Inequalities in Probability and Harmonic Analysis

提案：DMS-9870026主要研究者：Stephen J. Montgomery-Smith 翻译后摘要：本建议包含两个项目。首先，主要研究者计划研究随机变量总和的尾部分布。过去的大多数结果都集中在渐近解上，也就是说，它们得到的公式是有效的，例如，当求和的变量数量非常大时（大数定律或中心极限定理）。它旨在找到在所有情况下都有效的公式。第二个项目涉及获得一个不等式，其中我们希望表明，秩一的功能是准凸的。这将对调和分析和拟共形函数的研究产生深远的影响。它将解决Iwaniec关于Beurling-Ahlfors算子范数的最佳常数的猜想。它也有可能的后果，在获得弱型估计的奇异积分的维数无关，这将是相当出乎意料的调和分析。最后，研究将在应用数学中找到应用，在处理非线性均匀化问题。 第一个项目涉及以下问题。如果给我们一些随机数，我们能说它们的和是什么？例如，我们可以从人群中随机抽取10个人的平均身高吗？这个问题是概率论的基础，可以追溯到这门学科的起源。在这方面仍有许多问题没有得到解决。第二个项目研究凸性概念的性质。这是许多数学的基石性质，它已被用于许多科学领域，如物理和工程。这个特别的项目研究准凸性，这是一个试图理解弹性体在各种力和应力下如何变形的概念。主要研究者（沿着D.塔尔博特）已经用这些思想来研究另一种问题--即静电学问题。

Proposal: DMS-9870026 Principal Investigator: Stephen J. Montgomery-Smith Abstract: This proposal contains two projects. First, the principal investigator plans to investigate tail distributions of sums of random variables. Most results in the past have focused on asymptotic answers, that is, they obtain formulae that are valid, for example, when the number of variables summed is very large (the Law of Large Numbers, or the Central Limit Theorem). It is intended to find formulae that work in all situations. The second project concerns obtaining an inequality in which we wish to show that rank-one functions are quasi-convex. This would have profound consequences in harmonic analysis and the study quasi-conformal functions. It would settle the conjecture of Iwaniec concerning the best constant for the norm of the Beurling-Ahlfors Operator. It also has possible consequences in obtaining weak type estimates for singular integrals independent of dimension, which would be something quite unexpected in harmonic analysis. Finally, the research would find applications in applied mathematics, in problems dealing with homogenization of non-linear dielectrics. The first project deals with the following problem. If we are given a number of random quanitities, what can we say about their sum? For example, what can we say about the average height of ten people picked randomly from the population? This problem is fundamental in probability theory, tracing its historical roots back to the very beginnings of the discipline. There are still many unanswered problems in this area. The second project studies the nature of the concept of convexity. This is a cornerstone property of much mathematics, and it has been used in many areas of science, such as physics and engineering. This particular project studies quasi-convexity, a notion that emerged out of attempts to understand how elastic bodies are distorted under various forces and stresses. The principal investigator (along with D. Talbot) has already used t hese ideas to study problems of a different kind - namely, problems in electrostatics.