The Role of Gromov Hyperbolicity and Besov Spaces in Quasiconformal Analysis

本研究项目的灵感来自于普通欧几里得空间中使用的基本工具，涉及开发用于分析度量测量空间（具有良好定义的距离和体积概念的空间）的数学工具。这项工作对许多数学领域都有影响，包括动力系统、谐波分析和偏微分方程。几何分析中一个有用的测量是所谓的别索夫能量。该项目将探索紧致度量空间中的非局部贝索夫型能量与空间中编码的局部能量之间的联系，通过将紧致空间视为其边界的良好连接的超空间进行测量。这项工作的目的是利用这些联系来探索非局部能量是如何通过度量或距离的良好调节变化而转换的。该项目的部分工作将与研究生和其他早期职业数学家合作完成，从而也为未来的STEM劳动力提供培训。在本项目中，主要研究者将利用紧化度量空间的Gromov双曲填充来开发与Besov能量相关的潜在理论工具，从而实现紧化空间作为Gromov双曲空间的边界。这样的格罗莫夫双曲空间具有可控的几何形状，通过利用这种控制，首席研究员将寻求获得对非局部能量的控制，并获得有关这些能量如何通过准对称映射转换的信息。该项目还将通过发展分数阶p-拉普拉斯函数（分数阶拉普拉斯函数的非线性版本）的观点来增强Besov能量的潜在理论性质。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project concerns the development of mathematical tools for analysis on metric measure spaces (spaces with well-defined notions of distance and volume), inspired by basic tools used in ordinary Euclidean space. The work has implications for many fields of mathematics, including dynamical systems, harmonic analysis, and partial differential equations. One useful measurement in geometric analysis is the so-called Besov energy. This project will explore links between the nonlocal Besov-type energy in a compact metric measure space and local energy encoded in the space, as measured through a well-connected super-space that sees the compact space as its boundary. The work aims to use these links to explore how non-local energies are transformed by well-regulated changes in the metric, or distance. Parts of this project will be done in collaboration with graduate students and other early career mathematicians, thus also providing training of the future STEM workforce. In this project, the principal investigator will develop the potential theoretic tools related to Besov energies by using Gromov hyperbolic fillings of the compact metric measure space so that the compact space is realized as the boundary of the Gromov hyperbolic space. Such a Gromov hyperbolic space is of controlled geometry, and by exploiting this control, the principal investigator will seek to gain control of the nonlocal energies and to obtain information about how these energies are transformed by quasisymmetric maps. The project will also augment the potential theoretic properties of Besov energy by developing the perspective of fractional p-Laplacian (nonlinear versions of the fractional Laplacian).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.