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.

拟议的研究计划作为其目标是对障碍分析领域重要的功能空间及其应用。总的来说，分析中的许多问题都涉及表明该算法是有限的，并且必须指定哪些空间。尽管Lebesgue LP空间是基本的，但其他基本空间（例如Sobolev空间）已被证明对针对偏微分方程的规则性研究至关重要。现代谐波分析的主要领域之一是奇数积分运算符的thecalderon-zygmund理论，这也是在PDE解决方案中出现的。在此理论中，当P为1时，LP空间的尺度自然延伸至强壮的空间，当P是无限时，John和Nirenberg的空间BMO（有界平均振荡的函数）。这些空间本身就是感兴趣的，用于使用各种应用，例如Div-Curl引理。本地版本可以更好地适合于在域或歧管上工作。我们建议将这些和其他相关空间的研究扩展到更通用的设置，例如度量级数，并使用它们来证明其目前已知的结果类似物欧几里得设置。对措施空间是分析和几何领域中重要工作的主题。从欧几里得环境转移结果的目标是能够释放其依赖对图的结构的证据。例如，为了讨论诸如Div-Curl引理之类的结果，必须能够在不使用衍生物的情况下由差异几何形状产生的概念。最初，最好考虑存在额外结构（例如谎言组或图形），甚至具有绝对连续的欧几里得空间的额外结构，即通过重量给出的欧几里得空间。在这些设置中开发必要的工具本身就与功能空间的研究无关。特别是，需要确定某些不平等现象，例如庞加雷不平等。