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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635239
• 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 ofharmonic analysis and its applications. In general terms, many questions in analysis involve showing thatcertain operators are bounded, and one has to specify on which spaces. While the Lebesgue Lp spaces are themost basic, other fundamental spaces, such as Sobolev spaces, have proved essential to the study of regularityfor partial differential equations. One of the major areas of modern harmonic analysis is theCalderon-Zygmund theory of singular integral operators, which also arise in the solution of PDE. In thistheory, 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 interestin themselves and for their use is various applications, such as the div-curl lemma. Local versions can bedefined 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 metricmeasure 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. Thegoal in transferring results from the Euclidean setting is to be able to free the proofs of their dependence on thesmooth structure. For example, in order to discuss results such as the div-curl lemma, one has to be able todefine notions arising from differential geometry without using derivatives. Initially it is best to consider thecases where there is extra structure, such as Lie groups or graphs, or even Euclidean space with a measurewhich is absolutely continuous, i.e. given by a weight. Developing the necessary tools in these settings is ofimportance in itself, independently of the study of function spaces. In particular, one needs to determinewhether certain inequalities, such as the Poincare inequality, hold.

提出的研究计划的目标是研究在谐波分析及其应用领域中重要的函数空间。一般来说，分析中的许多问题涉及到证明某些算子是有界的，并且必须指定在哪些空间上。虽然Lebesgue Lp空间是最基本的，但其他基本空间，如Sobolev空间，已被证明对偏微分方程的正则性研究至关重要。现代谐波分析的主要领域之一是奇异积分算子的卡尔德隆-齐格蒙德理论，它也出现在偏微分方程的解中。在该理论中，Lp空间的尺度在p = 1时自然地扩展到Hardy空间，在p为无穷时扩展到John和Nirenberg的有界平均振荡函数（BMO）空间。这些空间本身就很有趣，因为它们有各种各样的应用，比如旋度引理。可以定义本地版本，它们更适合在域或流形上工作。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，如度量空间，并使用它们来证明目前在欧几里得设置中已知的结果的类似物。度量度量空间是最近在分析和几何领域重要工作的主题。从欧几里得集合转移结果的目标是能够摆脱它们对光滑结构的依赖证明。例如，为了讨论诸如旋度引理之类的结果，人们必须能够在不使用导数的情况下定义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，甚至具有绝对连续度量的欧几里得空间，即由权重给定。在这些设置中开发必要的工具本身就很重要，独立于功能空间的研究。特别是，人们需要确定某些不等式，如庞加莱不等式是否成立。