Mean oscillation and related function spaces

我的研究集中在谐和分析(傅立叶分析)的一般领域，它的核心思想是将一个函数(可以表示声音信号或图像)分解成在某种意义上更简单的基本分量。然后，使用运算符或变换来作用于这些分量就更容易了，例如在求解偏微分方程时出现的那些。 为了从部分重建整体，这些部分的数量通常是无限的，人们需要有一个收敛的概念，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接受我们的输入，并确定什么是适合输出的类。我们选择的函数空间越精细，我们对这些算子的行为的理解就越好。主要的挑战之一是能够识别不同伪装的函数空间，证明包含结果，并区分不同的空间。另一种是理解函数空间与问题的几何设置之间的关系。 这项拟议的研究主要是由以下问题推动的：我们对一个函数的平均振荡控制有什么了解？均值振荡衡量函数在给定集合上平均偏离其均值的程度。虽然已经知道了很多，但我们仍然更多地了解函数及其导数之间的关系，例如，比起平均振荡和函数之间的关系。 有界平均振荡函数空间(BMO)是由John和Nirenberg在1961年受弹性理论中的问题启发而提出的，它对所有子集的平均振荡施加一致控制。这种空间的变体最近得到了广泛的研究，可以在大小和光滑性方面给出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统计，它与布朗运动和随机微分方程有重要的联系。 C.费弗曼的一个著名结果将BMO与哈代太空H1联系在一起。自20世纪初以来，Hardy空间在调和分析中发挥了重要作用，最初是与傅立叶级数的收敛有关，最近又与偏微分方程有关。特别令人感兴趣的是Hardy空间的“局部”或非齐次版本，以及相应的BMO空间，它们非常适合某些类型的偏微分方程组，并允许在底层几何中提供更多的灵活性。在许多应用中，人们只考虑有界环境中的问题，例如在流体动力学的湖泊方程的情况下。区域的形状及其边界起着至关重要的作用。在其他情况下，来自欧几里得空间的结果需要扩展到不同的环境，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.