Function spaces in harmonic analysis

拟议的研究计划作为其目标的研究领域的重要功能空间， 谐波分析及其应用一般来说，分析中的许多问题都涉及到表明， 某些算子是有界的，并且必须指定在哪些空间上。而Lebesgue Lp空间是 最基本的，其他基本空间，如Sobolev空间，已被证明是必不可少的研究正则性 偏微分方程。现代谐波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的解决方案。在这 理论，当p为1时，Lp空间的标度自然地扩展到哈代空间，并且扩展到BMO空间 （有界平均振荡函数）的John和Nirenberg当p是无限的。这些空间很有趣 在它们自己和他们的使用是各种应用，如div-curl引理。本地版本可以是 定义，更好地适应域或流形上的工作。 我们建议将这些和其他相关空间的研究扩展到更一般的设置，如度量 测量空间，并使用它们来证明类似的结果是目前已知的欧几里德设置。 度量测度空间是分析和几何领域最近重要工作的主题。的 从欧几里德设置转移结果的目标是能够释放他们的依赖证明 平滑结构。例如，为了讨论诸如div-curl引理的结果，必须能够 定义不使用导数而从微分几何中产生的概念。最初，最好考虑 存在额外结构的情况，例如李群或图，甚至是具有测度的欧几里得空间 其是绝对连续的，即由权重给出。在这些环境中开发必要的工具， 它本身的重要性，独立于函数空间的研究。特别是，需要确定 某些不等式，如庞加莱不等式，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.