Mean oscillation and related function spaces

我的研究领域是谐波分析（傅里叶分析），其核心思想是将一个函数（可以代表声音信号或图像）分解成在某种意义上更简单的基本组件。这样就更容易用算子或变换作用于这些分量，比如在解偏微分方程时出现的那些。为了从部分重构整体，这些部分的数量通常是无限的，人们需要有收敛的概念，以及算子的有界性。这就是函数空间的重要性所在，即选择适当的函数类来作为输入，并确定适当的输出类是什么。我们选择的函数空间越精细，我们对这些运算符的行为的理解就越好。其中一个主要的挑战是能够识别不同形式的功能空间，证明包含结果并区分不同的空间。另一个是理解函数空间是如何与问题的几何环境相联系的。提出的研究在很大程度上是由以下问题驱动的：我们对控制其平均振荡的函数了解多少？平均振荡测量函数在给定集合上偏离其平均值的平均程度。虽然我们已经知道了很多，但我们对函数和它的导数之间的关系，例如，比平均振荡和函数之间的关系了解得多得多。有界平均振荡（BMO）的函数空间由John和Nirenberg于1961年在弹性理论的启发下引入，并对所有子集的平均振荡进行统一控制。这个空间的变体最近得到了广泛的研究，可以在大小和平滑度方面提供关于函数的更细微的信息。我们对平均振荡的理解也可以应用于概率和统计，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，Hardy空间在调和分析中起着至关重要的作用，最初与傅里叶级数的收敛有关，最近与偏微分方程有关。特别感兴趣的是Hardy空间的“局部”或非齐次版本，以及相应的BMO空间，它们非常适合于某些类型的偏微分方程，并且在底层几何中允许更多的灵活性。在许多应用中，人们只考虑有界环境下的问题，例如流体动力学的湖泊方程。区域的形状及其边界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。

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.