The unifying theme of this proposal is the study of the Hilbert transform and its closest mathematical relatives, the Cauchy transform and the Riesz transform, in non-homogeneous settings appearing in various applications. The applications included in this project were in the center of attention of analysts about 50 years ago -- the completeness and minimality problems, specifically the Beurling-Malliavin theory, the "gap and density" theorems of Beurling-Levinson type, the theory of Cartwright and Paley-Wiener spaces, etc. These areas contain some of the deepest results of linear complex analysis. Even modern expositions require hundreds of pagers with some proofs (like the proof of the Beurling-Malliavin multiplier theorem) still looking completely mysterious. The needs of spectral analysis call for a new approach to these problems and for an extension of the classical results. Powerful techniques of modern complex analysis should give rise to vast generalizations of the classical theory and effective applications to spectral problems.
This project focuses on complex analysis and its applications. Complex analysis is a classical area of mathematics that continues to play an important role in both pure and applied studies. One of the canonical objects of complex analysis is the so-called Hilbert transform. Studies of the Hilbert transform allow one to understand the behavior of complex differentiable functions near the boundary of their domains. Despite being one of the most studied objects in all of mathematics, Hilbert transform is far from being completely understood. Moreover, new developments in applications, such as mathematical models of solid state physics and differential equations, require considerable extensions of classical results. The goal of this project is to provide such extensions and to apply new results in several areas of analysis and mathematical physics. Among such applications are spectral problems for the string equation and the Schroedinger equation, which describes wave propagation in quantum mechanics.
