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.

我的研究在于谐波分析的一般领域（傅立叶分析），它的核心是将功能（可以代表声音或图像代表声音或图像）分解为基本组件的概念，在某种意义上更简单。然后，使用操作员或转换（例如在偏微分方程解决方案中产生的组件）对这些组件进行起作用更容易。为了从通常是无限数量的各个部分重建整体，需要有收敛的通知以及操作员的界限。这是函数空间的重要性进入图片的重要性，即选择适当的函数以获取我们的输入，并确定哪些适合输出的类别。我们选择的最终功能空间越好，我们对这些操作员行为的理解越好。主要挑战之一是能够识别不同吉他中的功能空间，证明纳入结果和不同的空间。另一个是了解该功能空间与问题的几何设置如何相关。提出的研究在很大程度上是出于以下问题的激励：我们对控制平均振荡的功能了解什么？平均振荡衡量该功能在给定集合上的平均值偏离了多少。尽管已经知道了很多，但我们仍然比平均振荡与函数之间的函数与其导数之间的关系更加了解。约翰和尼伦贝格在1961年引入了有限平均振荡（BMO）的功能空间，这是由弹性理论中的问题激励的，并且不可能统一控制所有子集的平均振荡。该空间的变体最近已被广泛研究，可以提供有关大小和光滑度方面的功能的更多细微信息。我们对平均振荡的理解也可以应用于概率和统计数据，在这种概率和统计中，它与布朗运动和随机微分方程具有重要联系。 C. Fefferman的著名结果将BMO与Hardy Space H1联系起来。自20世纪初以来，Hardy空间在谐波分析中起着至关重要的作用，最初与傅立叶系列的收敛有关，而最近与部分微分方程有关。特别有趣的是“局部”或非殖民版本的Hardy空间，以及相应的BMO空间，它们非常适合某些类型的部分微分方程，并且可以在基础几何形状中更灵活。在许多应用中，只有在界面环境中考虑了问题，例如在流体动力学的湖泊方程中。域及其边界的形状起着至关重要的作用。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。