Function spaces in harmonic analysis

所提议的研究方案的目标是研究在以下领域中重要的函数空间 调和分析及其应用。一般说来，分析中的许多问题都涉及到证明 某些运算符是有界的，必须指定在哪些空间上。而勒贝格Lp空间是 最基本的、其他的基本空间，如索博列夫空间，已被证明是正则性研究的基本要素 对于偏微分方程式。现代调和分析的主要领域之一是 奇异积分算子的Calderon-Zygmund理论，它也出现在偏微分方程解中。在这 理论上，当p为1时，Lp空间的尺度自然扩展到Hardy空间和BMO空间 John和Nirenberg的(有界平均振荡函数)。这些空间很有价值 它们本身和它们的用途是各种应用，例如div-curl引理。本地版本可以是 定义了哪些更适合在域或流形上工作。 我们建议将这些空间和其他相关空间的研究扩展到更一般的环境，如度量 测量空间，并用它们来证明与目前已知的欧几里得环境中的结果类似。 度量空间是分析和几何领域最近重要工作的主题。这个 从欧几里得环境中转移结果的目标是能够使他们的证明摆脱对 结构流畅。例如，为了讨论div-curl引理之类的结果，人们必须能够 定义由微分几何产生的概念，不使用导数。最初，最好是考虑 存在额外结构的情况，例如李群或图，甚至具有度量的欧几里德空间 它是绝对连续的，也就是由一个重量赋予的。在这些环境中开发必要的工具是 本身的重要性，独立于对函数空间的研究。特别是，我们需要确定 某些不平等，如庞加莱不平等，是否成立。

The proposed research program has as its objectives the study of function spaces of importance in the field of harmonic analysis and its applications. In general terms, many questions in analysis involve showing that certain operators are bounded, and one has to specify on which spaces. While the Lebesgue Lp spaces are the most basic, other fundamental spaces, such as Sobolev spaces, have proved essential to the study of regularity for partial differential equations. One of the major areas of modern harmonic analysis is the Calderon-Zygmund theory of singular integral operators, which also arise in the solution of PDE. In this theory, the scale of Lp spaces extends naturally to the Hardy space when p is 1, and to the space BMO (functions of bounded mean oscillation) of John and Nirenberg when p is infinite. These spaces are of interest in themselves and for their use is various applications, such as the div-curl lemma. Local versions can be defined which are better adapted to working on domains or on manifolds. We propose to extend the study of these and other related spaces to more general settings such as metric measure spaces, and use them to prove analogues of results which are currently known in the Euclidean setting. Metric measure spaces are the subject of important recent work in the areas of analysis and geometry. The goal in transferring results from the Euclidean setting is to be able to free the proofs of their dependence on the smooth structure. For example, in order to discuss results such as the div-curl lemma, one has to be able to define notions arising from differential geometry without using derivatives. Initially it is best to consider the cases where there is extra structure, such as Lie groups or graphs, or even Euclidean space with a measure which is absolutely continuous, i.e. given by a weight. Developing the necessary tools in these settings is of importance in itself, independently of the study of function spaces. In particular, one needs to determine whether certain inequalities, such as the Poincare inequality, hold.