Collaborative research: Universality phenomena and some hard problems of non-homogeneous Harmonic Analysis

合作研究：非齐次调和分析的普遍性现象和一些难题

• 批准号: 1301579
• 负责人: Serguei Treil
• 金额: $ 34.75万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301579&HistoricalAwards=false
• 关键词: Collaborative; research; Universality; phenomena; some

This collaborative mathematics research project by Fedor Nazarov, Serguei Treil and Alexander Volberg is in the area of harmonic analysis. Nazarov, Treil and Volberg will concentrate their efforts on several well-known hard problems in non-homogeneous harmonic analysis, geometric measure theory, and spectral theory. The common theme among majority of the problems is the so-called "universality" phenomenon, i.e. the fact that in many situations the boundedness of one operator (or a small collection of operators) implies that a much wider class of operators is bounded as well. Most of the problems lie in the realm of the non-homogeneous harmonic analysis, where underlying sets and measures are highly irregular. Singular integral operators with respect to singular measures and very irregular sets appear naturally in many problems of analysis. One of the motivations for the one-weight non-homogeneous case was the study of analytic capacity. The more sophisticated two-weight estimates of singular operators appear naturally in spectral theory and in the perturbation theory of self-adjoint operators. These problems are notoriously difficult, but using new techniques recently developed by Nazarov, Treil and Volberg and other researchers, they expect to make fundamental progress in the problems. This collaborative mathematics research project by Nazarov, Treil and Volberg is focused in the field of harmonic analysis, which is known to have fundamental applications to other disciplines, most notably to the analysis of large data sets, to image processing, and to the study of wave propagation. The results and mathematical tools that will be developed through this project could also have a bearing on other areas of mathematics, such as mathematical physics, partial differential equations, probability. The project will provide a good training ground for graduate students as well as for mathematicians at the beginning of their careers. Nazarov, Treil and Volberg anticipate an active involvement of their graduate students and postdocs in the project.

Fedor Nazarov，Serguei Treil和Alexander Volberg的合作数学研究项目正处于谐波分析领域。 Nazarov，Treil和Volberg将把他们的努力集中在非均匀谐波分析，几何测量理论和光谱理论中的几个众所周知的硬问题上。大多数问题之间的共同主题是所谓的“普遍性”现象，即，在许多情况下，一个运营商（或一小部分操作员）的界限意味着也意味着更广泛的运营商也有限。大多数问题都在于非均匀谐波分析的领域，其中基础集和措施高度不规则。在许多分析问题中，关于奇异措施和非常不规则的集合的单数积分运算符自然而然。一重的非均匀情况的动机之一是研究能力的研究。奇异运算符的更复杂的两重量估计在频谱理论和自我伴侣操作员的扰动理论中自然出现。众所周知，这些问题是困难的，但是使用Nazarov，Treil和Volberg以及其他研究人员最近开发的新技术，他们希望在问题上取得根本性的进展。 Nazarov，Treil和Volberg的这项合作数学研究项目集中在谐波分析领域，众所周知，该领域对其他学科具有基本应用，最著名的是对大型数据集的分析，图像处理，以及对波传播的研究。 通过该项目开发的结果和数学工具也可能与其他数学领域（例如数学物理学，部分微分方程，概率）有关。该项目将在职业生涯开始时为研究生以及数学家提供良好的培训理由。 Nazarov，Treil和Volberg预计他们的研究生和博士后会积极参与该项目。