Topics in Pure and Applied Harmonic Analysis

该项目致力于混合和缠绕方法，通常归因于现代谐波分析的应用或理论方面，以努力解决几个长期存在的问题，开发重要的跨领域技术，并产生有意义的应用。该项目的一部分涉及某些逼近/内插过程在其阶数趋于无穷时的行为，最近发展的涉及不规则样本的Poisson求和公式的变体，以及某些边值问题解的正则性。第二部分应用Bellman函数方法计算了非鞅环境下勒贝格空间上极大函数的算子范数，BMO空间上的Riesz变换，以及微分形式上的广义Beurling-Ahlfors变换；此外，还研究了John-Nirenberg不等式的早期尖锐结果的推广。该项目可能直接影响到图像和信号处理领域，包括在医学层析方面的应用。这项研究的成功完成将导致各种类型数据的积累、传输和表示方法的改进。研究偏微分方程解的正则性是用偏微分方程解的正则性来描述物理现象的基础，在物理、化学、生物、材料科学等领域都有广泛而必要的应用。反过来，计算重要算子的范数，如Riesz变换，可以估计各种偏微分方程解的大小。贝尔曼函数方法本身将这种计算与某些微分方程式的正解的存在性联系起来。另一方面，理论协同是该项目的核心，将导致新的强大技术的发展，影响傅立叶分析的应用和纯方面。这项研究的另一个重要影响是对教育的影响：从高级本科生到博士生，这项研究的结果将在不同层次的课程和研讨会上公布，内容多样且影响深远。

This project is devoted to blending and intertwining methods typically attributable to either applied or theoretical aspect of modern Harmonic Analysis, in an effort to solve several long-standing problems, develop important cross-field techniques, and produce meaningful applications. One part of the project deals with the behavior of certain approximation/interpolation processes as their order tends to infinity, a recently developed variant of the Poisson summation formula involving irregular samples, and the regularity properties of solutions of certain boundary value problems. The other part applies the Bellman function method to computing the operator norms of the maximal function on Lebesgue spaces in non-martingale settings, the Riesz transforms on the space BMO, and the generalized Beurling-Ahlfors transform on differential forms; in addition, an extension of earlier sharp results for the John-Nirenberg inequality is studied.Among the practical areas that may be directly affected by the project are the fields of image and signal processing, including applications to medical tomography. The successful completion of this research should lead to improved methods for accumulation, transmission, and representation of various types of data. Studying the regularity properties of solutions of partial differential equations (PDE) is a fundamental step on the way to using those PDE to describe physical phenomena; such use is widespread and necessary in physics, chemistry, biology, material science, etc. In turn, the computation of norms of important operators, such as the Riesz transforms, allows one to estimate the size of solutions of various PDE. The Bellman function method itself links such computation to the existence of positive solutions to certain differential equations. On the other hand, the theoretical synergy, which is at the heart of the project, will result in the development of novel powerful techniques, affecting applied as well as pure aspects of Fourier analysis. Another important impact of this research is on education: the results, diverse and far-reaching, will be presented in courses and seminars on different levels, from advanced undergraduate to the doctoral.