Sharp Inequalities

严重的不平等

• 批准号: 1854709
• 负责人: Rodrigo Banuelos
• 金额: $ 30万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854709&HistoricalAwards=false
• 关键词: Sharp; Inequalities

The thrust of this project concerns the introduction of new probabilisitic techniques to study properties of certain discrete transformations that play an important role in the theory of harmonic analysis and its applications in signal processing in the time/frequency domains. The transformations in question include several versions of the Hilbert transform on the integers in dimension one and on the lattice in several dimensions. These basic transformations were introduced by David Hilbert at the beginning of the 20th century as simpler models of their continuous counterparts. For many applications, discrete models are much simpler from the point of view of computations. One of the goals of the project is to show that the magnitudes of the discrete transformations, as measured by certain summability properties of the sequences they transform, coincide with those of their continuous counterparts. Related geometric problems concerning the long time behavior of Brownian motion and more general stochastic processes under certain natural constrains, will be studied as well.The project deals with several problems and conjectures for sharp inequalities which lie at the interface of probability, harmonic analysis and spectral, potential theoretic, and geometric properties of the Laplacian and the fractional Laplacian. A problem of significant interest in the early part of the 20th century was the question of how the size of a periodic function controls the size of its conjugate, where the size is measured by the Lebesgue Lp-norm, where p is strictly between 1 and infinity. In 1925, M. Riesz answered this question in his celebrated paper on the Lp boundedness of the Hilbert transform and showed that this implies the same for the discrete version acting on the space of doubly infinite sequences in little lp. Shortly thereafter, E.C. Titchmarsh gave a direct proof of the boundedness of the discrete Hilbert transform and showed that the norms of these two operators are equal. The following year Titchmarsh pointed out that his argument for equality was incorrect. The question of equality had been a long-standing open problem since 1927. In a recent publication, M. Kwasnicki and the PI solved this problem by identifying the sharp lp bound for the discrete Hilbert transform which turns out to be the same as the sharp Lp bound found for the continuous version found by Pichorides in the early 70's. The first part of this project discusses closely related problems for discrete operators in one and several dimensions. The second part of the project addresses questions of sharp inequalities, known as stability (or deficit) inequalities, for exit times of Brownian motion and symmetric stable processes from subsets of finite volume in Euclidean spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的重点是引入新的概率技术来研究某些离散变换的性质，这些离散变换在谐波分析理论及其在时/频域信号处理中的应用中发挥着重要作用。变换的问题包括几个版本的希尔伯特变换的整数在一个维度上，并在几个维度的格子。这些基本变换是由大卫希尔伯特在世纪初作为其连续对应物的简单模型引入的。对于许多应用，从计算的角度来看，离散模型要简单得多。该项目的目标之一是表明，离散变换的幅度，作为衡量的某些求和性质的序列，他们变换，与他们的连续同行。本课程还将研究布朗运动和更一般的随机过程在某些自然约束下的长时间行为的相关几何问题，涉及到位于概率、调和分析、谱、位势理论和拉普拉斯算子及分数拉普拉斯算子的几何性质之间的几个尖锐不等式的问题和证明。在世纪早期，一个重要的问题是周期函数的大小如何控制其共轭的大小，其中的大小是由勒贝格Lp-范数测量的，其中p严格介于1和无穷大之间。1925年，M. Riesz回答了这个问题，在他著名的文件对Lp有界的希尔伯特变换，并表明，这意味着同样的离散版本作用于空间的双重无限序列在小lp。不久之后，E。Titchmarsh直接证明了离散希尔伯特变换的有界性，并证明了这两个算子的范数相等。第二年Titchmarsh指出，他的论点平等是不正确的。自1927年以来，平等问题一直是一个长期悬而未决的问题。在最近的一份出版物中，M。Kwasnicki和PI通过确定离散Hilbert变换的尖锐Lp界来解决这个问题，事实证明，该界与Pichorides在70年代初发现的连续版本的尖锐Lp界相同。本项目的第一部分讨论了一维和多维离散算子的密切相关问题。该项目的第二部分解决了尖锐的不平等问题，被称为稳定性（或赤字）不平等，布朗运动和对称稳定过程的退出时间从有限体积的子集在欧几里德空间。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。