Sharp Inequalities

这个项目的主旨是引入新的概率技术来研究某些离散变换的性质，这些离散变换在谐和分析理论及其在时/频域信号处理中的应用中起着重要作用。所讨论的变换包括对一维整数和多维晶格的希尔伯特变换的几个版本。这些基本变换是由大卫·希尔伯特在20世纪初提出的，作为连续变换的更简单的模型。对于许多应用来说，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大小，通过它们变换的序列的某些可总和性质来衡量，与它们的连续对应序列的大小一致。与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有关的几何问题也将被研究。该项目涉及位于概率、调和分析和谱、位势理论以及拉普拉斯和分数拉普拉斯的几何性质之间的几个问题和关于尖锐不等式的猜想。20世纪早期的一个重要问题是周期函数的大小如何控制其共轭函数的大小，其中大小由勒贝格Lp范数来衡量，其中p严格地介于1和无穷大之间。1925年，M.Riesz在他著名的关于Hilbert变换的Lp有界性的论文中回答了这个问题，并证明了这对作用于小Lp中双无限序列空间的离散形式也是一样的。此后不久，E.C.Titchmarsh直接证明了离散Hilbert变换的有界性，并证明了这两个算子的范数相等。第二年，蒂奇马什指出，他的平等论点是不正确的。平等问题自1927年以来一直是一个长期悬而未决的问题。在最近的一篇文章中，M.Kuasicki和PI通过确定离散Hilbert变换的Lp界解决了这个问题，证明了这个Lp界与Pichorides在70年代初发现的连续形式的Lp界相同。这个项目的第一部分讨论了一维和几维离散算子的密切相关的问题。该项目的第二部分解决了关于布朗运动和欧氏空间中有限体积子集的对称稳定过程的退出时间的尖锐不等问题，称为稳定性(或赤字)不等式。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.