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

这个由Fedor Nazarov， Serguei Treil和Alexander Volberg合作的数学研究项目是在谐波分析领域。Nazarov、Treil和Volberg将集中精力研究非齐次谐波分析、几何测量理论和谱理论中几个众所周知的难题。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（或算子的一个小集合）的有界性意味着更广泛的算子类也是有界的。大多数问题属于非齐次谐波分析领域，其中底层集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出现在许多分析问题中。一权重非齐次情况的动机之一是分析能力的研究。在谱理论和自伴随算子的微扰理论中，很自然地出现了更复杂的奇异算子的二权估计。这些问题是出了名的困难，但利用Nazarov、Treil和Volberg以及其他研究人员最近开发的新技术，他们希望在这些问题上取得根本性的进展。这个由Nazarov， Treil和Volberg合作的数学研究项目集中在谐波分析领域，众所周知，谐波分析在其他学科中具有基础应用，最显著的是对大型数据集的分析，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、概率论。该项目将为研究生和初入职业生涯的数学家提供一个良好的训练基地。Nazarov、Treil和Volberg预计他们的研究生和博士后会积极参与这个项目。

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.