Singular Integrals and Maximal Functions

对于提案0098757I，计划研究定义在维度大于或等于2的欧氏空间上的函数上的某些积分算子的解析估计。在这些算子中，积分是在余维为正的曲面上进行的，我们寻求反映曲面曲率性质的估计。例如，对于欧氏空间中的每个点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上确界，然后定义了从欧氏空间上的函数到欧氏空间上的函数的变换。我们想知道对于什么曲线和p的什么值，这种变换在具有可积第p次方的函数的勒贝格空间上是有界的。这里的肯定结果暗示了勒贝格积分微分定理的变体。也就是说，如果从函数f到Mf的变换在这些勒贝格空间中的一个上有界，那么对于该勒贝格空间中的每个函数f和几乎每个点P，值f(P)可以作为f在通过P的一小部分曲线上的平均的极限来恢复。我也对这些算子的离散类似感兴趣，其中积分被离散的点集上的和所代替。一百多年来被称为经典分析的数学分支的一个基本问题是从该函数的平均中恢复一个函数。这个问题与用更简单的函数组合来逼近任意函数的问题密切相关，而这又是数学应用于现实世界问题的主要方式之一。我计划研究在至少两维欧几里得空间上从一维曲线小段上的平均值恢复函数的问题。对于连续函数，这是一个简单的问题，但对于极不连续的函数，这是一个取决于曲线的曲率性质的微妙问题。我还计划研究相关变换和这些变换的离散类似，在这些变换的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.