In Search of Sharp Inequalities

寻找尖锐的不平等

• 批准号: 0072037
• 负责人: Rodrigo Banuelos
• 金额: $ 19.2万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072037&HistoricalAwards=false
• 关键词: Search; Sharp; Inequalities

0072037Banuelos Firstly, ideas from martingales theory will be used to investigate sharp inequalities for certain singular integrals which have played a fundamental role in several different areas of harmonic analysis, partial differential equations and the calculus of variations. These investigations lead to connections of probability to a conjecture of Morrey which asserts that rank--one convexity does not imply quasiconvexity in dimensions larger than 2. Secondly, extremal problems for the hyperbolic metric will be investigated. These problems arise from the conformal invariance of Brownian motion and conditioned Brownian motion in simply connected domains in the plane. These investigations in turn lead to new problems for martingales and Brownian motion. These problems will also be explored. Morrey's conjecture is important because of its implications in constructing minimizers in several problems in the calculus of variations describing various physical systems. The connection of these problems to probability was discovered by the PI and his collaborator, G. Wang and A.J. Lindeman, while investigating sharp inequalities for stochastic integrals and more general martingales. These connections lead to several questions which if false prove Morrey's's conjecture, a long standing open problem, and if true prove a conjecture concerning the norm of certain singular integrals, another long standing open problem. The solution to the proposed problems on the hyperbolic metric will yield a deeper understanding of various isoperimetric (extremal) inequalities for Brownian motion and conditioned Brownian motion. As with previous research projects of the PI, graduate students will be (are already) involved.

[00:07 . 37]首先，我们将使用鞅理论的思想来研究某些奇异积分的尖锐不等式，这些奇异积分在调和分析、偏微分方程和变分学的几个不同领域中起着重要作用。这些研究将概率与Morrey的一个猜想联系起来，该猜想断言，在大于2的维度上，秩1的凸性并不意味着准凸性。其次，研究双曲度规的极值问题。这些问题是由平面上单连通域上布朗运动和条件布朗运动的共形不变性引起的。这些研究反过来又导致了鞅和布朗运动的新问题。这些问题也将进行探讨。Morrey的猜想很重要，因为它在描述各种物理系统的变分演算中的几个问题中构造极小值时具有重要意义。这些问题与概率的联系是由PI和他的合作者G. Wang和A.J. Lindeman在研究随机积分和更一般的鞅的尖锐不等式时发现的。这些联系导致了几个问题，如果假证明了莫雷猜想，这是一个长期存在的开放问题，如果真证明了一个关于某些奇异积分范数的猜想，这是另一个长期存在的开放问题。双曲度规问题的解决将使我们对布朗运动和条件布朗运动的各种等周（极值）不等式有更深的理解。与PI以前的研究项目一样，研究生将（已经）参与其中。