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

贝尔曼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的各种问题。PI计划进一步发展贝尔曼函数方法的适用性，以更广泛的问题，并研究其与其他常见的工具，出现在调和分析。这种方法在PI的工作中起着至关重要的作用，关于奇异积分算子在加权和unweighted空间。问题包括尖锐的数值界的Beurling算子在Lebesgue空间（著名的p-1问题），以及适当的尖锐的范数估计Riesz变换在n维加权Lebesgue空间，最好是独立的维度。PI感兴趣的一个相关方向涉及特定奇异积分算子的有界性，例如希尔伯特变换，如果源空间和目标空间具有不同的权重。这个项目在于调和分析，这一直是数学的中心领域很长一段时间。其他领域的许多问题都归结为在分析框架中提出和解决的最佳问题。这些问题包括存在特别适合于信号处理应用的特殊功能（小波）构建块。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非常重要。PI的工作包括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积木上的作用来研究奇异积分算子，使用从随机最优控制继承和修改的技术。这种方法，称为贝尔曼函数法，提供了一个更简单，但往往更强大的替代非常复杂的工具在谐波分析。此外，由于其相对简单，那些不太专业的背景知识，例如其他领域的科学家仍然可以使用。因此，它既作为链接到其他科学领域，促进跨学科交流，并提供了本科生或开始研究生早期参与研究的可能性。

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.