Mean oscillation and related function spaces

平均振荡和相关函数空间

• 批准号: RGPIN-2019-05510
• 负责人: Dafni, Galia
• 金额: $ 1.24万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758941
• 关键词: Mean; oscillation; related; function; spaces

My research lies in the general area of harmonic analysis (Fourier analysis), which has at its core the idea of decomposing a function (which can represent a sound signal or an image) into basic components which are in some sense simpler.  It is then easier to act on these components with operators, or transformations, such as those arising in the solution of partial differential equations.  In order to reconstruct the whole from its parts, which are usually infinite in number, one needs to have a notion of convergence, as well as boundedness of the operators.   This is where the importance of function spaces comes into the picture, namely choosing the appropriate class of functions in which to take our input, and determining what is the appropriate class for the output.  The finer the function spaces we choose, the better is our understanding of the behavior of these operators.  One of the main challenges is to be able to recognize a function space in different guises, prove inclusion results and distinguish different spaces. Another is to understand how the function spaces relate to the geometric setting of the problem. The proposed research is largely motivated by the following question: what do we know about a function from control of its mean oscillation? Mean oscillation measures how much, on the average, the function deviates from its mean on a given set. While much is already known, we still understand significantly more about the relationship between a function and its derivative, for example, than between the mean oscillation and the function. The space of functions of bounded mean oscillation (BMO) was introduced by John and Nirenberg in 1961, motivated by questions in elasticity theory, and imposes uniform control of the mean oscillation over all subsets. Variants of this space have been widely studied recently and can give more nuanced information about a function in terms of size and smoothness.  Our understanding of mean oscillation can also be applied to probability and statistics, where it has important connections to Brownian motion and stochastic differential equations. A celebrated result of C. Fefferman links BMO with the Hardy space H1.  Hardy spaces have played an essential role in harmonic analysis since the early 20th century, initially in relation to the convergence of Fourier series, and more recently in connection with partial differential equations.  Of particular interest are "local" or non-homogeneous versions of Hardy spaces, and the corresponding BMO spaces, which are well suited to certain types of partial differential equations, as well as allowing more flexibility in the underlying geometry.  In many applications, one only considers the problem in a bounded setting, for example in the case of the lake equations of fluid dynamics.  The shape of the domain and its boundary play a crucial role.  In other situations, results from Euclidean space need to be extended to a different setting, for example graphs.

我的研究属于谐波分析（傅里叶分析）的一般领域，其核心思想是将函数（可以表示声音信号或图像）分解为在某种意义上更简单的基本组件然后，使用算子或变换对这些分量进行操作会更容易，例如在偏微分方程的解中出现的算子或变换。为了从其数量通常是无限的部分重建整体，需要有一个。收敛的概念，以及​这就是函数空间的重要性所在，即选择合适的函数类别来获取输入，并确定合适的输出类别。​函数空间越精细。我们选择 ，我们对这些运算符的行为的理解就越好，主要挑战之一是能够识别不同形式的函数空间，证明包含结果并区分不同的空间。到问题的几何设置。研究主要是出于以下问题：通过控制函数的平均振荡，我们对函数了解多少？平均振荡衡量函数在给定集合上平均偏离其平均值的程度。例如，我们仍然更清楚地了解函数及其导数之间的关系，而不是平均振荡与函数之间的关系。有界平均振荡（BMO）的函数空间是由约翰和尼伦伯格在1961 年，受弹性理论问题的启发，并对所有子集的均值振荡进行了统一控制。最近，该空间的变体已得到广泛研究，可以在大小和平滑度方面提供有关函数的更细致的信息。振荡也可以应用于概率和统计，其中它与布朗运动和随机微分方程有重要的联系。 C. Fefferman 的一个著名结果将 BMO 与 Hardy 空间 H1 联系起来。自 20 世纪初以来，它在调和分析中发挥着重要作用，最初与傅立叶级数的收敛有关，最近与偏微分方程有关，特别令人感兴趣的是哈代空间的“局部”或非齐次版本，以及相应的 BMO 空间，它们非常适合某些类型的偏微分方程，并且允许基础几何具有更大的灵活性。在许多应用中，人们只考虑有界设置中的问题，例如在在其他情况下，欧几里得空间的结果需要扩展到不同的设置，例如图形。