Vector-Valued Analysis and Geometry of Banach Spaces

很大一部分的分析是用经典的Banach（函数）空间（如紧集合上的连续函数空间、Lebesgue函数空间和Hardy空间）以及这些空间之间的有界线性算子（如：傅里叶乘子算子、积分算子、伪微分算子和鞅变换）的语言构建的。PI将这些算子的结果从经典设置（即，在标量值函数空间之间）扩展到算子值设置（即，在巴拿赫空间值函数空间之间）。这种扩展在偏微分方程（如正则理论）和数学物理（如流体动力学）等领域都有应用。在这种扩展中，底层巴拿赫空间的几何（例如，傅里叶型和一致凸性）将发挥关键作用。PI还将探讨“渐近一致凸性”，这是Banach空间的一个几何性质，最近在线性化理论中得到了复兴和应用。巴拿赫空间是一个由向量组成的空间，其中，有一种方法可以测量两个向量之间的距离。一个鼓舞人心的例子是我们周围的物理三维空间。一个函数空间（其中一个函数被认为是一个向量），具有合适的距离（例如，由两个函数包围的区域），是巴拿赫空间的另一个例子。在科学应用（例如：物理、工程和信号处理）中，人们经常通过适当的巴拿赫空间来模拟现象；海洋中的波浪可以用巴拿赫空间中的函数来建模。人们可以通过函数的偏微分方程来研究这些自然现象。这些方程引出了巴拿赫空间之间的算子。最近的应用使专家们不再在实值巴拿赫函数空间中工作，而是在巴拿赫空间值巴拿赫函数空间中工作。在这些应用的激励下，PI将在这个巴拿赫空间值设置中探索这些算子。

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.