Function spaces in harmonic analysis

调和分析中的函数空间

• 批准号: 229655-2013
• 负责人: Dafni, Galia
• 金额: $ 1.09万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570371
• 关键词: Function; spaces; harmonic; analysis

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.

拟议的研究计划的目标是研究领域中重要的功能空间 谐波分析及其应用.一般来说，分析中的许多问题都涉及表明 某些运算符是有界的，必须指定哪些空格。虽然 Lebesgue Lp 空间是 最基本的其他基本空间，例如索博列夫空间，已被证明对于规律性的研究至关重要 对于偏微分方程。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，也出现在偏微分方程的求解中。在这个 理论上，当 p 为 1 时，Lp 空间的尺度自然延伸至 Hardy 空间，并延伸至 BMO 空间 当 p 无穷大时，John 和 Nirenberg 的（有界平均振荡函数）。这些空间很有趣 其本身及其用途有多种应用，例如 div-curl 引理。本地版本可以 定义了哪些更适合在域或流形上工作。 我们建议将对这些和其他相关空间的研究扩展到更一般的设置，例如度量 测量空间，并用它们来证明当前在欧几里得设置中已知的结果的类似物。 度量空间是分析和几何领域近期重要工作的主题。这 从欧几里得设置转移结果的目标是能够释放它们对 结构光滑。例如，为了讨论 div-curl 引理等结果，人们必须能够 定义由微分几何产生的概念，而不使用导数。最初最好考虑 存在额外结构的情况，例如李群或图，甚至具有测度的欧几里得空间 这是绝对连续的，即由权重给出。在这些环境中开发必要的工具非常重要 其本身的重要性，独立于功能空间的研究。特别是，需要确定 某些不等式（例如庞加莱不等式）是否成立。