Vector-Valued Analysis with a Flair from the Geometry of Banach Spaces

0600888 GirardiAbstract很大一部分的分析是在经典的Banach（函数）空间（如：Lebesgue空间和Sobolev空间），以及这些空间之间的有界线性算子（如：傅立叶乘子算子和Calderon-Zygmund算子）的语言框架。主要研究者将从经典设置（即，标量值函数空间之间）到算子值设置（即，Banach空值函数空间）。这样的扩展在谱理论和偏微分方程（例如，规则性理论）。这样的扩展也将导致鞅理论的结果，作为数学分析的几个领域之间的桥梁，如：调和分析，随机分析和Banach空间的几何。在这些扩展中，基本Banach空间的几何（例如，傅立叶型和一致凸性）将发挥关键作用。 一个Banach空间是一个向量空间，其中有一种方法来测量两个向量之间的距离。Banach空间最基本的例子是我们周围的三维空间。在科学应用中（例如，在：物理学、工程学和信号处理），空间中粒子或形状随时间的运动（如海洋中的波浪）由函数描述，函数与它们的距离一起产生更复杂的Banach函数空间。这些自然现象的建模功能的性质所描述的微分方程，它可以被看作是Banach空间之间的运营商。最近的应用导致专家工作在Banach空间值，而不是真正的价值，Banach空间。出于这样的应用程序，主要研究者将研究这些运营商在这个Banach空间值设置。

0600888 GirardiAbstractA 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.