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，它已被应用于调和分析中各种令人惊讶的问题。PI计划进一步发展Bellman函数方法对更广泛问题的适用性，并研究它与调和分析中其他常用工具的联系。该方法在PI关于加权和未加权空间中的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Beurling算子在勒贝格空间中的锐数值界(著名的p-1问题)，以及n维加权勒贝格空间中Riesz变换的合适的锐范数估计，最好与维度无关。PI感兴趣的一个相关方向是，如果源空间和目标空间具有不同的权重，则涉及特定奇异积分算子的有界性，例如希尔伯特变换。这个项目在于调和分析，这是很长一段时间以来数学的中心领域。其他领域的许多问题归结为在分析框架内提出和解决的最佳问题。这些问题包括特别适合于信号处理应用的函数(小波)的特殊构建块的存在。其他问题包括所谓的奇异积分算子的有界性，这在偏微分方程组的问题中非常重要，以及物理学。PI的工作包括详细地研究这类算子的连续性，我们经常利用继承和修改自随机最优控制的方法，通过它们在积木上的作用来研究奇异积分算子。这种方法被称为贝尔曼函数法，提供了一种比调和分析中非常复杂的工具更简单但往往更强大的替代方法。此外，由于它的相对简单性，其他领域的科学家等专业背景知识较少的人仍然可以使用它。因此，它既是与其他科学领域的联系，也是促进跨学科交流的纽带，并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.