Mathematical Sciences: Non-Standard Singular Integrals

本数学研究的目的是分析奇异积分变换及其在各种函数空间中的映射性质。特别是，在奇异积分的核缺乏经典技术所必需的光滑性的情况下，我们将努力建立这种映射的有界性。这些包括称为d-对易子的多线性奇异积分和不能将Hardy空间带入可积函数或有界函数到BMO的算子。已经取得了一些进展。例如，已经开发了一个类似于现在标准的David-Journe t1条件的结果，它允许考虑非常粗糙的核。本文的工作也适用于考虑奇异积分的混合齐次性，特别是抛物线齐次性。奇异积分变换是现代谐波分析中许多重要工作的基础。这些研究背后的推动力之一是寻找解决大类偏微分方程的综合技术的目标。这些方程是许多重要的非线性物理现象的模型。

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.