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

Fedor Nazarov、Serguei Treil 和 Alexander Volberg 的合作数学研究项目属于调和分析领域。 Nazarov、Treil 和 Volberg 将集中精力解决非齐次调和分析、几何测度理论和谱理论中的几个众所周知的难题。大多数问题的共同主题是所谓的“普遍性”现象，即在许多情况下，一个运算符（或一小部分运算符）的有界性意味着更广泛的一类运算符也是有界的。大多数问题都存在于非齐次调和分析领域，其中基础集合和测度是高度不规则的。关于奇异测度和非常不规则集的奇异积分算子自然地出现在许多分析问题中。单权重非齐次案例的动机之一是分析能力的研究。奇异算子的更复杂的二权估计自然地出现在谱理论和自伴算子的微扰理论中。这些问题是出了名的困难，但利用纳扎罗夫、特雷伊和沃尔伯格等研究人员最近开发的新技术，他们期望在这些问题上取得根本性进展。 Nazarov、Treil 和 Volberg 的合作数学研究项目重点关注调和分析领域，众所周知，调和分析在其他学科中具有基础应用，尤其是对大数据集的分析、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。该项目将为研究生以及职业生涯初期的数学家提供良好的训练场地。 纳扎罗夫、特雷尔和沃尔伯格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.