Two Weight Inequalities for Singular Integrals

奇异积分的两个权重不等式

• 批准号: 1265570
• 负责人: Michael Lacey
• 金额: $ 32.7万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-05-15 至 2017-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265570&HistoricalAwards=false
• 关键词: Two; Weight; Inequalities; Singular; Integrals

This mathematics research project by Michael Lacey is in the general area of harmonic analysis. The Shannon Sampling Theorem asserts that a function in the Paley-Wiener class can be recovered by its values at the integers. This foundational fact has many quantifications, and extensions. Among the most delicate is to what extent the integers can be replaced by other discrete sets: For which other sets is sampling also well behaved. In the square-integrable case, this is just one instance of the profound two-weight inequality for the Hilbert transform. It asks for a real-variable characterization of those pairs of measures on the real line for which the Hilbert transform is bounded between the Hilbert spaces associated with the pair of measures (weights). Lacey has recently solved this problem, in complete generality. This and a family of related extensions will be under investigation by Lacey.This mathematics research project by Michael Lacey is in the field of harmonic analysis, which has deep applications to other fields such as engineering, for instance to signal transmission. The hallmark of transmission of information is rapid and regular sampling of a signal, followed by a synthesis of the samples into, for instance, a voice conversation over a cell phone. The system is robust, with small perturbations of the sampling not affecting the synthesis, in part because the most obvious problems with the sampling are all accounted for. The abstraction of this sampling leads to fascinating mathematical problems: A dramatic oversampling of the signal increases the observed energy. How bad can the oversampling become, while preserving the energy of the signal? Just how degenerate of a sampling is permitted before the signal is lost? Lacey has provided a mathematical solution to the first problem, an advance that will open the door to a range of related questions. These techniques, disseminated across fields of science and engineering, will impact the underlying theories of sampling and information. Some of the problems in this project will be studied by Ph.D. students under Lacey's direction.

迈克尔·莱西的这项数学研究项目属于调和分析的一般领域。香农抽样定理断言，Paley-Wiener类中的函数可以通过它在整数处的值恢复。这一基本事实有许多量化和延伸。其中最微妙的问题是整数可以在多大程度上被其他离散集取代：对于哪些其他集也表现良好。在平方可积的情况下，这只是希尔伯特变换的深刻的双权不等式的一个例子。它要求对实线上希尔伯特变换在与该度量对(权重)相关联的希尔伯特空间之间有界的那些度量对进行实变量表征。莱西最近完全笼统地解决了这个问题。迈克尔·莱西的这一数学研究项目是在调和分析领域进行的，该领域在其他领域有深入的应用，如工程，例如信号传输。信息传输的特点是快速而有规律地对信号进行采样，然后将采样合成为例如通过手机进行的语音交谈。该系统是稳健的，采样的微小扰动不会影响合成，部分原因是采样最明显的问题都被考虑在内。这种采样的抽象导致了有趣的数学问题：信号的戏剧性过采样增加了观察到的能量。在保留信号能量的同时，过采样会变得多糟糕？在信号丢失之前，究竟允许多大程度的采样退化？莱西为第一个问题提供了一个数学解决方案，这一进步将为一系列相关问题打开大门。这些技术在科学和工程领域传播，将影响抽样和信息的基本理论。这个项目中的一些问题将由博士生在莱西的指导下进行研究。