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

这一建议的内容涉及广泛的问题阵列的奇异积分算子，在谐波分析感兴趣的中心区域。区分函数的正、负频率的算子是奇异积分算子的原型。在各种空间中对其进行良好的估计，并且在此过程中开发的工具可以应用于近年来经历了快速增长和广泛认可的现代谐波分析的许多部分。研究者将解决著名的关于Lebesgue空间中Beurling算子的尖锐边界问题，以及它的多维模拟形式的无维边界问题。在经典核算子和非核算子的情况下，我们还将考虑一系列广泛的加权问题。最后，对产品域中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，近年来有了显著的增长。直到最近，谐波分析工具才被用来解决这些难题。研究者所追求的问题范围将需要谐波分析新技术的发展。

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.