Nonreflexive Function Spaces: Fourier v. Martingale Approach

This project investigates a list of exceptionally hard problems at the crossroads of harmonic analysis (singular integral operators), partial differential equations (Sobolev spaces, singularities of solutions) and Probability (Brownian motion, discrete time martingales and general Haar systems). Harmonic analysis is a branch of mathematics designed for investigating physical phenomena of oscillatory nature. Its applications to other disciplines of mathematics include beautiful results in number theory, additive combinatorics, probability theory, differential equations, dynamical systems, potential theory, geometry of infinite dimensional spaces, and topology. The recent, explosive growth of Harmonic Analysis is due (in part) to its role played in singular integral operators operators (of Calderon-Zygmund type) and Wavelet theory. Martingale theory is a central and important branch of probability theory. Initially developed as a tool for analyzing winning strategies in game theory, it turned out to be extremely useful in many branches of mathematical analysis - in particular in harmonic analysis and in the construction of solutions of partial differential equations. One of the prime examples of martingales, the Wiener process also called Brownian motion, found a wide range of applications outside mathematics, e.g., in theoretical physics, social sciences and economy. Within mathematics, the classical martingale theory is well suited to study spaces of integrable functions; more recently it became a promising tool in the analysis of singularities of measures, arising as generalized solutions of partial differential equations. In this project, we use modern martingale theory to study Banach spaces of differentiable functions, which play a central role in analysis. They are crucial in the theory of differential equations, dynamical systems, probability theory etc. However their inner structure, especially in the non-reflexive case, remained largely elusive. This is true in particular when comparing Banach spaces of differentiable functions to Banach spaces of analytic functions. Hence one of the main goals of this project is to expand the knowledge of the structure of non-reflexive Banach spaces of differentiable functions and description of singularities of their elements. In the project we would like to adapt and develop martingale methods to study singularities of differentiable functions and their generalizations. We want to connect these with embeddings and trace theorems. Moreover we want to exploit the possible similarities and differences between the theories of analytic and smooth functions. Harmonic analysis of operators in non-reflexive spaces is not only important and crucial in applications, but it is also a task full of mathematical beauty and leads to deep often unexpected links, and therefore it should most certainly be studied and developed intensively.