space of dimension greater than or equal to two. In these operators the integration is over surfaces of positive codimension, and we seek estimates reflecting curvature properties of the surface. Suppose for example, for each point, P, inthe Euclidean space we have a one dimensional curve emanating from P. From a given function, f, we form a new function Mf, called the maximal function, whose value at the point P is the supremum of the averages of f over the curve emanating from P. This then defines a transformation from functions on the Euclidean space to functions onthe Euclidean space. We want to know for what curves and what values of p this transformation is bounded on the Lebesgue space of functions with integrable pth power. Positive results here imply variants of Lebesgue's theorem on the differentiation of the integral. Namely ifthe transformation from a function f to Mf is bounded on one of these Lebesgue spaces, then for every function f in that Lebesgue space and almost every point P, the value f(P) may be recovered as a limit of averages of f over small portions of the curve through P. I am alsointerested in discrete analogues of these operators in which integration is replaced by sums over discrete sets of points.A basic problem for over a hundred years of the branch of mathematics known as Classical Analysis is that of recovering a function from averages of that function. This problem has been intimately connected with that of approximating an arbitrary function by a combination ofsimpler functions which in turn has been one of the main ways mathematics is applied to real world problems. I plan to study the problem of recovering functions on Euclidean space of at least two dimensions, from averages over small pieces of one dimensional curves. For a continuous function this is an easy question, but for wildly discontinuous functions it is a subtle problem depending on curvatureproperties of the curves. I also plan to study relatedtransformations and discrete analogues of these transformations where the averaging process is over discrete sets of points.