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.

Harmonic analysis originated as a mathematical framework for studying oscillatory phenomena arising in physical contexts. It has evolved into a rich theory continuing to prove remarkably effective in addressing a wide range of problems across mathematics. Its applications --- often surprising --- span number theory, additive combinatorics, probability theory, partial differential equations, dynamical systems, potential theory, the geometry of infinite-dimensional spaces, and topology. A significant advancement in the twentieth century was the development of the theory of singular integral operators, particularly those of Caldern-Zygmund type, which has profoundly shaped modern harmonic analysis and yielded numerous applications. Martingale theory constitutes a central pillar of modern probability theory. Originally conceived to analyze strategies in games of chance, it has since found deep connections with various areas of analysis, notably harmonic analysis and the theory of partial differential equations. Among the most celebrated martingales is the Wiener process (Brownian motion), which has had a wide-ranging impact not only within mathematics but also in theoretical physics, economics, and the social sciences. Classical martingale theory is well-suited to study integrable functions, however, it's full potential in the analysis of singular measures --- such as those arising as solutions to partial differential equations and their generalizations --- has not yet been realized. This project applied contemporary martingale techniques to the study of Banach spaces of differentiable functions, central to many areas of analysis, including PDEs, dynamical systems, and stochastic processes. The internal structure of these spaces --- particularly in the non-reflexive setting --- was the subject of our investigations. A primary objective achieved by our research efforts is to deepen the understanding of non-reflexive Banach spaces of differentiable functions, with a particular focus on characterizing singularities of their elements. The harmonic analysis of operators acting on non-reflexive spaces is both mathematically rich and crucial for applications. Our study of those operators revealed both deep and unexpected connections across several mathematical disciplines.