Mathematical Sciences: Non-Standard Singular Integrals

数学科学：非标准奇异积分

• 批准号: 9203930
• 负责人: Steven Hofmann
• 金额: $ 3.34万
• 依托单位: Wright State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-05-15 至 1995-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203930&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Non; Standard; Singular

The object of this mathematical research is the analysis of certain singular integral transforms and their mapping properties relative to various spaces of functions. In particular, work will be done to establishing the boundedness of such mappings in those cases where the kernel of the singular integral lacks the smoothness necessary to be treated by classical techniques. These include multilinear singular integrals called d-commutators and operators which fail to carry the Hardy space into integrable functions or bounded functions into BMO. Some progress has already been made. For example, a result similar to the now standard David-Journe T1-condition has been developed which permits consideration of very rough kernels. The work also lends itself to the consideration of commutators of singular integrals with mixed homogeneity, in particular parabolic homogeneity. Singular integral transformations form the basis for much of the important work under active consideration in modern harmonic analysis. Among the motivating forces behind these studies is the goal for finding comprehensive techniques for solving broad classes of partial differential equations. These equations arise as models of many important, nonlinear physical phenomena.

这项数学研究的目的是分析某些奇异积分变换及其相对于不同函数空间的映射性质。特别是，在奇异积分的核缺乏经典技巧所必需的光滑性的情况下，将致力于建立这种映射的有界性。其中包括称为d-交换子的多线性奇异积分，以及不能将Hardy空间化为可积函数或有界函数化为BMO的算子。已经取得了一些进展。例如，一个类似于现在标准的David-Journe T1条件的结果已经被开发出来，它允许考虑非常粗糙的核。这项工作也适用于考虑具有混合齐次的奇异积分的交换子，特别是抛物齐次。奇异积分变换构成了现代调和分析中许多正在积极考虑的重要工作的基础。在这些研究背后的驱动力中，有一个目标是寻找解决大类偏微分方程组的综合技术。这些方程是作为许多重要的、非线性物理现象的模型出现的。