Mathematical Sciences: Presidential Young Investigator Award

本研究主要集中在 概率和分析，其中鞅，调和分析， 解析函数和布朗运动重叠。 的一部分 研究涉及一类算子的研究， 叫做鞅变换的投影。 基本 这些算子的性质与它们的 与概率积和奇异积分的联系。 这项研究是在分析和 概率，表现出广泛和平衡的知识， 这两种方法都能产生交叉作用， 在一个领域应用于另一个领域。 研究进行了很长时间 Gundy在1983年提出了一个猜想， 对于鞅成立的迭代对数也应该 对调和函数成立。 这是一个根本问题 它要求精确描述这两个基本的 调和函数上的非切极大算子 函数和Lusin区域函数，相互控制。

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.