Mathematical Sciences: Presidential Young Investigator Award

数学科学：总统青年研究员奖

• 批准号: 8958449
• 负责人: Rodrigo Banuelos
• 金额: $ 13.69万
• 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: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8958449&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Presidential; Young; Investigator

This research is concentrated at the interface of probability and analysis, where martingales, harmonic analysis, analytic functions, and Brownian motion overlap. Part of the research involves the study of a class of operators which are called the projections of martingale transforms. Basic properties of these operators ares examined together with their connections to probabilistic products and to singular integrals. This research conducted at the border between analysis and probability, demonstrates an extensive and balanced knowledge of both and results in a cross-fertilization where fundamental tools in one area are applied to the other. The research goes a long ways toward answering Gundy's 1983 conjecture that the law of the iterated logarithm, which holds for martingales, should also hold for harmonic functions. This is a fundamental question which asks for a precise description of how the two basic operators on harmonic functions, the nontangential maximal function and the Lusin area function, control each other.

这项研究集中在概率与分析的交界处，在这一交界处，鞅、调和分析、解析函数和布朗运动交叠在一起。研究的一部分涉及到一类算子的研究，这类算子被称为鞅变换的投影。研究了这些算子的基本性质以及它们与概率积和奇异积分之间的联系。这项研究在分析和概率之间进行，展示了对两者的广泛和平衡的知识，并导致了一种交叉受精，即一个领域的基本工具应用于另一个领域。这项研究在很大程度上回答了Gundy 1983年的猜想，即重对数律适用于鞅，也适用于调和函数。这是一个基本问题，它要求精确地描述调和函数上的两个基本算子--非切极大函数和卢辛面积函数--是如何相互控制的。