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

合作研究：非齐次调和分析的普遍性现象及若干难题

• 批准号: 1265623
• 负责人: Fedor Nazarov
• 金额: $ 17.1万
• 依托单位: Kent State 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 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265623&HistoricalAwards=false
• 关键词: Collaborative; research; Universality; phenomena; several

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，多数较广泛的操作员也有限。大多数问题都在于非谐音和衡量标准的范围内，并且在这些领域中，衡量了衡量标准和衡量标准。非常不规则的集合在许多分析问题中出现。一个重量的非殖民案例的动机之一是对分析能力的研究。纳扎罗夫，Treil和Volberg的这一协作数学研究项目的基本进展集中在谐波分析领域，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，用于大型数据集，用于图像处理和波浪传播研究。 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。