Problems in Weighted Inequalities, Phase Plane Analysis

希尔伯特变换是一个与电荷分布等物理情况密切相关的数学对象。它和相关的对象，如柯西变换，出现在一系列分析问题中，从正交多项式到动力学，再到解析函数空间和偏微分方程。希尔伯特变换的双权问题涉及到这些度量对的特征，以便变换将一个度量的希尔伯特空间映射到另一个度量的希尔伯特空间。在国家科学基金会赞助的工作中，提出者Eric T Sawyer和Ignacio Uriate-Tuero已经解决了这个问题。例如，这对算子论中存在已久的Sarason猜想有直接的应用。这一建议的目标是在这一成功的基础上，转向将这种刻画推广到其他自然问题，如柯西变换的刻画，这是解析函数空间理论的基础。这项工作试图详细描述精确模拟物理情况的变换的微妙属性，例如电荷分布。因此，这些技术将影响可用于解决这些问题的分析工具的范围，这些问题需要对这些操作员有很好的了解。此外，提倡者还在指导和培训下一代科学劳动力方面发挥了各种作用。这包括：(1)研究人员在MCTP补助金上的工作，以在佐治亚理工学院招聘和培训有才华的本科专业。(2)指导两名研究生的毕业论文工作。(3)指导一批博士后研究员。(4)在美国、加拿大和欧洲的研究中心组织会议。(5)《美国数学学会会报》和《几何分析杂志》的编辑工作。(6)传播研究成果和目标，包括在世界各地举办讲座。

The Hilbert transform is an mathematical object intimately related to physical situations such as charge distribution. It and related objects such as the Cauchy transform, occur in a range of analytical questions, from orthogonal polynomials to dynamics, to analytic function spaces and partial differential equations. The two-weight problem for the Hilbert transform concerns a characterization of those pairs of measures so that the transform maps Hilbert space of one measure into Hilbert space of the other. This problem has been solved by the proposer, Eric T Sawyer and Ignacio Uriate-Tuero, in work sponsored by the National Science Foundation. This has immediate application to for instance a long-standing conjecture of Sarason in operator theory. The goal of this proposal, is to build upon this success, turning to extensions of this characterization for other natural questions, such as that for the Cauchy transform, which is fundamental for the theory of analytic function spaces. This work seeks to detail subtle properties of transforms which closely model physical situations, such as electrical charge distributions. As such, the techniques will impact the range of analytical tools that can be brought to bear on these questions that entail fine knowledge about these operators. In addition, the proposer carries out a variety of roles in mentoring and training a next generation of scientific workforce. This includes: (1) The work of the investigator on an MCTP grant to recruit and train talented undergraduate majors at the Georgia Institute of Technology. (2) Directing two graduate students in their thesis work. (3) Mentoring a number of postdoctoral fellows. (4) Conference organization at research centers in the US, Canada and Europe. (5) Editorial work for the Proceedings of the American Mathematical Society and the Journal of Geometric Analysis. (6) Dissemination of research accomplishments and goals, including lectures at venues around the world.