Mathematical Sciences: Non-Standard Singular Integrals

数学研究的目的是分析 一类奇异积分变换及其映射性质 相对于各种功能空间。 特别是，工作 将这样做，以建立这种映射的有界性， 这些情况下，其中的核心奇异积分缺乏 需要用经典技术处理的光滑度。 其中包括多线性奇异积分称为d-积分器 不能使哈代空间成为可积空间的算子 函数或有界函数到BMO。 了一些进展 已经做了。 例如，类似于现在的结果 已经开发了标准的David-Quinne T1条件，其 允许考虑非常粗糙的内核。 这项工作还有助于 奇异积分的分解子的考虑 具有混合均匀性，特别是抛物线均匀性。 奇异积分变换形成了许多 现代和声学中积极思考的重要工作 分析. 这些研究背后的动力之一是 目标是找到全面的技术来解决广泛的 一类偏微分方程 这些方程产生 作为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.