A large part of analysis is framed in the language of classical Banach (function) spaces (such as: Lebesgue spaces and Sobolev spaces), as well as the bounded linear operators between these spaces (such as: Fourier multiplier operators and Calderon-Zygmund operators). The Principle Investigator will extend results for such operators from the classical setting (i.e., between scalar-valued function spaces) to operator-valued setting (i.e., between Banach space-valued functions spaces). Such extensions have applications in, among others, spectral theory and partial differential equations (e.g., regularity theory). Such extensions will also lead to results in martingale theory, which serves as a bridge between several areas of mathematical analysis, such as: harmonic analysis, stochastic analysis, and the geometry of Banach spaces. In these extensions, the geometry of the underlying Banach spaces (e.g., Fourier type and uniform convexity) will play a key role.
A Banach space is a space of vectors that has, among other things, a way to measure the distance between two vectors. The most basic example of a Banach space is the three-dimensional space around us. In scientific applications (e.g., in: physics, engineering, and signal processing) the movement of particles or shapes in space over time (such as waves in the ocean) is described by functions, which together with their distances, give rise to more sophisticated Banach function spaces. The properties of such functions modeling these natural phenomena are described by differential equations, which can be viewed as operators between Banach spaces. Recent applications have led the experts to work in Banach space-valued, rather than real-valued, Banach spaces. Motivated by such applications, the Principle Investigator will research such operators in this Banach space-valued setting.
