Mathematical Sciences: Presidential Young Investigator Award

本研究集中在概率与分析的界面，其中鞅、调和分析、解析函数和布朗运动重叠。部分研究涉及到一类被称为鞅变换投影的算子的研究。研究了这些算子的基本性质及其与概率积和奇异积分的联系。这项研究在分析和概率之间的边界进行，展示了对两者的广泛和平衡的知识，并导致交叉施肥，其中一个领域的基本工具应用于另一个领域。这项研究对解答Gundy在1983年提出的猜想大有帮助，该猜想认为，适用于鞅的迭代对数定律也适用于调和函数。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，即非切极大函数和Lusin面积函数是如何相互控制的。

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.