Mathematical Sciences: Singular Integrals and Averages of Functions

谐波分析的核心是两个基本的 问题：如何将函数分解为基本成分 以及如何从这些部分恢复功能。 的 这个数学框架的起源是傅立叶级数。 在试图理解函数如何从其 傅立叶级数，一个是导致问题的平均函数 奇异积分算子 它是研究这些 这些概念构成了本研究的基础。 这个特殊项目的起源在于经典的 Lebesgue的实变量结果，它给出了精确解 什么时候可以对积分求微分， 被积函数 没有完全类似的情况， 更高的维度，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 其中将功能限制在各种尺寸的表面上 要么设定极限，要么最大化估计 近似导数（平均值）。 这些任务相当于 曲面上奇异积分的范数估计 而 在过去两年中， 几十年来，描述正确的类的主要问题 曲线或曲面，沿着这些曲线或曲面，微分理论可以 建造仍然令人生畏。 表面的形状起着 关键作用，这项工作将继续澄清 这是产生一个全面理论所需要的条件。 虽然这项研究的一般描述是 用和声分析的语言来说， 偏微分应用的影响 方程

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.