Sharp Estimates in Harmonic Analysis via Wavelets, Paraproducts and Bellman Functions

这项建议的内容涉及一系列关于奇异积分算子的问题，这是调和分析中感兴趣的一个中心领域。区分函数正负频率的算子是奇异积分算子的原型。在各种空间对它的良好估计和在此过程中开发的工具在现代调和分析的许多部分都得到了应用，这些部分近年来经历了快速的增长和广泛的认可。这位研究者将解决著名的问题，即Beurling算子在勒贝格空间中的锐界，以及它在形式上的多维模拟的无量纲界。无论是在经典的内核运算符设置中，还是在非内核情况下，都将考虑一系列广泛的加权问题。最后，对PI在产品领域的谐波分析及其应用进行了研究。这一领域提出了一系列广泛的问题，并在最近几年取得了显著增长。直到最近，调和分析工具才被用来承担这些困难的问题。研究人员追求的问题范围将要求在谐波分析中开发新的技术。

The content of this proposal concerns a broad array of questions on singular integral operators, a central area of interest in harmonic analysis. The operator which distinguishes positive and negative frequencies of a function is the prototype of a singular integral operator. Good estimates for it in a variety of spaces and the tools developed in the process enjoy applications to many parts of modern harmonic analysis that have experienced rapid growth and widespread recognition in recent years. The investigator is going to tackle the famous problem concerning sharp bounds for the Beurling operator in Lebesgue spaces as well as dimension-free bounds for its multidimensional analog on forms. A broad array of weighted questions will be considered as well, both in the classical setting of kernel oparators as well as in the non-kernel case. Finally, the PI conducts research in the area of harmonic analysis in product domains and its applications. This area has opened up a broad array of questions and has enjoyed significant growth in recent years. Only recently have harmonic analysis tools been brought to bear these difficult questions. The range of questions pursued by the investigator will require the development of new techniques in Harmonic Analysis.