Grant: Discovery Grants Program - Individual

The proposed research lies in the broad mathematical area known as analysis. More specifically, it concerns harmonic analysis and its application to partial differential equations and several complex variables. In particular, I am interested in developing and using the theory of Hardy spaces and related function spaces. Classical harmonic analysis is an old field of mathematics, dating back to the time of Fourier and his contemporaries, who, in the early 1800's, used trigonometric series (i.e. expansion in terms of ""harmonics"", a term coming from music) to solve such physical problems as the conduction of heat and the position of a vibrating string. These processes are governed by what is known as partial differential equations. While harmonic analysis has undergone a major modernization in the 20th century, following the development of Lebesgue integration and the theory of functional analysis (related to the ideas of quantum mechanics in physics), it has always been motivated by the same problem, namely the decomposition of a ""signal"", given as a mathematical function, into some basic components, and the reconstruction of that signal given those components. In fact, the widespread use of wavelets today in such applications as the storage of fingerprints is an example of the concrete nature of harmonic analysis. Function spaces provide a way of measuring various features of functions (representing, for example, signals or the solutions of a differential equation) and grouping them according to these features. Being able to prove that for a certain type of input, the output of a process (a transform or operator solving a differential equation, for example) lies in a certain function space provides control of the process. Developing the right kind of function spaces to measure desired properties is therefore of great importance. The significance of this proposal lies in furthering the understanding of the existing theory of Hardy spaces and BMO and its applications to partial differential equations, as well as in the introduction of new types of function spaces which can provide finer tools for these applications.