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

通过小波、副积和贝尔曼函数进行谐波分析的精确估计

• 批准号: 0701304
• 负责人: William Beckner
• 金额: $ 11.9万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701304&HistoricalAwards=false
• 关键词: Sharp; Estimates; Harmonic; Analysis; via

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.

该提案的内容涉及有关奇异积分算子的广泛问题，这是调和分析感兴趣的中心领域。区分函数正负频率的算子是奇异积分算子的原型。对它在各种空间中的良好估计以及在此过程中开发的工具在现代谐波分析的许多部分中得到了应用，这些部分近年来经历了快速增长和广泛认可。 研究人员将解决有关 Lebesgue 空间中 Beurling 算子的锐界及其形式上多维模拟的无维界的著名问题。还将考虑一系列广泛的加权问题，无论是在内核操作符的经典设置中还是在非内核情况下。最后，PI对产品领域的谐波分析及其应用进行研究。这一领域提出了一系列广泛的问题，并且近年来取得了显着的增长。直到最近，谐波分析工具才被用来解决这些难题。 研究者所追求的一系列问题将需要开发谐波分析新技术。