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

这项研究将集中于附属von Neumann代数的正则性理论的发展及其在几个分析领域和偏微分方程组理论中的应用。由于谱乘子和傅立叶乘子的正则性在许多领域的重要性，几十年来它一直是一个密集而持续研究的主题。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、相对论克莱因-戈登方程和许多其他方程的解都可以写成谱乘子，即支配方程的算符的函数(例如拉普拉斯算子)。乘子定理及其对大时间的进一步依赖(衰减)一直是所谓的色散估计的基石，意味着进一步的Strichartz估计，如今是研究非线性方程的全局时间适定性的主要工具。由于数学物理的不同设置，该方案有许多变种，不同的算符取代拉普拉斯算子，不同类型的势，以及不同类型的非线性。事实上，在最简单的欧几里得环境中已知的几个结果允许根据附属的von Neumann代数的功能子空间，或者更确切地说，附属于该空间的von Neumann代数的(密集定义的)算子的空间来解释它们。如果基本空间具有群结构，这可以是群von Neumann代数，或者是由空间上的给定算子生成的von Neumann代数，例如非对易几何的Dirac算子或量子群环境中的算子。在这种方法中，我们可以将乘子视为附属于给定von Neumann代数的那些算子(从属关系是包含的扩展，从而建立了一个严格的框架，用于进行具有复杂谱结构的无界算子的谱分析或函数演算)。我们感兴趣的是发展一种新的方法来证明不同函数空间上的算子的乘子定理，方法是研究它们在相关的von Neumann代数的正则性尺度上的正则性。该项目的目标有两个：在一般理论方面取得进展，但同时记住环境(群、流形、分形图和许多其他包含在该框架中的)和演化偏微分方程组的所有特别重要的激励例子，并将其应用于全局时间适定性的初始和初始边界问题。因此，它将提供一种新的方法来为它们的解建立色散估计，这是一个在变系数偏微分方程组或复杂几何领域中长期存在且出了名的困难的问题。在非对易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方面，这是一项重要的、具有挑战性的及时的研究。

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.