Mathematical Sciences: Singular Integrals and Averages of Functions

谐波分析的核心是两个基本问题：如何将函数分解成基本组成部分以及如何从这些组成部分中恢复函数。这个数学框架的起源是傅里叶级数。在试图理解一个函数如何从它的傅立叶级数中恢复时，人们会遇到函数的平均和奇异积分算子的问题。正是对这些概念的研究构成了本研究的基础。这个特殊项目的起源在于勒贝格的经典实变量结果，该结果给出了何时可以微分并恢复被积函数的精确解。在高维中没有完全类似的情况，因为传递到极限的方法允许许多解释。这项工作的重点是一个限制程序，其中一个限制函数到各种尺寸的表面，要么取极限，要么对近似导数（平均值）进行最大化估计。这些任务相当于估计曲面上奇异积分的范数。虽然在过去的二十年里已经取得了相当大的进展，但描述可以建立微分理论的曲线或曲面的正确类别的主要问题仍然令人生畏。表面的形状起着至关重要的作用，这项工作将继续澄清产生一个全面理论所需的条件。虽然这项研究的一般描述是用调和分析的语言表达的，但它的大部分影响来自于偏微分方程的应用。

At the heart of harmonic analysis are two fundamental issues: how to decompose a function into elementary constituent parts and how to recover a function from those parts. The genesis for this mathematical framework is the Fourier series. In trying to understand how a function can be recovered from its Fourier series, one is led to questions of averages of functions and singular integral operators. It is the study of these concepts which forms the basis for this research. The genesis of this particular project lies in the classical real-variable result of Lebesgue which gives a precise solution of when one can differentiate an integral and recover the integrand. There is no complete analogue to this situation in higher dimensions since the method of passing to the limit allows many interpretations. This work focuses on a limiting procedure where one restricts functions to surfaces of various dimensions and either takes limits or makes maximizing estimates of approximate derivatives (averages). These tasks amount to estimating norms of singular integrals over surfaces. While there has been considerable progress made on over the past two decades, the main problems of describing the correct class of curves or surfaces along which a differentiation theory can be built remains daunting. The shape of the surface plays a critical role and this work will continue to clarify the conditions needed to produce a comprehensive theory. Although the general description of this research is phrased in the language of harmonic analysis, much of its influence derives from applications to partial differential equations.