Hardy spaces, related function spaces and applications

拟议的研究在于广泛的数学领域称为分析。 更具体地说，它涉及调和分析及其在偏微分方程和多个复变量中的应用。 特别是，我感兴趣的发展和使用理论的哈代空间和相关的功能空间。 经典谐波分析是一个古老的数学领域，可以追溯到傅立叶和他的同时代人的时代，他们在19世纪初使用三角级数（即“谐波”的展开，一个来自音乐的术语）来解决热传导和振动弦的位置等物理问题。 这些过程是由所谓的偏微分方程控制的。 虽然谐波分析在世纪经历了重大的现代化，随着勒贝格积分和泛函分析理论（与物理学中的量子力学思想有关）的发展，它一直受到同一问题的激励，即将作为数学函数给出的“信号”分解为一些基本分量，并在给定这些分量的情况下重建该信号。 事实上，今天小波在指纹存储等应用中的广泛使用是谐波分析具体性质的一个例子。 函数空间提供了一种测量函数的各种特征（例如，表示信号或微分方程的解）并根据这些特征对它们进行分组的方法。 能够证明对于某种类型的输入，过程的输出（例如，解微分方程的变换或算子）位于某个函数空间中提供了对过程的控制。 因此，开发正确的函数空间来度量所需的属性是非常重要的。 这一建议的意义在于进一步理解现有的理论的哈代空间和BMO及其应用偏微分方程，以及在引进新类型的功能空间，可以提供更好的工具，这些应用。

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.