A Further Development of the Theory of Bellman Functions and Applications to Estimates for Singular Integral Operators

Bellman函数法起源于最优控制理论。近年来，它已被应用于谐波分析中各种各样的问题。PI计划进一步发展Bellman函数方法在更广泛问题上的适用性，并研究它与谐波分析中出现的其他常用工具的联系。该方法在PI关于加权和非加权空间中奇异积分算子的工作中起着至关重要的作用。这些问题包括Lebesgue空间中Beurling算子的明确数值界限（著名的p-1问题）以及n维加权Lebesgue空间中Riesz变换的合适的明确范数估计，最好与维数无关。PI感兴趣的一个相关方向是关注特定奇异积分算子的有界性，如希尔伯特变换，如果源空间和目标空间具有不同的权值。本课题的研究方向是谐波分析，长期以来，谐波分析一直是数学研究的中心领域。其他领域的许多问题都可以归结为在分析框架中提出和解决的问题。这些问题包括是否存在特别适合于信号处理应用的函数的特殊构建块（小波）。其他问题包括所谓的奇异积分算子的有界性，这在偏微分方程和物理问题中是极其重要的。PI的工作就是详细研究这些算子的连续性。我们经常通过奇异积分算子在构造块上的作用来研究它们，这是一种继承和改进了随机最优控制的技术。这种方法被称为Bellman函数法，它为谐波分析中非常复杂的工具提供了一种更简单但更强大的替代方法。此外，由于它的相对简单，那些没有专业背景知识的人仍然可以使用，例如其他领域的科学家。因此，它既是与其他科学领域的联系，促进了跨学科的交流，也为本科生或研究生早期参与研究提供了可能性。

The method of Bellman functions originated in the theory of optimal control. In recent years it has been applied to a surprising variety of problems in harmonic analysis. The PI plans to further develop the applicability of the method of Bellman functions to a broader range of problems and study its connection with other common tools that appear in harmonic analysis.This method plays a crucial role in the PI's work concerning singular integral operators in weighted and unweighted spaces. The problems include sharp numerical bounds for the Beurling operator in Lebesgue spaces (the famous p-1 problem) as well as suitable sharp norm estimates for Riesz transforms in n dimensional weighted Lebesgue spaces, preferrably independent of the dimension. A related direction the PI is interested in is concerned with boundedness of a particular singular integral operator such as the Hilbert transform if the source and the target space have a different weight. This project lies in harmonic analysis, which has been a central area of mathematics for a long time. Many questions in other fields reduce to questions best posed and solved in the framework of analysis. Such questions include the existence of special building blocks of functions (wavelets) that are particularly well-suited for applications in signal processing. Other questions include the boundedness property of so-called singular integral operators, which are of extreme importance in problems in partial differential equations, and physics. The PI's work consists of studying the continuity properties of such operators in detail.We often study singular integral operators through their actions on the building blocks, using a technique inherited and modified from stochastic optimal control. This method, called method of Bellman functions, provides a simpler yet often more powerful alternative to very involved tools in harmonic analysis. In addition, through its relative simplicity still being accessible to those with less specialized background knowledge, for example scientists in other fields. As such it serves both as link to other scientific fields, furthering interdisciplinary communication and provides a possibility of early involvement of undergraduates or beginning graduate students into research.