Collaborative Research: Universality Phenomena and Some Hard Problems of Non-homogeneous Harmonic Analysis

这个合作数学研究项目的费多尔Nazarov，谢尔盖Treil和亚历山大沃尔伯格是在该地区的谐波分析。 纳扎罗夫，特雷和沃尔伯格将集中精力在几个著名的非齐次谐波分析，几何测度理论和谱理论的难题。大多数问题的共同主题是所谓的“普适性”现象，即在许多情况下，一个算子（或一小部分算子）的有界性意味着更广泛的一类算子也是有界的。大多数的问题在于领域的非齐次调和分析，其中基本的集合和措施是非常不规则的。关于奇异测度和非常不规则集合的奇异积分算子自然地出现在许多分析问题中。单权非齐次情形的动机之一是分析能力的研究。奇异算子的更复杂的双权估计自然出现在谱理论和自伴算子的扰动理论中。这些问题是出了名的困难，但使用最近开发的新技术Nazarov，Treil和Volberg和其他研究人员，他们希望在这些问题上取得根本性的进展。Nazarov，Treil和Volberg的这个合作数学研究项目专注于谐波分析领域，众所周知，谐波分析在其他学科中具有基本应用，最值得注意的是大型数据集的分析，图像处理和波传播的研究。 通过该项目开发的结果和数学工具也可能对数学的其他领域产生影响，例如数学物理，偏微分方程，概率。该项目将提供一个良好的培训基地，研究生以及数学家在他们的职业生涯的开始。 Nazarov，Treil和Volberg预计他们的研究生和博士后将积极参与该项目。

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.