Two Weight Inequalities for Singular Integrals

这个数学研究项目由迈克尔莱西是在一般地区的谐波分析。香农采样定理断言Paley-Wiener类中的函数可以通过其整数值恢复。这一基本事实有许多量化和扩展。 其中最微妙的是在何种程度上的整数可以被其他离散集所取代：对于其他集的采样也表现良好。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。它要求在真实的线上的那些测度对的实变量表征，对于这些测度对，希尔伯特变换在与测度对（权重）相关联的希尔伯特空间之间有界。莱西最近解决了这个问题，在完全的一般性。这个数学研究项目由迈克尔·莱西在谐波分析领域，它有很深的应用到其他领域，如工程，例如信号传输。信息传输的标志是快速和定期的信号采样，然后将采样合成为例如手机上的语音对话。该系统是鲁棒的，采样的小扰动不影响合成，部分原因是采样的最明显的问题都被考虑在内。这种采样的抽象导致了迷人的数学问题：信号的戏剧性过采样增加了观察到的能量。在保持信号能量的同时，过采样会变得多糟糕？在信号丢失之前，允许采样退化到什么程度？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，将影响采样和信息的基本理论。本项目中的一些问题将由博士研究。学生在莱西的指导下。

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.