Problems in Weighted Inequalities, Phase Plane Analysis

加权不等式、相平面分析中的问题

• 批准号: 0968499
• 负责人: Michael Lacey
• 金额: $ 29.2万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0968499&HistoricalAwards=false
• 关键词: Problems; Weighted; Inequalities; Phase; Plane

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.

希尔伯特变换是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，出现在一系列的分析问题中，从正交多项式到动力学，再到解析函数空间和偏微分方程。希尔伯特变换的二权问题涉及到这些测度对的表征，以便变换将一个测度的希尔伯特空间映射到另一个测度的希尔伯特空间。这个问题已经被提议者埃里克·T·索耶和伊格纳西奥·乌里亚特-图罗在国家科学基金会的资助下解决了。这可以直接应用于例如算符理论中Sarason的一个长期存在的猜想。这个建议的目标，是建立在这个成功的基础上，将这个特征扩展到其他自然问题，比如柯西变换，这是解析函数空间理论的基础。这项工作旨在详细描述变换的微妙特性，这些特性与物理情况（如电荷分布）密切相关。因此，这些技术将影响分析工具的范围，这些工具可以用来解决这些问题，这些问题需要对这些操作人员有很好的了解。此外，申请人在指导和培训下一代科学劳动力方面发挥各种作用。这包括：(1)研究人员在MCTP资助下招募和培训乔治亚理工学院有才华的本科专业学生。(2)指导2名研究生的毕业论文工作。(3)培养一批博士后。(4)在美国、加拿大和欧洲的研究中心组织会议。(5)《美国数学学会学报》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。