Two Weight Inequalities for Singular Integrals

迈克尔·莱西的这个数学研究项目是在谐波分析的一般领域。香农抽样定理断言paly - wiener类中的函数可以通过其在整数处的值恢复。这个基本事实有许多量化和扩展。其中最微妙的是整数在多大程度上可以被其他离散集所取代：对于其他集合的抽样也表现良好。在平方可积的情况下，这只是希尔伯特变换中深奥的双权不等式的一个例子。它要求在实数线上希尔伯特变换被限定在与这对测度（权重）相关的希尔伯特空间之间的那些测度对的实变量表征。莱西最近全面地解决了这个问题。这和一系列相关的扩展将由莱西进行调查。Michael Lacey的这个数学研究项目是在谐波分析领域，它在工程等其他领域有深入的应用，例如信号传输。信息传输的标志是对信号进行快速和有规律的采样，然后将采样合成，例如，通过手机进行语音对话。该系统是鲁棒的，采样的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。这种采样的抽象化导致了引人入胜的数学问题：信号的急剧过采样增加了观测到的能量。在保持信号能量的情况下，过采样会变得多糟糕？在信号丢失之前，采样的简并程度是允许的？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广泛传播，将影响抽样和信息的基础理论。这个项目中的一些问题将由莱西指导下的博士生进行研究。

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.