Topics in Pure and Applied Harmonic Analysis

该项目致力于混合和交织方法，通常归因于现代谐波分析的应用或理论方面，以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近似/插值过程的行为，因为它们的顺序趋于无穷大，最近开发的泊松求和公式的变体涉及不规则样本，以及某些边值问题的解的正则性。第二部分应用Bellman函数方法计算非鞅情形下Lebesgue空间上极大函数的算子范数，BMO空间上的Riesz变换和微分形式上的广义Beurling-Ahlfors变换;此外，本发明还涉及一种用于该方法，一个扩展的早期夏普结果的约翰-研究了Nirenberg不等式，该项目可能直接影响的实际领域包括图像和信号处理领域，包括在医学层析成像中的应用。这项研究的成功完成应该会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（PDE）解的正则性是使用这些PDE来描述物理现象的基本步骤;这种使用在物理学，化学，生物学，材料科学等领域是广泛和必要的，反过来，重要算子的范数的计算，如Riesz变换，允许人们估计各种PDE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正解的存在性联系起来。另一方面，理论协同作用，这是该项目的核心，将导致新的强大技术的发展，影响应用以及傅立叶分析的纯方面。这项研究的另一个重要影响是对教育的影响：研究结果，多样和深远的，将在不同层次的课程和研讨会上提出，从高级本科到博士。

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.