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

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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.