Vector-Valued Analysis and Geometry of Banach Spaces

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AbstractGirardi A large part of analysis is framed in the language of classical Banach (function) spaces (such as: the space of continuous functions on a compact set, Lebesgue functions spaces, and Hardy spaces), as well as the bounded linear operators between these spaces (such as: Fourier multiplier operators, integral operators, pseudo-differential operators, and martingale transforms). The PI 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, partial differential equations (e.g., regularity theory) and mathematical physics (e.g., fluid dynamics). In such extensions, the geometry of the underlying Banach spaces (e.g., Fourier type and uniform convexity) will play a key role. The PI will also explore "asymptotic uniform convexity", a geometric property of a Banach space that has recently been revitalized and has applications in linearization theory. A Banach space is a space of vectors that has, among other things, a way to measure the distance between two vectors. A motivating example is the physical three-dimensional space around us. A space of functions (where a function is consider as a vector), with a suitable distance (for example, the area bounded by two functions), are other examples of Banach spaces. In scientific applications (e.g. in: physics, engineering, and signal processing) one often models a phenomena via an appropriate Banach space; a wave in the ocean can be model by a function in a Banach space. One can then study these natural phenomena via partial differential equations of the functions. These equations lead to operators between Banach spaces. Recent applications have led the experts to work not in real-valued Banach function spaces but rather in Banach space-valued Banach function spaces. Motivated by such applications, the PI will explore such operators in this Banach space-valued setting.