Regularity in affiliated von Neumann algebras and applications to partial differential equations

拟议的研究将集中在附属冯诺依曼代数的正则性理论的发展及其随后的应用到几个领域的分析和偏微分方程的理论。频谱和傅立叶乘数的正则性的主题，现在一直是一个主题的密集持续研究了几十年，由于其重要性，许多领域。事实上，数学物理的主要方程，如薛定谔方程、波动方程、克莱因-戈登方程、相对论性克莱因-戈登方程和许多其他方程的解都可以写成谱乘数，即控制方程的算子的函数（例如拉普拉斯算子）。乘子定理及其在大时间内的进一步依赖性（衰减）一直是所谓的色散估计的基石，这意味着进一步的Eschenhartz估计，现在是研究非线性方程在时间上的全局适定性的主要工具。该方案有许多变种的动机，各种设置的数学物理，不同的运营商取代拉普拉斯算子，不同类型的潜力，和不同类型的非线性。本项目的目的是把冯诺依曼代数的现代技术到这些调查。事实上，在最简单的欧几里德设置中已知的几个结果允许它们在附属冯诺依曼代数的函数子空间方面的解释，或者更确切地说，附属于空间的冯诺依曼代数的（密集定义的）算子空间。如果基础空间具有群结构，则这可以是群冯诺依曼代数，或者由空间上的给定算子生成的冯诺依曼代数，例如非交换几何中的狄拉克算子或量子群中的算子。在这种方法中，我们可以将乘子视为附属于给定冯诺依曼代数的那些算子（隶属关系是包含的延伸，建立了一个严格的框架，约翰冯诺依曼后，做谱分析或功能演算的无界算子与复杂的谱结构）。我们有兴趣在开发一种新的方法来证明乘数定理的运营商在不同的功能空间，看看他们的规律性在相关的规模的正规性在附属冯诺依曼代数。该项目的目的是双重的：在一般理论中取得进展，但要记住所有特别重要的激励设置的例子（组，流形，分形，和许多其他包括在这个框架中）和演化偏微分方程，与应用程序的全球时间适定性为他们的初始和初始边界问题。因此，它将提供一种新的方法来建立色散估计的解决方案，这是一个长期存在的问题，众所周知的困难，在该地区的偏微分方程的变系数或复杂的几何形状。这是重要的，具有挑战性的和及时的研究，在非交换算子分析和偏微分方程的理论，以及它们与其他领域和应用的关系具有深刻的影响。

The proposed research will concentrate on the development of the regularity theory in affiliated von Neumann algebras and its subsequent applications to several areas of analysis and the theory of partial differential equations.The subject of the regularity of spectral and Fourier multipliers has been now a topic of intensive continuous research over many decades due to its importance to many areas. Indeed, solutions to main equations of mathematical physics such as Schrödinger, wave, Klein-Gordon, relativistic Klein-Gordon, and many other equations can be written as spectral multipliers, i.e. functions of the operator governing the equation (e.g. the Laplacian). Multiplier theorems and their further dependence (decay) for large times has been a building block of the so-called dispersive estimates, implying further Strichartz estimates, nowadays being the main tool for investigating the global in time well-posedness of nonlinear equations. This scheme has many variants motivated by a variety of settings of the mathematical physics, with different operators replacing the Laplacian, different types of potentials, and different types of nonlinearities.The present project aims at bringing the modern techniques of von Neumann algebras into these investigations. Indeed, several results known in the simplest Euclidean setting allow for their interpretation in terms of the functional subspaces of affiliated von Neumann algebras, or rather of spaces of (densely defined) operators affiliated to the von Neumann algebra of the space. This can be the group von Neumann algebra if the underlying space has a group structure, or von Neumann algebras generated by given operators on the space, such as the Dirac operator of noncommutative geometry or the one in the setting of quantum groups.In this approach we can think of multipliers as those operators that are affiliated to the given von Neumann algebra (the affiliation is an extension of the inclusion, setting up a rigorous framework, after John von Neumann, for doing spectral analysis or functional calculus of unbounded operators with complicated spectral structure). We are interested in developing a new approach to proving multiplier theorems for operators on different function spaces by looking at their regularity in the relevant scales of regularity in the affiliated von Neumann algebras. The aim of the project is two-fold: to make advances in a general theory, but keeping in mind all the particular important motivating examples of settings (groups, manifolds, fractals, and many others that are included in this framework) and of evolution PDEs, with applications to the global in time well-posedness for their initial and initial-boundary problems. As such, it will provide a new approach to establishing dispersive estimates for their solutions, the problem that is long-standing and notoriously difficult in the area of partial differential equations with variable coefficients or in complicated geometry. This is important, challenging and timely research with deep implications in theories of noncommutative operator analysis and partial differential equations, as well as their relation to other areas and applications.